diff --git a/dfols/__init__.py b/dfols/__init__.py index 74d7e05..5ae8a0e 100644 --- a/dfols/__init__.py +++ b/dfols/__init__.py @@ -7,8 +7,7 @@ It solves the nonlinear least-squares problem: min_{x} f(x) = r1(x)**2 + ... + rm(x)**2, -subject to the (optional) bounds - lb <= x <= ub, +(optionally) subject to finitely many convex constraints, where each function ri(x) is differentiable, possibly nonconvex. Since the derivatives of ri(x) are never required or approximated, the solver works when the evaluation of ri(x) is noisy. diff --git a/dfols/controller.py b/dfols/controller.py index 0796858..32e3e37 100644 --- a/dfols/controller.py +++ b/dfols/controller.py @@ -43,6 +43,7 @@ 'EXIT_INPUT_ERROR', 'EXIT_TR_INCREASE_ERROR', 'EXIT_LINALG_ERROR', 'EXIT_FALSE_SUCCESS_WARNING', 'EXIT_AUTO_DETECT_RESTART_WARNING'] +EXIT_TR_INCREASE_WARNING = 5 # warning, TR increase in proj constrained case - likely due to multiple active constraints EXIT_AUTO_DETECT_RESTART_WARNING = 4 # warning, auto-detected restart criteria EXIT_FALSE_SUCCESS_WARNING = 3 # warning, maximum fake successful steps reached EXIT_SLOW_WARNING = 2 # warning, maximum number of slow (successful) iterations reached @@ -70,6 +71,8 @@ def message(self, with_stem=True): return "Warning (slow progress): " + self.msg elif self.flag == EXIT_MAXFUN_WARNING: return "Warning (max evals): " + self.msg + elif self.flag == EXIT_TR_INCREASE_WARNING: + return "Warning (trust region increase): " + self.msg elif self.flag == EXIT_INPUT_ERROR: return "Error (bad input): " + self.msg elif self.flag == EXIT_TR_INCREASE_ERROR: @@ -82,7 +85,7 @@ def message(self, with_stem=True): return "Unknown exit flag: " + self.msg def able_to_do_restart(self): - if self.flag in [EXIT_TR_INCREASE_ERROR, EXIT_LINALG_ERROR, EXIT_SLOW_WARNING, EXIT_AUTO_DETECT_RESTART_WARNING]: + if self.flag in [EXIT_TR_INCREASE_ERROR, EXIT_TR_INCREASE_WARNING, EXIT_LINALG_ERROR, EXIT_SLOW_WARNING, EXIT_AUTO_DETECT_RESTART_WARNING]: return True elif self.flag in [EXIT_MAXFUN_WARNING, EXIT_INPUT_ERROR]: return False @@ -92,13 +95,13 @@ def able_to_do_restart(self): class Controller(object): - def __init__(self, objfun, args, x0, r0, r0_nsamples, xl, xu, npt, rhobeg, rhoend, nf, nx, maxfun, params, + def __init__(self, objfun, args, x0, r0, r0_nsamples, xl, xu, projections, npt, rhobeg, rhoend, nf, nx, maxfun, params, scaling_changes, do_logging): self.do_logging = do_logging self.objfun = objfun self.args = args self.maxfun = maxfun - self.model = Model(npt, x0, r0, xl, xu, r0_nsamples, precondition=params("interpolation.precondition"), + self.model = Model(npt, x0, r0, xl, xu, projections, r0_nsamples, precondition=params("interpolation.precondition"), abs_tol = params("model.abs_tol"), rel_tol = params("model.rel_tol"), do_logging=do_logging) self.nf = nf self.nx = nx @@ -137,6 +140,107 @@ def initialise_coordinate_directions(self, number_of_samples, num_directions, pa assert self.model.num_pts <= (self.n() + 1) * (self.n() + 2) // 2, "prelim: must have npt <= (n+1)(n+2)/2" assert 1 <= num_directions < self.model.num_pts, "Initialisation: must have 1 <= ndirs_initial < npt" + + if self.model.projections: + D = np.zeros((self.n(),self.n())) + k = 0 + while k < self.n(): + ek = np.zeros(self.n()) + ek[k] = 1 + p = np.dot(ek,min(1,self.delta)) + yk = dykstra(self.model.projections, self.model.xbase + p, max_iter=params("dykstra.max_iters"), tol=params("dykstra.d_tol")) + D[k,:] = yk - self.model.xbase + + k += 1 # move on to next point + + # Have at least one L.D. vector, try negative direction on bad one first + k = 0 + mr_tol = params("matrix_rank.r_tol") + D_rank, diag = qr_rank(D,tol=mr_tol) + while D_rank != num_directions and k < self.n(): + if diag[k] < mr_tol: + ek = np.zeros(self.n()) + ek[k] = 1 + p = -np.dot(ek,min(1,self.delta)) + yk = dykstra(self.model.projections, self.model.xbase + p, max_iter=params("dykstra.max_iters"), tol=params("dykstra.d_tol")) + dk = D[k,:].copy() + D[k,:] = yk - self.model.xbase + D_rank2, _diag2 = qr_rank(D,tol=params("matrix_rank.r_tol")) + if D_rank2 <= D_rank: + # Did not improve rank, revert change + D[k,:] = dk + # rank was improved, update D_rank for next comparison + D_rank = D_rank2 + k += 1 + + # Try random combination of negatives... + k = 0 + slctr = np.random.randint(0, 1+1, self.n()) # generate rand binary "selector" array + D_rank, diag = qr_rank(D,tol=params("matrix_rank.r_tol")) + while D_rank != num_directions and k < 100*self.n(): + if slctr[k%self.n()] == 1: # if selector says make -ve, make -ve + ek = np.zeros(self.n()) + ek[k%self.n()] = 1 + p = -np.dot(ek,min(1,self.delta)) + yk = dykstra(self.model.projections, self.model.xbase + p, max_iter=params("dykstra.max_iters"), tol=params("dykstra.d_tol")) + dk = D[k%self.n(),:].copy() + D[k%self.n(),:] = yk - self.model.xbase + D_rank2, _diag2 = qr_rank(D,tol=params("matrix_rank.r_tol")) + if D_rank2 <= D_rank: + # Did not improve rank, revert change + D[k%self.n(),:] = dk + # rank was improved, update D_rank for next comparison + D_rank = D_rank2 + + # Go again + slctr = np.random.randint(0, 1+1, self.n()) + k += 1 + + # Set still not L.I? Try random directions + i = 0 + D_rank, diag = qr_rank(D,tol=params("matrix_rank.r_tol")) + while D_rank != num_directions and i <= 100*num_directions: + k = 0 + while k < self.n(): + if diag[k] < mr_tol: + p = np.random.normal(size=self.n()) + p = p/np.linalg.norm(p) + p = np.dot(p,min(1,self.delta)) + yk = dykstra(self.model.projections, self.model.xbase + p, max_iter=params("dykstra.max_iters"), tol=params("dykstra.d_tol")) + dk = D[k,:].copy() + D[k,:] = yk - self.model.xbase + D_rank2, _diag2 = qr_rank(D,tol=params("matrix_rank.r_tol")) + if D_rank2 <= D_rank: + # Did not improve rank, revert change + D[k,:] = dk + # rank was improved, update D_rank for next comparison + D_rank = D_rank2 + k += 1 + i += 1 + + if D_rank != num_directions: + raise RuntimeError("Unable to generate suitable initial directions") + + # we have a L.I set of interpolation points + for k in range(0,self.n()): + # Evaluate objective at this new point + x = self.model.as_absolute_coordinates(D[k, :]) + rvec_list, f_list, num_samples_run, exit_info = self.evaluate_objective(x, number_of_samples, params) + + # Handle exit conditions (f < min obj value or maxfun reached) + if exit_info is not None: + if num_samples_run > 0: + self.model.save_point(x, np.mean(rvec_list[:num_samples_run, :], axis=0), num_samples_run, + x_in_abs_coords=True) + return exit_info # return & quit + + # Otherwise, add new results (increments model.npt_so_far) + self.model.change_point(k+1, x - self.model.xbase, rvec_list[0, :]) # expect step, not absolute x + for i in range(1, num_samples_run): + self.model.add_new_sample(k+1, rvec_extra=rvec_list[i, :]) + + return None # return & continue + at_lower_boundary = (self.model.sl > -0.01 * self.delta) # sl = xl - x0, should be -ve, actually < -rhobeg at_upper_boundary = (self.model.su < 0.01 * self.delta) # su = xu - x0, should be +ve, actually > rhobeg @@ -147,17 +251,19 @@ def initialise_coordinate_directions(self, number_of_samples, num_directions, pa # k = 2n+1, ..., (n+1)(n+2)/2 --> off-diagonal directions if 1 <= k < self.n() + 1: # first step along coord directions dirn = k - 1 # direction to move in (0,...,n-1) - stepa = self.delta if not at_upper_boundary[dirn] else -self.delta + stepa = self.delta if not at_upper_boundary[dirn] else -self.delta # take a +delta step if at lower, -delta if at upper stepb = None - xpts_added[k, dirn] = stepa + xpts_added[k, dirn] = stepa # set new (relative) point to the step since we haven't done any moving, so relative point is all zeros. elif self.n() + 1 <= k < 2 * self.n() + 1: # second step along coord directions dirn = k - self.n() - 1 # direction to move in (0,...,n-1) - stepa = xpts_added[k - self.n(), dirn] - stepb = -self.delta + stepa = xpts_added[k - self.n(), dirn] # previous step + stepb = -self.delta # new step if at_lower_boundary[dirn]: + # if at lower boundary, set the second step to be +ve stepb = min(2.0 * self.delta, self.model.su[dirn]) # su = xu - x0, should be +ve if at_upper_boundary[dirn]: + # if at upper boundary, set the second step to be -ve stepb = max(-2.0 * self.delta, self.model.sl[dirn]) # sl = xl - x0, should be -ve xpts_added[k, dirn] = stepb @@ -325,10 +431,13 @@ def get_new_direction_for_growing(self, step_length): return dirn * (step_length / LA.norm(dirn)) - def trust_region_step(self): + def trust_region_step(self, params): # Build model for full least squares objectives gopt, H = self.model.build_full_model() - d, gnew, crvmin = trsbox(self.model.xopt(), gopt, H, self.model.sl, self.model.su, self.delta) + if self.model.projections: + d, gnew, crvmin = ctrsbox(self.model.xopt(abs_coordinates=True), gopt, H, self.model.projections, self.delta, d_max_iters=params("dykstra.max_iters"), d_tol=params("dykstra.d_tol")) + else: + d, gnew, crvmin = trsbox(self.model.xopt(), gopt, H, self.model.sl, self.model.su, self.delta) return d, gopt, H, gnew, crvmin def geometry_step(self, knew, adelt, number_of_samples, params): @@ -337,8 +446,13 @@ def geometry_step(self, knew, adelt, number_of_samples, params): try: c, g = self.model.lagrange_gradient(knew) # c = 1.0 if knew == self.model.kopt else 0.0 # based at xopt, just like d - # Solve problem: bounds are sl <= xnew <= su, and ||xnew-xopt|| <= adelt - xnew = trsbox_geometry(self.model.xopt(), c, g, np.minimum(self.model.sl, 0.0), np.maximum(self.model.su, 0.0), adelt) + if self.model.projections: + # Solve problem: use projection onto arbitrary constraints, and ||xnew-xopt|| <= adelt + step = ctrsbox_geometry(self.model.xopt(abs_coordinates=True), c, g, self.model.projections, adelt, d_max_iters=params("dykstra.max_iters"), d_tol=params("dykstra.d_tol")) + xnew = self.model.xopt() + step + else: + # Solve problem: bounds are sl <= xnew <= su, and ||xnew-xopt|| <= adelt + xnew = trsbox_geometry(self.model.xopt(), c, g, np.minimum(self.model.sl, 0.0), np.maximum(self.model.su, 0.0), adelt) except LA.LinAlgError: exit_info = ExitInformation(EXIT_LINALG_ERROR, "Singular matrix encountered in geometry step") return exit_info # didn't fix geometry - return & quit @@ -499,13 +613,16 @@ def reduce_rho(self, current_iter, params): def calculate_ratio(self, current_iter, rvec_list, d, gopt, H): exit_info = None f = sumsq(np.mean(rvec_list, axis=0)) # estimate actual objective value - pred_reduction = - model_value(gopt, H, d) + pred_reduction = - model_value(gopt, H, d) # negative of m since m(0) = 0 actual_reduction = self.model.fopt() - f self.diffs = [abs(actual_reduction - pred_reduction), self.diffs[0], self.diffs[1]] if min(sqrt(sumsq(d)), self.delta) > self.rho: # if ||d|| >= rho, successful! self.last_successful_iter = current_iter if pred_reduction < 0.0: - exit_info = ExitInformation(EXIT_TR_INCREASE_ERROR, "Trust region step gave model increase") + if len(self.model.projections) > 1: # if we are using multiple projections, only warn since likely due to constraint intersection + exit_info = ExitInformation(EXIT_TR_INCREASE_WARNING, "Either multiple constraints are active or trust region step gave model increase") + else: + exit_info = ExitInformation(EXIT_TR_INCREASE_ERROR, "Either rust region step gave model increase") ratio = actual_reduction / pred_reduction return ratio, exit_info diff --git a/dfols/model.py b/dfols/model.py index 20373a9..745f2e6 100644 --- a/dfols/model.py +++ b/dfols/model.py @@ -36,12 +36,12 @@ import scipy.linalg as LA from .trust_region import trsbox_geometry -from .util import sumsq +from .util import sumsq, dykstra __all__ = ['Model'] class Model(object): - def __init__(self, npt, x0, r0, xl, xu, r0_nsamples, n=None, m=None, abs_tol=1e-12, rel_tol=1e-20, precondition=True, + def __init__(self, npt, x0, r0, xl, xu, projections, r0_nsamples, n=None, m=None, abs_tol=1e-12, rel_tol=1e-20, precondition=True, do_logging=True): if n is None: n = len(x0) @@ -63,6 +63,7 @@ def __init__(self, npt, x0, r0, xl, xu, r0_nsamples, n=None, m=None, abs_tol=1e- self.xbase = x0.copy() self.sl = xl - self.xbase # lower bound w.r.t. xbase (require xpt >= sl) self.su = xu - self.xbase # upper bound w.r.t. xbase (require xpt <= su) + self.projections = projections self.points = np.zeros((npt, n)) # interpolation points w.r.t. xbase # Function values @@ -123,6 +124,8 @@ def xpt(self, k, abs_coordinates=False): return np.minimum(np.maximum(self.sl, self.points[k, :].copy()), self.su) else: # Apply bounds and convert back to absolute coordinates + if self.projections: + return dykstra(self.projections, self.xbase + self.points[k,:]) return self.xbase + np.minimum(np.maximum(self.sl, self.points[k, :]), self.su) def rvec(self, k): @@ -133,8 +136,10 @@ def fval(self, k): assert 0 <= k < self.npt(), "Invalid index %g" % k return self.fval[k] - def as_absolute_coordinates(self, x): + def as_absolute_coordinates(self, x, full_dykstra=False): # If x were an interpolation point, get the absolute coordinates of x + if self.projections: + return dykstra(self.projections, self.xbase + x) return self.xbase + np.minimum(np.maximum(self.sl, x), self.su) def xpt_directions(self, include_kopt=True): diff --git a/dfols/params.py b/dfols/params.py index e3e2622..af7b93c 100644 --- a/dfols/params.py +++ b/dfols/params.py @@ -109,6 +109,11 @@ def __init__(self, n, npt, maxfun, objfun_has_noise=False): self.params["growing.full_rank.min_sing_val"] = 1e-6 # absolute floor on singular values self.params["growing.full_rank.svd_max_jac_cond"] = 1e8 # maximum condition number of Jacobian self.params["growing.perturb_trust_region_step"] = False # add random direction onto TRS solution? + # Dykstra's algorithm + self.params["dykstra.d_tol"] = 1e-10 + self.params["dykstra.max_iters"] = 100 + # Matrix rank algorithm + self.params["matrix_rank.r_tol"] = 1e-18 self.params_changed = {} for p in self.params: @@ -257,6 +262,12 @@ def param_type(self, key, npt): type_str, nonetype_ok, lower, upper = 'float', True, 1.0, None elif key == "growing.perturb_trust_region_step": type_str, nonetype_ok, lower, upper = 'bool', False, None, None + elif key == "dykstra.d_tol": + type_str, nonetype_ok, lower, upper = 'float', False, 0.0, None + elif key == "dykstra.max_iters": + type_str, nonetype_ok, lower, upper = 'int', False, 0, None + elif key == "matrix_rank.r_tol": + type_str, nonetype_ok, lower, upper = 'float', False, 0.0, None else: assert False, "ParameterList.param_type() has unknown key: %s" % key return type_str, nonetype_ok, lower, upper diff --git a/dfols/solver.py b/dfols/solver.py index f6279ba..0c7ee60 100644 --- a/dfols/solver.py +++ b/dfols/solver.py @@ -93,7 +93,7 @@ def __str__(self): return output -def solve_main(objfun, x0, args, xl, xu, npt, rhobeg, rhoend, maxfun, nruns_so_far, nf_so_far, nx_so_far, nsamples, params, +def solve_main(objfun, x0, args, xl, xu, projections, npt, rhobeg, rhoend, maxfun, nruns_so_far, nf_so_far, nx_so_far, nsamples, params, diagnostic_info, scaling_changes, r0_avg_old=None, r0_nsamples_old=None, default_growing_method_set_by_user=None, do_logging=True, print_progress=False): # Evaluate at x0 (keep nf, nx correct and check for f < 1e-12) @@ -160,7 +160,7 @@ def solve_main(objfun, x0, args, xl, xu, npt, rhobeg, rhoend, maxfun, nruns_so_f params('growing.delta_scale_new_dirns', new_value=0.1) # Initialise controller - control = Controller(objfun, args, x0, r0_avg, num_samples_run, xl, xu, npt, rhobeg, rhoend, nf, nx, maxfun, + control = Controller(objfun, args, x0, r0_avg, num_samples_run, xl, xu, projections, npt, rhobeg, rhoend, nf, nx, maxfun, params, scaling_changes, do_logging) # Initialise interpolation set @@ -271,7 +271,7 @@ def solve_main(objfun, x0, args, xl, xu, npt, rhobeg, rhoend, maxfun, nruns_so_f # Trust region step - d, gopt, H, gnew, crvmin = control.trust_region_step() + d, gopt, H, gnew, crvmin = control.trust_region_step(params) if do_logging: logging.debug("Trust region step is d = " + str(d)) xnew = control.model.xopt() + d @@ -851,7 +851,7 @@ def solve_main(objfun, x0, args, xl, xu, npt, rhobeg, rhoend, maxfun, nruns_so_f return x, rvec, f, jacmin, nsamples, control.nf, control.nx, nruns_so_far, exit_info, diagnostic_info -def solve(objfun, x0, args=(), bounds=None, npt=None, rhobeg=None, rhoend=1e-8, maxfun=None, nsamples=None, user_params=None, +def solve(objfun, x0, args=(), bounds=None, projections=[], npt=None, rhobeg=None, rhoend=1e-8, maxfun=None, nsamples=None, user_params=None, objfun_has_noise=False, scaling_within_bounds=False, do_logging=True, print_progress=False): x0 = x0.astype(float) n = len(x0) @@ -869,6 +869,10 @@ def solve(objfun, x0, args=(), bounds=None, npt=None, rhobeg=None, rhoend=1e-8, scaling_within_bounds = False warnings.warn("Ignoring scaling_within_bounds=True for unconstrained problem/1-sided bounds", RuntimeWarning) + if projections and scaling_within_bounds: + scaling_within_bounds = False + warnings.warn("Ignoring scaling_within_bounds=True for problem with arbitrary constraints", RuntimeWarning) + if xl is None: xl = -1e20 * np.ones((n,)) # unconstrained if xu is None: @@ -882,6 +886,18 @@ def solve(objfun, x0, args=(), bounds=None, npt=None, rhobeg=None, rhoend=1e-8, if nsamples is None: nsamples = lambda delta, rho, iter, nruns: 1 # no averaging + # If using arbitrary constraints, create projection from bounds + if projections: + xlb = xl.copy() + xub = xu.copy() + bproj = lambda w: pbox(w,xlb,xub) + projections = projections.copy() + projections.append(bproj) + + # since using arbitrary constraints, don't constrain otherwise + xl = -1e20 * np.ones((n,)) + xu = 1e20 * np.ones((n,)) + # Set parameters params = ParameterList(int(n), int(npt), int(maxfun), objfun_has_noise=objfun_has_noise) # make sure int, not np.int if user_params is not None: @@ -976,6 +992,13 @@ def solve(objfun, x0, args=(), bounds=None, npt=None, rhobeg=None, rhoend=1e-8, results = OptimResults(None, None, None, None, 0, 0, 0, exit_flag, exit_msg) return results + # Enforce arbitrary constraint bounds on x0 + if projections: + xp = dykstra(projections,x0,max_iter=params("dykstra.max_iters"),tol=params("dykstra.d_tol")) + if not np.allclose(xp,x0): + warnings.warn("x0 not feasible w.r.t given constraints, adjusting", RuntimeWarning) + x0 = xp.copy() + # Enforce lower & upper bounds on x0 idx = (x0 <= xl) if np.any(idx): @@ -993,7 +1016,7 @@ def solve(objfun, x0, args=(), bounds=None, npt=None, rhobeg=None, rhoend=1e-8, nf = 0 nx = 0 xmin, rmin, fmin, jacmin, nsamples_min, nf, nx, nruns, exit_info, diagnostic_info = \ - solve_main(objfun, x0, args, xl, xu, npt, rhobeg, rhoend, maxfun, nruns, nf, nx, nsamples, params, + solve_main(objfun, x0, args, xl, xu, projections, npt, rhobeg, rhoend, maxfun, nruns, nf, nx, nsamples, params, diagnostic_info, scaling_changes, default_growing_method_set_by_user=default_growing_method_set_by_user, do_logging=do_logging, print_progress=print_progress) @@ -1012,12 +1035,12 @@ def solve(objfun, x0, args=(), bounds=None, npt=None, rhobeg=None, rhoend=1e-8, % (fmin, nf, rhobeg, rhoend)) if params("restarts.hard.use_old_rk"): xmin2, rmin2, fmin2, jacmin2, nsamples2, nf, nx, nruns, exit_info, diagnostic_info = \ - solve_main(objfun, xmin, args, xl, xu, npt, rhobeg, rhoend, maxfun, nruns, nf, nx, nsamples, params, + solve_main(objfun, xmin, args, xl, xu, projections, npt, rhobeg, rhoend, maxfun, nruns, nf, nx, nsamples, params, diagnostic_info, scaling_changes, r0_avg_old=rmin, r0_nsamples_old=nsamples_min, do_logging=do_logging, print_progress=print_progress) else: xmin2, rmin2, fmin2, jacmin2, nsamples2, nf, nx, nruns, exit_info, diagnostic_info = \ - solve_main(objfun, xmin, args, xl, xu, npt, rhobeg, rhoend, maxfun, nruns, nf, nx, nsamples, params, + solve_main(objfun, xmin, args, xl, xu, projections, npt, rhobeg, rhoend, maxfun, nruns, nf, nx, nsamples, params, diagnostic_info, scaling_changes, do_logging=do_logging, print_progress=print_progress) if fmin2 < fmin or np.isnan(fmin): diff --git a/dfols/tests/test_model.py b/dfols/tests/test_model.py index 9b6a384..66ed89f 100644 --- a/dfols/tests/test_model.py +++ b/dfols/tests/test_model.py @@ -46,7 +46,7 @@ def runTest(self): x0 = np.array([-1.2, 1.0]) xl = -1e20 * np.ones((n,)) xu = 1e20 * np.ones((n,)) - model = Model(npt, x0, rosenbrock(x0), xl, xu, 1) + model = Model(npt, x0, rosenbrock(x0), xl, xu, [], 1) self.assertEqual(model.npt(), 1, 'Wrong npt after initialisation') self.assertTrue(array_compare(model.xopt(abs_coordinates=True), x0), 'Wrong xopt after initialisation') self.assertTrue(array_compare(model.ropt(), rosenbrock(x0)), 'Wrong ropt after initialisation') @@ -97,7 +97,7 @@ def runTest(self): x0 = np.array([-1.2, 1.0]) xl = -1e20 * np.ones((n,)) xu = 1e20 * np.ones((n,)) - model = Model(npt, x0, rosenbrock(x0), xl, xu, 1) + model = Model(npt, x0, rosenbrock(x0), xl, xu, [], 1) # Now add better point x1 = np.array([1.0, 0.9]) rvec = rosenbrock(x1) @@ -124,7 +124,7 @@ def runTest(self): x0 = np.array([-1.2, 1.0]) xl = -1e2 * np.ones((n,)) xu = 1e2 * np.ones((n,)) - model = Model(npt, x0, rosenbrock(x0), xl, xu, 1) + model = Model(npt, x0, rosenbrock(x0), xl, xu, [], 1) self.assertTrue(array_compare(model.sl, xl - x0), 'Wrong sl after initialisation') self.assertTrue(array_compare(model.su, xu - x0), 'Wrong su after initialisation') x1 = np.array([1.0, 0.9]) @@ -204,7 +204,7 @@ def runTest(self): x0 = np.array([-1.2, 1.0]) xl = -1e2 * np.ones((n,)) xu = 1e2 * np.ones((n,)) - model = Model(npt, x0, rosenbrock(x0), xl, xu, 1) + model = Model(npt, x0, rosenbrock(x0), xl, xu, [], 1) x1 = np.array([1.0, 0.9]) model.change_point(1, x1 - model.xbase, rosenbrock(x1)) x2 = np.array([1.0, 1.0]) @@ -224,17 +224,17 @@ def runTest(self): x0 = np.array([-1.2, 1.0]) xl = -1e2 * np.ones((n,)) xu = 1e2 * np.ones((n,)) - model = Model(npt, x0, rosenbrock(x0), xl, xu, 1) + model = Model(npt, x0, rosenbrock(x0), xl, xu, [], 1) x1 = np.array([1.0, 0.9]) model.change_point(1, x1 - model.xbase, rosenbrock(x1)) x2 = np.array([2.0, 0.9]) model.change_point(2, x2 - model.xbase, rosenbrock(x2)) self.assertAlmostEqual(model.min_objective_value(), 1e-12, 'Wrong min obj value') - model = Model(npt, x0, rosenbrock(x0), xl, xu, 1, rel_tol=1e-2) + model = Model(npt, x0, rosenbrock(x0), xl, xu, [], 1, rel_tol=1e-2) self.assertAlmostEqual(model.min_objective_value(), 1e-2 * sumsq(rosenbrock(x0)), 'Wrong min obj value 2') - model = Model(npt, x0, rosenbrock(x0), xl, xu, 1, abs_tol=1.0) + model = Model(npt, x0, rosenbrock(x0), xl, xu, [], 1, abs_tol=1.0) self.assertAlmostEqual(model.min_objective_value(), 1.0, 'Wrong min obj value 3') - model = Model(npt, x0, rosenbrock(x0), xl, xu, 1, abs_tol=1.0, rel_tol=1e-2) + model = Model(npt, x0, rosenbrock(x0), xl, xu, [], 1, abs_tol=1.0, rel_tol=1e-2) self.assertAlmostEqual(model.min_objective_value(), 1.0, 'Wrong min obj value 4') @@ -245,7 +245,7 @@ def runTest(self): x0 = np.array([-1.2, 1.0]) xl = -1e2 * np.ones((n,)) xu = 1e2 * np.ones((n,)) - model = Model(npt, x0, rosenbrock(x0), xl, xu, 1) + model = Model(npt, x0, rosenbrock(x0), xl, xu, [], 1) model.add_new_sample(0, rosenbrock(x0)) x1 = np.array([1.0, 0.9]) model.change_point(1, x1 - model.xbase, rosenbrock(x1)) @@ -295,7 +295,7 @@ def runTest(self): # x0 = np.array([1.0, 2.9]) xl = -1e2 * np.ones((n,)) xu = 1e2 * np.ones((n,)) - model = Model(npt, x0, rosenbrock(x0), xl, xu, 1) + model = Model(npt, x0, rosenbrock(x0), xl, xu, [], 1) model.add_new_sample(0, rosenbrock(x0)) x1 = np.array([1.0, 0.9]) model.change_point(1, x1 - model.xbase, rosenbrock(x1)) @@ -326,7 +326,7 @@ def runTest(self): x0 = np.array([-1.2, 1.0]) xl = -1e2 * np.ones((n,)) xu = 1e2 * np.ones((n,)) - model = Model(npt, x0, rosenbrock(x0), xl, xu, 1) + model = Model(npt, x0, rosenbrock(x0), xl, xu, [], 1) model.add_new_sample(0, rosenbrock(x0)) x1 = np.array([-1.2, 0.9]) model.change_point(1, x1 - model.xbase, rosenbrock(x1)) @@ -354,7 +354,7 @@ def runTest(self): delta = 0.5 xl = -1e2 * np.ones((n,)) xu = 1e2 * np.ones((n,)) - model = Model(npt, x0, rosenbrock(x0), xl, xu, 1) + model = Model(npt, x0, rosenbrock(x0), xl, xu, [], 1) model.add_new_sample(0, rosenbrock(x0)) x1 = x0 + delta * np.array([1.0, 0.0]) model.change_point(1, x1 - model.xbase, rosenbrock(x1)) @@ -373,7 +373,7 @@ def runTest(self): x0 = np.array([-1.2, 1.0]) xl = -1e2 * np.ones((n,)) xu = 1e2 * np.ones((n,)) - model = Model(npt, x0, rosenbrock(x0), xl, xu, 1) + model = Model(npt, x0, rosenbrock(x0), xl, xu, [], 1) x1 = np.array([1.0, 0.9]) model.change_point(1, x1 - model.xbase, rosenbrock(x1)) x2 = np.array([2.0, 0.9]) @@ -407,7 +407,7 @@ def runTest(self): # x0 = np.zeros((n,)) xl = -1e2 * np.ones((n,)) xu = 1e2 * np.ones((n,)) - model = Model(npt, A[0,:], np.array([b[0]]), xl, xu, 1) + model = Model(npt, A[0,:], np.array([b[0]]), xl, xu, [], 1) for i in range(1,npt): xi = A[i,:] @@ -450,7 +450,7 @@ def runTest(self): xl = -1e2 * np.ones((n,)) xu = 1e2 * np.ones((n,)) # model = Model(n+1, x0, np.array([0.0]), xl, xu, 1) - model = Model(n+1, A[0, :], np.array([b[0]]), xl, xu, 1) + model = Model(n+1, A[0, :], np.array([b[0]]), xl, xu, [], 1) for i in range(1,npt): xi = A[i,:] diff --git a/dfols/tests/test_solver.py b/dfols/tests/test_solver.py index 9c43445..b0601ad 100644 --- a/dfols/tests/test_solver.py +++ b/dfols/tests/test_solver.py @@ -41,6 +41,13 @@ def rosenbrock_jacobian(x): return np.array([[-20.0*x[0], 10.0], [-1.0, 0.0]]) +def p_box(x,l,u): + return np.minimum(np.maximum(x,l), u) + +def p_ball(x,c,r): + return c + (r/np.max([np.linalg.norm(x-c),r]))*(x-c) + + class TestNans(unittest.TestCase): # Generic objective that only returns NaNs (like optclim code) # Verify get a sensible termination @@ -185,3 +192,21 @@ def runTest(self): self.assertTrue(array_compare(soln.jacobian, jac(soln.x), thresh=1e-1), "Wrong Jacobian") self.assertTrue(abs(soln.f) < 1e-10, "Wrong fmin") + +class TestRosenbrockBoxBall(unittest.TestCase): + # Minimise the (2d) Rosenbrock function, where x[1] hits the upper bound + def runTest(self): + # n, m = 2, 2 + x0 = np.array([-1.2, 0.7]) # standard start point does not satisfy bounds + lower = np.array([0.7, -2.0]) + upper = np.array([1.0, 2]) + boxproj = lambda x: p_box(x,lower,upper) + ballproj = lambda x: p_ball(x,np.array([0.5,1]),0.25) + xmin = np.array([0.70424386, 0.85583188]) # approximate + fmin = np.dot(rosenbrock(xmin), rosenbrock(xmin)) + soln = dfols.solve(rosenbrock, x0, projections=[boxproj,ballproj]) + print(soln.x) + self.assertTrue(array_compare(soln.x, xmin, thresh=1e-2), "Wrong xmin") + self.assertTrue(array_compare(soln.resid, rosenbrock(soln.x), thresh=1e-10), "Wrong resid") + self.assertTrue(array_compare(soln.jacobian, rosenbrock_jacobian(soln.x), thresh=1e-2), "Wrong Jacobian") + self.assertTrue(abs(soln.f - fmin) < 1e-4, "Wrong fmin") diff --git a/dfols/tests/test_trust_region.py b/dfols/tests/test_trust_region.py index 64524e3..b9561c6 100644 --- a/dfols/tests/test_trust_region.py +++ b/dfols/tests/test_trust_region.py @@ -27,7 +27,7 @@ import numpy as np import unittest -from dfols.trust_region import trsbox, trsbox_geometry +from dfols.trust_region import ctrsbox, ctrsbox_geometry, trsbox, trsbox_geometry from dfols.util import model_value @@ -70,6 +70,12 @@ def cauchy_pt_box(g, H, delta, lower, upper): crvmin = -1.0 return s, red, crvmin +def p_box(x,l,u): + return np.minimum(np.maximum(x,l), u) + +def p_ball(x,c,r): + return c + (r/np.max([np.linalg.norm(x-c),r]))*(x-c) + class TestUncInternal(unittest.TestCase): def runTest(self): @@ -340,3 +346,135 @@ def runTest(self): # print(x) # print(xtrue) self.assertTrue(np.max(np.abs(x - xtrue)) < 1e-10, 'Wrong step') + + +# DFO-LS with arbitrary constraints +class TestGeom3CDFO(unittest.TestCase): + def runTest(self): + xbase = np.array([0.0, 0.0]) + 1 + g = np.array([1.0, -1.0]) + a = np.array([-2.0, -2.0]) + 1 + b = np.array([1.0, 2.0]) + 1 + proj = lambda x: p_box(x,a,b) + delta = 5.0 + c = 3.0 # may want to max instead + x = ctrsbox_geometry(xbase, c, g, [proj], delta) + xtrue = np.array([1.0, -2.0]) + self.assertTrue(np.max(np.abs(x - xtrue)) < 1e-10, 'Wrong step') + +class TestUncInternalCDFO(unittest.TestCase): + def runTest(self): + n = 3 + g = np.array([1.0, 0.0, 1.0]) + H = np.array([[1.0, 0.0, 0.0], [0.0, 2.0, 0.0], [0.0, 0.0, 2.0]]) + Delta = 2.0 + xopt = np.ones((n,)) # trying nonzero (since bounds inactive) + d, gnew, _crvmin = ctrsbox(xopt, g, H, [], Delta) + true_d = np.array([-1.0, 0.0, -0.5]) + est_min = model_value(g, H, d) + true_min = model_value(g, H, true_d) + s_cauchy, red_cauchy, _crvmin_cauchy = cauchy_pt(g, H, Delta) + self.assertTrue(est_min <= red_cauchy, 'Cauchy reduction not achieved') + self.assertTrue(np.all(gnew == g + H.dot(d)), 'Wrong gnew') + +class TestUncBdryCDFO(unittest.TestCase): + def runTest(self): + n = 3 + g = np.array([1.0, 0.0, 1.0]) + H = np.array([[1.0, 0.0, 0.0], [0.0, 2.0, 0.0], [0.0, 0.0, 2.0]]) + Delta = 5.0 / 12.0 + xopt = np.zeros((n,)) + d, gnew, _crvmin = ctrsbox(xopt, g, H, [], Delta) + true_d = np.array([-1.0 / 3.0, 0.0, -0.25]) + est_min = model_value(g, H, d) + true_min = model_value(g, H, true_d) + s_cauchy, red_cauchy, _crvmin_cauchy = cauchy_pt(g, H, Delta) + self.assertTrue(est_min <= red_cauchy, 'Cauchy reduction not achieved') + self.assertTrue(np.allclose(gnew, g + H.dot(d)), 'Wrong gnew') + + +class TestUncHardCDFO(unittest.TestCase): + def runTest(self): + n = 3 + g = np.array([0.0, 0.0, 1.0]) + H = np.array([[-2.0, 0.0, 0.0], [0.0, -1.0, 0.0], [0.0, 0.0, -1.0]]) + Delta = sqrt(2.0) + xopt = np.zeros((n,)) + d, gnew, _crvmin = ctrsbox(xopt, g, H, [], Delta) + true_d = np.array([1.0, 0.0, -1.0]) # non-unique solution + est_min = model_value(g, H, d) + true_min = model_value(g, H, true_d) + s_cauchy, red_cauchy, _crvmin_cauchy = cauchy_pt(g, H, Delta) + self.assertTrue(est_min <= red_cauchy, 'Cauchy reduction not achieved') + self.assertTrue(np.allclose(gnew, g + H.dot(d)), 'Wrong gnew') + +class TestConInternalCDFO(unittest.TestCase): + def runTest(self): + n = 3 + g = np.array([1.0, 0.0, 1.0]) + H = np.array([[1.0, 0.0, 0.0], [0.0, 2.0, 0.0], [0.0, 0.0, 2.0]]) + Delta = 2.0 + xopt = np.ones((n,)) # trying nonzero (since bounds inactive) + sl = xopt + np.array([-0.5, -10.0, -10.0]) + su = xopt + np.array([10.0, 10.0, 10.0]) + proj = lambda x: p_box(x,sl,su) + d, gnew, _crvmin = ctrsbox(xopt, g, H, [proj], Delta) + true_d = np.array([-1.0, 0.0, -0.5]) + est_min = model_value(g, H, d) + true_min = model_value(g, H, true_d) + s_cauchy, red_cauchy, _crvmin_cauchy = cauchy_pt_box(g, H, Delta, sl-xopt, su-xopt) + self.assertTrue(est_min <= red_cauchy, 'Cauchy reduction not achieved') + self.assertTrue(np.all(gnew == g + H.dot(d)), 'Wrong gnew') + +class TestConBdryCDFO(unittest.TestCase): + def runTest(self): + n = 3 + g = np.array([1.0, 0.0, 1.0]) + H = np.array([[1.0, 0.0, 0.0], [0.0, 2.0, 0.0], [0.0, 0.0, 2.0]]) + Delta = 5.0 / 12.0 + xopt = np.zeros((n,)) + sl = xopt + np.array([-0.3, -0.01, -0.1]) + su = xopt + np.array([10.0, 1.0, 10.0]) + proj = lambda x: p_box(x,sl,su) + d, gnew, _crvmin = ctrsbox(xopt, g, H, [proj], Delta) + true_d = np.array([-1.0 / 3.0, 0.0, -0.25]) + est_min = model_value(g, H, d) + true_min = model_value(g, H, true_d) + s_cauchy, red_cauchy, _crvmin_cauchy = cauchy_pt_box(g, H, Delta, sl - xopt, su - xopt) + self.assertTrue(est_min <= red_cauchy, 'Cauchy reduction not achieved') + self.assertTrue(np.max(np.abs(gnew - g - H.dot(d))) < 1e-10, 'Wrong gnew') + +class TestBoxBallInternalCDFO(unittest.TestCase): + def runTest(self): + n = 3 + g = np.array([1.0, 0.0, 1.0]) + H = np.array([[1.0, 0.0, 0.0], [0.0, 2.0, 0.0], [0.0, 0.0, 2.0]]) + Delta = 2.0 + xopt = np.ones((n,)) # trying nonzero (since bounds inactive) + sl = xopt + np.array([-0.5, -10.0, -10.0]) + su = xopt + np.array([10.0, 10.0, 10.0]) + boxproj = lambda x: p_box(x,sl,su) + ballproj = lambda x: p_ball(x,xopt,5) + d, gnew, _crvmin = ctrsbox(xopt, g, H, [boxproj,ballproj], Delta) + true_d = np.array([-0.5, 0.0, -0.5]) + est_min = model_value(g, H, d) + true_min = model_value(g, H, true_d) + self.assertTrue(est_min <= true_min + 1e-3, 'Sufficient decrease not achieved') + + +class TestBoxBallBdryCDFO(unittest.TestCase): + def runTest(self): + n = 3 + g = np.array([1.0, 0.0, 1.0]) + H = np.array([[1.0, 0.0, 0.0], [0.0, 2.0, 0.0], [0.0, 0.0, 2.0]]) + Delta = 5.0 / 12.0 + xopt = np.zeros((n,)) + sl = xopt + np.array([-0.3, -0.01, -0.1]) + su = xopt + np.array([10.0, 1.0, 10.0]) + boxproj = lambda x: p_box(x,sl,su) + ballproj = lambda x: p_ball(x,xopt,0.25) + d, gnew, _crvmin = ctrsbox(xopt, g, H, [boxproj,ballproj], Delta) + true_d = np.array([-0.22913085, 0.0, -0.09999527]) + est_min = model_value(g, H, d) + true_min = model_value(g, H, true_d) + self.assertTrue(est_min <= true_min + 1e-3, 'Sufficient decrease not achieved') diff --git a/dfols/tests/test_util.py b/dfols/tests/test_util.py index f497904..51c1b30 100644 --- a/dfols/tests/test_util.py +++ b/dfols/tests/test_util.py @@ -165,4 +165,135 @@ def runTest(self): self.assertTrue(np.linalg.norm(dirns[i, :]) <= delta + 1e-10, "Unconstrained: dirn %i too long" % i) self.assertTrue(np.all(dirns[i, :] >= lower), "Direction %i below lower bound" % i) self.assertTrue(np.all(dirns[i, :] <= upper), "Direction %i above upper bound" % i) - # self.assertTrue(False, "bad") \ No newline at end of file + # self.assertTrue(False, "bad") + +# Trivial case of full rank +class TestMatrixRankQR1(unittest.TestCase): + def runTest(self): + mr_tol = 1e-18 + A = np.array([ + [1,0,0,0], + [0,1,0,0], + [0,0,1,0], + [0,0,0,1]]) + rank, D = qr_rank(A,mr_tol) + self.assertTrue(np.all(D > mr_tol), "Incorrect diagonal matrix output") + self.assertTrue(rank == 4, "Incorrect rank output") + +# Full rank but QR has negative entries for diag(R) +class TestMatrixRankQR2(unittest.TestCase): + def runTest(self): + mr_tol = 1e-18 + A = np.array([ + [1,2,3,4], + [0,6,7,8], + [-1,-2,-2,-1], + [4,2,2,1]]) + rank, D = qr_rank(A,mr_tol) + self.assertTrue(np.all(D > mr_tol), "Incorrect diagonal matrix output") + self.assertTrue(rank == 4, "Incorrect rank output") + + +# Not full rank +class TestMatrixRankQR3(unittest.TestCase): + def runTest(self): + mr_tol = 1e-18 + A = np.array([ + [1,2,3,4], + [2,6,4,8], + [0,0,0,0], + [0,0,0,0]]) + rank, D = qr_rank(A,mr_tol) + self.assertTrue(np.all(D[0:2] > mr_tol), "Incorrect diagonal matrix output (rows 1,2)") + self.assertTrue(np.all(D[2:4] <= mr_tol), "Incorrect diagonal matrix output (rows 3,4)") + self.assertTrue(rank == 2, "Incorrect rank output") + + +class TestDykstraBoxInt(unittest.TestCase): + def runTest(self): + x0 = np.array([0,0]) + lower = np.array([-0.01, -0.1]) + upper = np.array([0.01, 0.5]) + boxproj = lambda x: pbox(x,lower,upper) + P = [boxproj] + xproj = dykstra(P,x0) + self.assertTrue(np.all(xproj == x0), "Incorrect point returned by Dykstra") + + +class TestDykstraBoxExt(unittest.TestCase): + def runTest(self): + x0 = np.array([-2,5]) + lower = np.array([-1, -1]) + upper = np.array([0.5, 0.9]) + boxproj = lambda x: pbox(x,lower,upper) + P = [boxproj] + xproj = dykstra(P,x0) + xtrue = np.array([-1,0.9]) + self.assertTrue(np.allclose(xproj, xtrue), "Incorrect point returned by Dykstra") + +class TestDykstraBallInt(unittest.TestCase): + def runTest(self): + x0 = np.array([0,0]) + ballproj = lambda x: pball(x,x0+1,2) + P = [ballproj] + xproj = dykstra(P,x0) + self.assertTrue(np.all(xproj == x0), "Incorrect point returned by Dykstra") + + +class TestDykstraBallExt(unittest.TestCase): + def runTest(self): + x0 = np.array([-3,5]) + ballproj = lambda x: pball(x,np.array([-0.5,1]),1) + P = [ballproj] + xproj = dykstra(P,x0) + xtrue = np.array([-1.02999894, 1.8479983]) + self.assertTrue(np.allclose(xproj, xtrue), "Incorrect point returned by Dykstra") + + +class TestDykstraBoxBallInt(unittest.TestCase): + def runTest(self): + x0 = np.array([0.72,1.1]) + lower = np.array([0.7, -2.0]) + upper = np.array([1.0, 2]) + boxproj = lambda x: pbox(x,lower,upper) + ballproj = lambda x: pball(x,np.array([0.5,1]),0.25) + P = [boxproj,ballproj] + xproj = dykstra(P,x0) + self.assertTrue(np.all(xproj == x0), "Incorrect point returned by Dykstra") + +class TestDykstraBoxBallExt1(unittest.TestCase): + def runTest(self): + x0 = np.array([0,4]) + lower = np.array([0.7, -2.0]) + upper = np.array([1.0, 2]) + boxproj = lambda x: pbox(x,lower,upper) + ballproj = lambda x: pball(x,np.array([0.5,1]),0.25) + P = [boxproj,ballproj] + xproj = dykstra(P,x0) + xtrue = np.array([0.6940582, 1.1576116]) + self.assertTrue(np.allclose(xproj, xtrue), "Incorrect point returned by Dykstra") + + +class TestDykstraBoxBallExt2(unittest.TestCase): + def runTest(self): + x0 = np.array([0.8,-3]) + lower = np.array([0.7, -2.0]) + upper = np.array([1.0, 2]) + boxproj = lambda x: pbox(x,lower,upper) + ballproj = lambda x: pball(x,np.array([0.5,1]),0.25) + P = [boxproj,ballproj] + xproj = dykstra(P,x0) + xtrue = np.array([0.68976232, 0.8372417]) + self.assertTrue(np.allclose(xproj, xtrue), "Incorrect point returned by Dykstra") + + +class TestDykstraBoxBallBdry(unittest.TestCase): + def runTest(self): + x0 = np.array([0.7,0.85]) + lower = np.array([0.7, -2.0]) + upper = np.array([1.0, 2]) + boxproj = lambda x: pbox(x,lower,upper) + ballproj = lambda x: pball(x,np.array([0.5,1]),0.25) + P = [boxproj,ballproj] + xproj = dykstra(P,x0) + self.assertTrue(np.allclose(xproj, x0), "Incorrect point returned by Dykstra") diff --git a/dfols/trust_region.py b/dfols/trust_region.py index ec8c975..74ce269 100644 --- a/dfols/trust_region.py +++ b/dfols/trust_region.py @@ -11,6 +11,15 @@ The other outputs: gnew is the gradient of the model at d, and crvmin has information about the curvature of the model at the solution. +For handling arbitrary constraints, the call is + d, gnew, crvmin = ctrsbox(xopt, g, H, projections, delta) +which produces a new vector d approximately solving the constrained trust region subproblem: + min_{d} g'*d + 0.5*d'*H*d + s.t. ||d|| <= delta + xopt + d is feasible w.r.t. the constraint set C +The other outputs: gnew is the gradient of the model at d, and crvmin has +information about the curvature of the model at the solution. + We also provide a function for maximising the absolute value of a linear function inside a similar trust region - this is useful for geometry steps. The call @@ -23,6 +32,13 @@ min_s abs(c + g' * d) s.t. lower <= xbase + d <= upper ||d|| <= delta +Again, we have a version of this for handling arbitrary constraints +The call + x = ctrsbox_geometry(xbase, c, g, projections, Delta) +Solves + min_s abs(c + g' * d) + s.t. xbase + d is feasible w.r.t. the constraint set C + ||d|| <= delta Notes ---- @@ -63,13 +79,77 @@ # Fall back to Python implementation USE_FORTRAN = False +from .util import dykstra, pball, pbox, sumsq, model_value -from .util import sumsq +__all__ = ['ctrsbox', 'ctrsbox_geometry', 'trsbox', 'trsbox_geometry'] +ZERO_THRESH = 1e-14 -__all__ = ['trsbox', 'trsbox_geometry'] +def ctrsbox(xopt, g, H, projections, delta, d_max_iters=100, d_tol=1e-10, use_fortran=USE_FORTRAN): + n = xopt.size + assert xopt.shape == (n,), "xopt has wrong shape (should be vector)" + assert g.shape == (n,), "g and xopt have incompatible sizes" + assert len(H.shape) == 2, "H must be a matrix" + assert H.shape == (n,n), "H and xopt have incompatible sizes" + assert np.allclose(H, H.T), "H must be symmetric" + assert delta > 0.0, "delta must be strictly positive" -ZERO_THRESH = 1e-14 + d = np.zeros((n,)) + gnew = g.copy() + gy = g.copy() + crvmin = -1.0 + y = d.copy() + eta = 1.2 # L backtrack scaling factor + t = 1 + + # Initial guess of L is norm(Hessian) + L = np.linalg.norm(H, 2) + + # trust region is a ball of radius delta around xopt + trproj = lambda w: pball(w, xopt, delta) + + # combine trust region constraints with user-entered constraints + P = projections.copy() + P.append(trproj) + def proj(d0): + p = dykstra(P, xopt+d0, max_iter=d_max_iters, tol=d_tol) + # we want the step only, so we subtract xopt + # from the new point: proj(xk+d) - xk + return p - xopt + + MAX_LOOP_ITERS = 100 * n ** 2 + + # projected GD loop + for ii in range(MAX_LOOP_ITERS): + w = y - (1/L)*gy + prev_d = d.copy() + d = proj(w) + + # size of step taken + s = d - prev_d + stplen = np.linalg.norm(s) + + # update true gradient + gnew += H.dot(s) + + # update CRVMIN + crv = s.dot(H).dot(s)/sumsq(s) if sumsq(s) >= ZERO_THRESH else crvmin + crvmin = min(crvmin, crv) if crvmin != -1.0 else crv + + # exit condition + if stplen <= ZERO_THRESH: + break + + # momentum update + prev_t = t + t = (1 + np.sqrt(1 + 4 * t ** 2))/2 + prev_y = y.copy() + y = d + s*(prev_t - 1)/t + + # update gradient w.r.t y + gy += H.dot(y - prev_y) + + return d, gnew, crvmin def trsbox(xopt, g, H, sl, su, delta, use_fortran=USE_FORTRAN): @@ -405,8 +485,63 @@ def ball_step(x0, g, Delta): if sqrt(gsqnorm) < ZERO_THRESH: # Error catching: if g=0, make no step return 0.0 else: - return (sqrt(gdotx0**2 + gsqnorm*(Delta**2 - x0sqnorm)) - gdotx0) / gsqnorm + # Sqrt had negative input on prob 46 in OG DFOLS with noise + # print("Inside of the sqrt:", gdotx0**2 + gsqnorm*(Delta**2 - x0sqnorm)) + # Got Inside of the sqrt: -3.608971127647144e-42 + # Added max(0,...) here + return (sqrt(np.maximum(0,gdotx0**2 + gsqnorm*(Delta**2 - x0sqnorm))) - gdotx0) / gsqnorm + +def ctrsbox_linear(xbase, g, projections, Delta, d_max_iters=100, d_tol=1e-10, use_fortran=USE_FORTRAN): + # Solve the convex program: + # min_d g' * d + # s.t. xbase + d is feasible w.r.t. constraint set C + # ||d||^2 <= Delta^2 + + n = g.size + d = np.zeros((n,)) + y = d.copy() + t = 1 + dirn = -g + cons_dirns = [] + + # If g[i] = 0, never step along this direction + constant_directions = np.where(np.abs(dirn) < ZERO_THRESH)[0] + dirn[constant_directions] = 0.0 + + # trust region is a ball of radius delta centered around xbase + trproj = lambda w: pball(w, xbase, Delta) + # combine trust region constraints with user-entered constraints + P = projections.copy() + P.append(trproj) + def proj(d0): + p = dykstra(P, xbase + d0, max_iter=d_max_iters, tol=d_tol) + # we want the step only, so we subtract + # xbase from the new point: proj(xk + d) - xk + return p - xbase + + MAX_LOOP_ITERS = 100 * n ** 2 + + # projected GD loop + for ii in range(MAX_LOOP_ITERS): + w = y + dirn + prev_d = d.copy() + d = proj(w) + + s = d - prev_d + stplen = np.linalg.norm(s) + + # exit condition + if stplen <= ZERO_THRESH: + break + + # 'momentum' update + prev_t = t + t = (1 + np.sqrt(1 + 4 * t ** 2))/2 + prev_y = y.copy() + y = d + s*(prev_t - 1)/t + + return d def trsbox_linear(g, a_in, b_in, Delta, use_fortran=USE_FORTRAN): # Solve the convex program: @@ -466,6 +601,22 @@ def trsbox_linear(g, a_in, b_in, Delta, use_fortran=USE_FORTRAN): dirn[idx_hit] = 0.0 # no more searching this direction return x +def ctrsbox_geometry(xbase, c, g, projections, Delta, d_max_iters=100, d_tol=1e-10, use_fortran=USE_FORTRAN): + # Given a Lagrange polynomial defined by: L(x) = c + g' * (x - xbase) + # Maximise |L(x)| in a box + trust region - that is, solve: + # max_x abs(c + g' * (x - xbase)) + # s.t. x is feasible w.r.t constraint set C + # ||x-xbase|| <= Delta + # Setting s = x-xbase (or x = xbase + s), this is equivalent to: + # max_s abs(c + g' * s) + # s.t. xbase + s is is feasible w.r.t constraint set C + # ||s|| <= Delta + smin = ctrsbox_linear(xbase, g, projections, Delta, d_max_iters=100, d_tol=1e-10, use_fortran=use_fortran) # minimise g' * s + smax = ctrsbox_linear(xbase, -g, projections, Delta, d_max_iters=100, d_tol=1e-10, use_fortran=use_fortran) # maximise g' * s + if abs(c + np.dot(g, smin)) >= abs(c + np.dot(g, smax)): # choose the one with largest absolute value + return smin + else: + return smax def trsbox_geometry(xbase, c, g, lower, upper, Delta, use_fortran=USE_FORTRAN): # Given a Lagrange polynomial defined by: L(x) = c + g' * (x - xbase) diff --git a/dfols/util.py b/dfols/util.py index 6107e09..02c9884 100644 --- a/dfols/util.py +++ b/dfols/util.py @@ -27,11 +27,12 @@ import logging import numpy as np +import scipy.linalg as LA import sys __all__ = ['sumsq', 'eval_least_squares_objective', 'model_value', 'random_orthog_directions_within_bounds', - 'random_directions_within_bounds', 'apply_scaling', 'remove_scaling'] + 'random_directions_within_bounds', 'apply_scaling', 'remove_scaling', 'pbox', 'pball', 'dykstra', 'qr_rank'] def sumsq(x): @@ -207,3 +208,50 @@ def remove_scaling(x_scaled, scaling_changes): shift, scale = scaling_changes return shift + x_scaled * scale + +def dykstra(P,x0,max_iter=100,tol=1e-10): + x = x0.copy() + p = len(P) + y = np.zeros((p,x0.shape[0])) + + n = 0 + cI = float('inf') + while n < max_iter and cI >= tol: + cI = 0 + for i in range(0,p): + # Update iterate + prev_x = x.copy() + x = P[i](prev_x - y[i,:]) + + # Update increment + prev_y = y[i,:].copy() + y[i,:] = x - (prev_x - prev_y) + + # Stop condition + cI += np.linalg.norm(prev_y - y[i,:])**2 + + n += 1 + + return x + + +def pball(x,c,r): + return c + (r/np.max([np.linalg.norm(x-c),r]))*(x-c) + + +def pbox(x,l,u): + return np.minimum(np.maximum(x,l), u) + +''' +Calculates rank of square matrix with QR. +We use the fact that the rank of a square matrix A +can be given by the number of nonzero diagonal elements of +R in the QR factorization of A. +''' +def qr_rank(A,tol=1e-15): + m,n = A.shape + assert m == n, "Input matrix must be square" + Q,R = LA.qr(A) + D = np.abs(np.diag(R)) + rank = np.sum(D > tol) + return rank, D diff --git a/dfols/version.py b/dfols/version.py index 94b452c..dc26bd5 100644 --- a/dfols/version.py +++ b/dfols/version.py @@ -22,4 +22,4 @@ """ -__version__ = '1.2.3' +__version__ = '1.3.0' diff --git a/docs/advanced.rst b/docs/advanced.rst index 6073888..1303bee 100644 --- a/docs/advanced.rst +++ b/docs/advanced.rst @@ -103,6 +103,15 @@ Dynamically Growing Initial Set * :code:`growing.gamma_dec` - Trust region decrease parameter during the growing phase. Default is :code:`tr_radius.gamma_dec`. * :code:`growing.num_new_dirns_each_iter` - Number of new search directions to add with each iteration where we do not have a full set of search directions. Default is 0, as this approach is not recommended. +Dykstra's Algorithm +------------------------------- +* :code:`dykstra.d_tol` - Tolerance on the stopping conditions of Dykstra's algorithm. Default is :math:`10^{-10}`. +* :code:`dykstra.max_iters` - The maximum number of iterations Dykstra's algorithm is allowed to take before stopping. Default is :math:`100`. + +Checking Matrix Rank +------------------------------- +* :code:`matrix_rank.r_tol` - Tolerance on what is the smallest posisble diagonal entry value in the QR factorization before being considered zero. Default is :math:`10^{-18}`. + References ---------- diff --git a/docs/build/doctrees/advanced.doctree b/docs/build/doctrees/advanced.doctree index 06e7653..678b5b6 100755 Binary files a/docs/build/doctrees/advanced.doctree and b/docs/build/doctrees/advanced.doctree differ diff --git a/docs/build/doctrees/diagnostic.doctree b/docs/build/doctrees/diagnostic.doctree index 8d214f7..a3635b0 100755 Binary files a/docs/build/doctrees/diagnostic.doctree and b/docs/build/doctrees/diagnostic.doctree differ diff --git a/docs/build/doctrees/environment.pickle b/docs/build/doctrees/environment.pickle index 0740510..bd34af3 100755 Binary files a/docs/build/doctrees/environment.pickle and b/docs/build/doctrees/environment.pickle differ diff --git a/docs/build/doctrees/history.doctree b/docs/build/doctrees/history.doctree index e0c1d81..665337c 100755 Binary files a/docs/build/doctrees/history.doctree and b/docs/build/doctrees/history.doctree differ diff --git a/docs/build/doctrees/index.doctree b/docs/build/doctrees/index.doctree index c4b2eb2..9fc32b2 100755 Binary files a/docs/build/doctrees/index.doctree and b/docs/build/doctrees/index.doctree differ diff --git a/docs/build/doctrees/info.doctree b/docs/build/doctrees/info.doctree index 8f40b31..a431aac 100755 Binary files a/docs/build/doctrees/info.doctree and b/docs/build/doctrees/info.doctree differ diff --git a/docs/build/doctrees/install.doctree b/docs/build/doctrees/install.doctree index 8aab039..05b2139 100755 Binary files a/docs/build/doctrees/install.doctree and b/docs/build/doctrees/install.doctree differ diff --git a/docs/build/doctrees/userguide.doctree b/docs/build/doctrees/userguide.doctree index e30357b..5f8acd3 100755 Binary files a/docs/build/doctrees/userguide.doctree and b/docs/build/doctrees/userguide.doctree differ diff --git a/docs/build/html/.buildinfo b/docs/build/html/.buildinfo index 3bc5cd9..987e588 100644 --- a/docs/build/html/.buildinfo +++ b/docs/build/html/.buildinfo @@ -1,4 +1,4 @@ # Sphinx build info version 1 # This file hashes the configuration used when building these files. When it is not found, a full rebuild will be done. -config: 7851303ebe6ed71f013a86e86fbeadbc +config: f48a4434832128cee1c37b97e0d08ba7 tags: 645f666f9bcd5a90fca523b33c5a78b7 diff --git a/docs/build/html/_sources/advanced.rst.txt b/docs/build/html/_sources/advanced.rst.txt index 6073888..1303bee 100644 --- a/docs/build/html/_sources/advanced.rst.txt +++ b/docs/build/html/_sources/advanced.rst.txt @@ -103,6 +103,15 @@ Dynamically Growing Initial Set * :code:`growing.gamma_dec` - Trust region decrease parameter during the growing phase. Default is :code:`tr_radius.gamma_dec`. * :code:`growing.num_new_dirns_each_iter` - Number of new search directions to add with each iteration where we do not have a full set of search directions. Default is 0, as this approach is not recommended. +Dykstra's Algorithm +------------------------------- +* :code:`dykstra.d_tol` - Tolerance on the stopping conditions of Dykstra's algorithm. Default is :math:`10^{-10}`. +* :code:`dykstra.max_iters` - The maximum number of iterations Dykstra's algorithm is allowed to take before stopping. Default is :math:`100`. + +Checking Matrix Rank +------------------------------- +* :code:`matrix_rank.r_tol` - Tolerance on what is the smallest posisble diagonal entry value in the QR factorization before being considered zero. Default is :math:`10^{-18}`. + References ---------- diff --git a/docs/build/html/_sources/history.rst.txt b/docs/build/html/_sources/history.rst.txt index fd4d8ef..0d1938c 100644 --- a/docs/build/html/_sources/history.rst.txt +++ b/docs/build/html/_sources/history.rst.txt @@ -45,3 +45,8 @@ Version 1.2.2 (26 Feb 2021) Version 1.2.3 (1 Jun 2021) --------------------------- * Minor update to customise handling of NaNs in objective evaluations - no changes to the DFO-LS algorithm. + +Version 1.3.0 (16 Oct 2021) +--------------------------- +* Handle finitely many arbitrary convex constraints in addition to simple bound constraints. +* Only new functionality is added, so there is no change to the solver for unconstrained/bound-constrained problems. diff --git a/docs/build/html/_sources/index.rst.txt b/docs/build/html/_sources/index.rst.txt index a0e467e..031d6c9 100644 --- a/docs/build/html/_sources/index.rst.txt +++ b/docs/build/html/_sources/index.rst.txt @@ -11,16 +11,17 @@ DFO-LS: Derivative-Free Optimizer for Least-Squares Minimization **Author:** `Lindon Roberts `_ -DFO-LS is a flexible package for finding local solutions to nonlinear least-squares minimization problems (with optional bound constraints), without requiring any derivatives of the objective. DFO-LS stands for Derivative-Free Optimizer for Least-Squares. +DFO-LS is a flexible package for finding local solutions to nonlinear least-squares minimization problems (with optional constraints), without requiring any derivatives of the objective. DFO-LS stands for Derivative-Free Optimizer for Least-Squares. That is, DFO-LS solves .. math:: \min_{x\in\mathbb{R}^n} &\quad f(x) := \sum_{i=1}^{m}r_{i}(x)^2 \\ - \text{s.t.} &\quad a \leq x \leq b + \text{s.t.} &\quad x \in C\\ + &\quad a \leq x \leq b -The upper and lower bounds on the variables are non-relaxable (i.e. DFO-LS will never ask to evaluate a point outside the bounds). +The constraint set :math:`C` is the intersection of multiple convex sets provided as input by the user. All constraints are non-relaxable (i.e. DFO-LS will never ask to evaluate a point that is not feasible). Full details of the DFO-LS algorithm are given in our paper: C. Cartis, J. Fiala, B. Marteau and L. Roberts, `Improving the Flexibility and Robustness of Model-Based Derivative-Free Optimization Solvers `_, *ACM Transactions on Mathematical Software*, 45:3 (2019), pp. 32:1-32:41 [`preprint `_] . DFO-LS is a more flexible version of `DFO-GN `_. diff --git a/docs/build/html/_sources/info.rst.txt b/docs/build/html/_sources/info.rst.txt index 824a232..e4e0b29 100644 --- a/docs/build/html/_sources/info.rst.txt +++ b/docs/build/html/_sources/info.rst.txt @@ -3,14 +3,16 @@ Overview When to use DFO-LS ------------------ -DFO-LS is designed to solve the nonlinear least-squares minimization problem (with optional bound constraints) +DFO-LS is designed to solve the nonlinear least-squares minimization problem (with optional convex constraints). .. math:: \min_{x\in\mathbb{R}^n} &\quad f(x) := \sum_{i=1}^{m}r_{i}(x)^2 \\ - \text{s.t.} &\quad a \leq x \leq b + \text{s.t.} &\quad x \in C\\ + &\quad a \leq x \leq b We call :math:`f(x)` the objective function and :math:`r_i(x)` the residual functions (or simply residuals). +:math:`C` is the intersection of multiple convex sets given as input by the user. DFO-LS is a *derivative-free* optimization algorithm, which means it does not require the user to provide the derivatives of :math:`f(x)` or :math:`r_i(x)`, nor does it attempt to estimate them internally (by using finite differencing, for instance). diff --git a/docs/build/html/_sources/userguide.rst.txt b/docs/build/html/_sources/userguide.rst.txt index 6804892..e8a7b54 100644 --- a/docs/build/html/_sources/userguide.rst.txt +++ b/docs/build/html/_sources/userguide.rst.txt @@ -9,9 +9,9 @@ DFO-LS is designed to solve the local optimization problem .. math:: \min_{x\in\mathbb{R}^n} &\quad f(x) := \sum_{i=1}^{m}r_{i}(x)^2 \\ - \text{s.t.} &\quad a \leq x \leq b + \text{s.t.} &\quad x \in C -where the bound constraints :math:`a \leq x \leq b` are optional. The upper and lower bounds on the variables are non-relaxable (i.e. DFO-LS will never ask to evaluate a point outside the bounds). +where the set :math:`C` is an optional non-empty, closed and convex constraint set. The constraints are non-relaxable (i.e. DFO-LS will never ask to evaluate a point that is not feasible). DFO-LS iteratively constructs an interpolation-based model for the objective, and determines a step using a trust-region framework. For an in-depth technical description of the algorithm see the paper [CFMR2018]_. @@ -48,6 +48,7 @@ The possible values of :code:`soln.flag` are defined by the following variables: * :code:`soln.EXIT_MAXFUN_WARNING` - maximum allowed objective evaluations reached. This is the most likely return value when using multiple restarts. * :code:`soln.EXIT_SLOW_WARNING` - maximum number of slow iterations reached. * :code:`soln.EXIT_FALSE_SUCCESS_WARNING` - DFO-LS reached the maximum number of restarts which decreased the objective, but to a worse value than was found in a previous run. +* :code:`soln.EXIT_TR_INCREASE_WARNING` - model increase when solving the trust region subproblem with multiple arbitrary constraints. * :code:`soln.EXIT_INPUT_ERROR` - error in the inputs. * :code:`soln.EXIT_TR_INCREASE_ERROR` - error occurred when solving the trust region subproblem. * :code:`soln.EXIT_LINALG_ERROR` - linear algebra error, e.g. the interpolation points produced a singular linear system. @@ -75,6 +76,7 @@ These arguments are: * :code:`args` - a tuple of extra arguments passed to the objective function. * :code:`bounds` - a tuple :code:`(lower, upper)` with the vectors :math:`a` and :math:`b` of lower and upper bounds on :math:`x` (default is :math:`a_i=-10^{20}` and :math:`b_i=10^{20}`). To set bounds for either :code:`lower` or :code:`upper`, but not both, pass a tuple :code:`(lower, None)` or :code:`(None, upper)`. +* :code:`projections` - a list :code:`[f1,f2,...,fn]` of functions that each take as input a point :code:`x` and return a new point :code:`y`. The new point :code:`y` should be given by the projection of :code:`x` onto a closed convex set. The intersection of all sets corresponding to a function must be non-empty. * :code:`npt` - the number of interpolation points to use (default is :code:`len(x0)+1`). If using restarts, this is the number of points to use in the first run of the solver, before any restarts (and may be optionally increased via settings in :code:`user_params`). * :code:`rhobeg` - the initial value of the trust region radius (default is :math:`0.1\max(\|x_0\|_{\infty}, 1)`, or 0.1 if :code:`scaling_within_bounds`). * :code:`rhoend` - minimum allowed value of trust region radius, which determines when a successful termination occurs (default is :math:`10^{-8}`). @@ -144,7 +146,7 @@ This and all following problems can be found in the `examples - + + - Advanced Usage — DFO-LS v1.2.3 documentation + Advanced Usage — DFO-LS v1.3.0 documentation @@ -61,7 +62,7 @@
- 1.2.3 + 1.3.0
@@ -85,7 +86,7 @@ -

Contents:

+

Contents:

@@ -171,20 +174,20 @@
-
+

Advanced Usage

This section describes different optional user parameters available in DFO-LS.

In the last section (Using DFO-LS), we introduced dfols.solve(), which has the optional input user_params. This is a Python dictionary of user parameters. We will now go through the settings which can be changed in this way. More details are available in the paper [CFMR2018].

The default values, used if no override is given, in some cases vary depending on whether objfun has stochastic noise; that is, whether evaluating objfun(x) several times at the same x gives the same result or not. Whether or not this is the case is determined by the objfun_has_noise input to dfols.solve() (and not by inspecting objfun, for instance).

-
+

General Algorithm Parameters

  • general.rounding_error_constant - Internally, all interpolation points are stored with respect to a base point \(x_b\); that is, we store \(\{y_t-x_b\}\), which reduces the risk of roundoff errors. We shift \(x_b\) to \(x_k\) when \(\|s_k\| \leq \text{const}\|x_k-x_b\|\), where ‘const’ is this parameter. Default is 0.1.

  • general.safety_step_thresh - Threshold for when to call the safety step, \(\|s_k\| \leq \gamma_S \rho_k\). Default is \(\gamma_S =0.5\).

  • general.check_objfun_for_overflow - Whether to cap the value of \(r_i(x)\) when they are large enough that an OverflowError will be encountered when trying to evaluate \(f(x)\). Default is True.

-
-
+
+

Logging and Output

-
-
+ +

Initialization of Points

  • init.random_initial_directions - Build the initial interpolation set using random directions (as opposed to coordinate directions). Default as of version 1.2 is False.

  • init.random_directions_make_orthogonal - If building initial interpolation set with random directions, whether or not these should be orthogonalized. Default is True.

  • init.run_in_parallel - If using random directions, whether or not to ask for all objfun to be evaluated at all points without any intermediate processing. Default is False.

-
- -
+ +

Termination on Small Objective Value

  • model.abs_tol - Tolerance on \(f(x_k)\); quit if \(f(x_k)\) is below this value. Default is \(10^{-12}\).

  • model.rel_tol - Relative tolerance on \(f(x_k)\); quit if \(f(x_k)/f(x_0)\) is below this value. Default is \(10^{-20}\).

-
-
+ +

Termination on Slow Progress

  • slow.history_for_slow - History used to determine whether the current iteration is ‘slow’. Default is 5.

  • slow.thresh_for_slow - Threshold for objective decrease used to determine whether the current iteration is ‘slow’. Default is \(10^{-4}\).

  • slow.max_slow_iters - Number of consecutive slow successful iterations before termination (or restart). Default is 20*len(x0).

-
-
+ +

Stochastic Noise Information

-
-
+ +

Interpolation Management

  • interpolation.precondition - whether or not to scale the interpolation linear system to improve conditioning. Default is True.

  • interpolation.throw_error_on_nans - whether or not to throw numpy.linalg.LinAlgError if trying to interpolate to NaN objective values. If False, DFO-LS should terminate gracefully with an error flag. Default is False.

-
-
+ +

Regression Model Management

  • regression.num_extra_steps - In successful iterations, the number of extra points (other than accepting the trust region step) to move, useful when \(|Y_k|>n+1\) (\(n\) is len(x0)). Default is 0.

  • regression.increase_num_extra_steps_with_restart - The amount to increase regression.num_extra_steps by with each restarts, for instance if increasing the number of points with each restart. Default is 0.

  • regression.momentum_extra_steps - If moving extra points in successful iterations, whether to use the ‘momentum’ method. If not, uses geometry-improving steps. Default is False.

-
-
+ +

Multiple Restarts

-
-
+ +

Dynamically Growing Initial Set

-
-
+ +
+

Dykstra’s Algorithm

+
    +

  • dykstra.d_tol - Tolerance on the stopping conditions of Dykstra’s algorithm. Default is \(10^{-10}\).

    • +

  • dykstra.max_iters - The maximum number of iterations Dykstra’s algorithm is allowed to take before stopping. Default is \(100\).

    • +
+
+
+

Checking Matrix Rank

+
    +

  • matrix_rank.r_tol - Tolerance on what is the smallest posisble diagonal entry value in the QR factorization before being considered zero. Default is \(10^{-18}\).

    • +
+
+

References

CFMR2018

Coralia Cartis, Jan Fiala, Benjamin Marteau and Lindon Roberts, Improving the Flexibility and Robustness of Model-Based Derivative-Free Optimization Solvers, ACM Transactions on Mathematical Software, 45:3 (2019), pp. 32:1-32:41 [preprint]

-
-
+ +
diff --git a/docs/build/html/diagnostic.html b/docs/build/html/diagnostic.html index f4e5a4c..90a8f83 100644 --- a/docs/build/html/diagnostic.html +++ b/docs/build/html/diagnostic.html @@ -5,10 +5,11 @@ - + + - Diagnostic Information — DFO-LS v1.2.3 documentation + Diagnostic Information — DFO-LS v1.3.0 documentation @@ -61,7 +62,7 @@
- 1.2.3 + 1.3.0
@@ -85,7 +86,7 @@ -

Contents:

+

Contents:

  • Installing DFO-LS
  • Overview
    • @@ -164,7 +165,7 @@
    -
    +

    Diagnostic Information

    In Using DFO-LS, we saw that the output of DFO-LS returns a container which includes diagnostic information about the progress of the algorithm (soln.diagnostic_info). This object is a Pandas DataFrame, with one row per iteration of the algorithm. In this section, we explain the meaning of each type of output (the columns of the DataFrame).

    To save this information to a CSV file, use:

    @@ -183,23 +184,23 @@

    Diagnostic Information

    Depending on exactly how DFO-LS terminates, the last row of results may not be fully populated.

    -
    +

    Current Iterate

    • xk - Best point found so far (current iterate). This is only saved if user_params['logging.save_xk'] = True.

    • rk - The vector of residuals at the current iterate. This is only saved if user_params['logging.save_rk'] = True.

    • fk - The value of \(f\) at the current iterate.

    -
    -
    + +

    Trust Region

    • rho - The lower bound on the trust region radius \(\rho_k\).

    • delta - The trust region radius \(\Delta_k\).

    • norm_sk - The norm of the trust region step \(\|s_k\|\).

    -
    -
    + +

    Model Interpolation

    -
    -
    + +

    Iteration Count

    -
    -
    + +

    Algorithm Progress

    • iter_type - A text description of what type of iteration we had (e.g. Successful, Safety, etc.)

    • ratio - The ratio of actual to predicted objective reduction in the trust region step.

    • slow_iter - Equal to 1 if the current iteration is successful but slow, 0 if is successful but not slow, and -1 if was not successful.

    -
    -
    + +
    diff --git a/docs/build/html/genindex.html b/docs/build/html/genindex.html index ab3ee15..3f9829a 100644 --- a/docs/build/html/genindex.html +++ b/docs/build/html/genindex.html @@ -9,7 +9,7 @@ - Index — DFO-LS v1.2.3 documentation + Index — DFO-LS v1.3.0 documentation @@ -60,7 +60,7 @@
    - 1.2.3 + 1.3.0
    @@ -84,7 +84,7 @@ -

    Contents:

    +

    Contents:

    @@ -167,42 +169,42 @@
    -
    +

    Version History

    This section lists the different versions of DFO-LS and the updates between them.

    -
    +

    Version 1.0 (6 Feb 2018)

    • Initial release of DFO-LS

    -
    -
    +
    +

    Version 1.0.1 (20 Feb 2018)

    • Minor bug fix to trust region subproblem solver (the output crvmin is calculated correctly) - this has minimal impact on the performance of DFO-LS.

    -
    -
    + +

    Version 1.0.2 (20 Jun 2018)

    • Extra optional input args which passes through arguments for objfun.

    • Bug fixes: default parameters for reduced initialization cost regime, returning correct value if exiting from within a safety step, retrieving dependencies during installation.

    -
    -
    + +

    Version 1.1 (16 Jan 2019)

    • Use different default reduced initialization cost method for inverse problems to ensure whole space is searched correctly.

    • Bug fixes: default trust region radius when scaling feasible region, exit correctly when no Jacobian returned, handling overflow at initial value

    -
    -
    + +

    Version 1.1.1 (5 Apr 2019)

    • Link code to Zenodo, to create DOI - no changes to the DFO-LS algorithm.

    -
    -
    + +

    Version 1.2 (12 Feb 2020)

    • Use deterministic initialisation by default (so it is no longer necessary to set a random seed for reproducibility of DFO-LS results).

      • @@ -213,26 +215,33 @@

      Version 1.2 (12 Feb 2020) +

    +

    Version 1.2.1 (13 Feb 2020)

    • Make the use of the trustregion package optional, not installed by default.

    -
    -
    + +

    Version 1.2.2 (26 Feb 2021)

    • Minor update to remove NumPy deprecation warnings - no changes to the DFO-LS algorithm.

    -
    -
    + +

    Version 1.2.3 (1 Jun 2021)

    • Minor update to customise handling of NaNs in objective evaluations - no changes to the DFO-LS algorithm.

    -
    -
    + +
    +

    Version 1.3.0 (16 Oct 2021)

    +
      +

    • Handle finitely many arbitrary convex constraints in addition to simple bound constraints.

      • +

    • Only new functionality is added, so there is no change to the solver for unconstrained/bound-constrained problems.

      • +
    +
    +
    diff --git a/docs/build/html/index.html b/docs/build/html/index.html index 76457b0..022ad20 100644 --- a/docs/build/html/index.html +++ b/docs/build/html/index.html @@ -5,10 +5,11 @@ - + + - DFO-LS: Derivative-Free Optimizer for Least-Squares Minimization — DFO-LS v1.2.3 documentation + DFO-LS: Derivative-Free Optimizer for Least-Squares Minimization — DFO-LS v1.3.0 documentation @@ -60,7 +61,7 @@
    - 1.2.3 + 1.3.0
    @@ -84,7 +85,7 @@ -

    Contents:

    +

    Contents:

    -
    +

    Acknowledgements

    This software was developed under the supervision of Coralia Cartis, and was supported by the EPSRC Centre For Doctoral Training in Industrially Focused Mathematical Modelling (EP/L015803/1) in collaboration with the Numerical Algorithms Group.

    -
    - + + diff --git a/docs/build/html/info.html b/docs/build/html/info.html index f3867f0..72e907a 100644 --- a/docs/build/html/info.html +++ b/docs/build/html/info.html @@ -5,10 +5,11 @@ - + + - Overview — DFO-LS v1.2.3 documentation + Overview — DFO-LS v1.3.0 documentation @@ -61,7 +62,7 @@
    - 1.2.3 + 1.3.0
    @@ -85,7 +86,7 @@ -

    Contents:

    +

    Contents:

    • Installing DFO-LS
    • Overview
        @@ -164,22 +165,24 @@
        -
        +

        Overview

        -
        +

        When to use DFO-LS

        -

        DFO-LS is designed to solve the nonlinear least-squares minimization problem (with optional bound constraints)

        +

        DFO-LS is designed to solve the nonlinear least-squares minimization problem (with optional convex constraints).

        \[\begin{split}\min_{x\in\mathbb{R}^n} &\quad f(x) := \sum_{i=1}^{m}r_{i}(x)^2 \\ -\text{s.t.} &\quad a \leq x \leq b\end{split}\]
        -

        We call \(f(x)\) the objective function and \(r_i(x)\) the residual functions (or simply residuals).

        +\text{s.t.} &\quad x \in C\\ + &\quad a \leq x \leq b\end{split}\]
        +

        We call \(f(x)\) the objective function and \(r_i(x)\) the residual functions (or simply residuals). +\(C\) is the intersection of multiple convex sets given as input by the user.

        DFO-LS is a derivative-free optimization algorithm, which means it does not require the user to provide the derivatives of \(f(x)\) or \(r_i(x)\), nor does it attempt to estimate them internally (by using finite differencing, for instance).

        There are two main situations when using a derivative-free algorithm (such as DFO-LS) is preferable to a derivative-based algorithm (which is the vast majority of least-squares solvers).

        If the residuals are noisy, then calculating or even estimating their derivatives may be impossible (or at least very inaccurate). By noisy, we mean that if we evaluate \(r_i(x)\) multiple times at the same value of \(x\), we get different results. This may happen when a Monte Carlo simulation is used, for instance, or \(r_i(x)\) involves performing a physical experiment.

        If the residuals are expensive to evaluate, then estimating derivatives (which requires \(n\) evaluations of each \(r_i(x)\) for every point of interest \(x\)) may be prohibitively expensive. Derivative-free methods are designed to solve the problem with the fewest number of evaluations of the objective as possible.

        However, if you have provide (or a solver can estimate) derivatives of \(r_i(x)\), then it is probably a good idea to use one of the many derivative-based solvers (such as one from the SciPy library).

        -
        -
        + +

        Parameter Fitting

        A very common problem in many quantitative disciplines is fitting parameters to observed data. Typically, this means that we have developed a model for some proccess, which takes a vector of (known) inputs \(\mathrm{obs}\in\mathbb{R}^N\) and some model parameters \(x=(x_1, \ldots, x_n)\in\mathbb{R}^n\), and computes a (predicted) quantity of interest \(y\in\mathbb{R}\):

        @@ -193,8 +196,8 @@

        Parameter Fitting +

        +

        Solving Nonlinear Systems of Equations

        Suppose we wish to solve the system of nonlinear equations: find \(x\in\mathbb{R}^n\) satisfying

        @@ -211,23 +214,23 @@

        Solving Nonlinear Systems of Equations\(f=0\) at the solution).

        Which solution? DFO-LS, and most similar software, will only find one solution to a set of nonlinear equations. Which one it finds is very difficult to predict, and depends very strongly on the point where the solver is started from. Often it finds the closest solution, but there are no guarantees this will be the case. If you need to find all/multiple solutions for your problem, consider techniques such as deflation.

        -

        -
        +
        +

        Details of the DFO-LS Algorithm

        DFO-LS is a type of trust-region method, a common category of optimization algorithms for nonconvex problems. Given a current estimate of the solution \(x_k\), we compute a model which approximates the objective \(m_k(s)\approx f(x_k+s)\) (for small steps \(s\)), and maintain a value \(\Delta_k>0\) (called the trust region radius) which measures the size of \(s\) for which the approximation is good.

        At each step, we compute a trial step \(s_k\) designed to make our approximation \(m_k(s)\) small (this task is called the trust region subproblem). We evaluate the objective at this new point, and if this provided a good decrease in the objective, we take the step (\(x_{k+1}=x_k+s_k\)), otherwise we stay put (\(x_{k+1}=x_k\)). Based on this information, we choose a new value \(\Delta_{k+1}\), and repeat the process.

        In DFO-LS, we construct our approximation \(m_k(s)\) by interpolating a linear approximation for each residual \(r_i(x)\) at several points close to \(x_k\). To make sure our interpolated model is accurate, we need to regularly check that the points are well-spaced, and move them if they aren’t (i.e. improve the geometry of our interpolation points).

        -

        A complete description of the DFO-LS algorithm is given in our paper [CFMR2018].

        -
        -
        +

        A complete description of the DFO-LS algorithm is given in our paper [CFMR2018].

        + +

        References

        CFMR2018

        Coralia Cartis, Jan Fiala, Benjamin Marteau and Lindon Roberts, Improving the Flexibility and Robustness of Model-Based Derivative-Free Optimization Solvers, ACM Transactions on Mathematical Software, 45:3 (2019), pp. 32:1-32:41 [preprint]

        -
        -
        + +
        diff --git a/docs/build/html/install.html b/docs/build/html/install.html index d74c88f..e12aafc 100644 --- a/docs/build/html/install.html +++ b/docs/build/html/install.html @@ -5,10 +5,11 @@ - + + - Installing DFO-LS — DFO-LS v1.2.3 documentation + Installing DFO-LS — DFO-LS v1.3.0 documentation @@ -61,7 +62,7 @@
        - 1.2.3 + 1.3.0
        @@ -85,7 +86,7 @@ -

        Contents:

        +

        Contents:

        • Installing DFO-LS
          • Requirements
            • @@ -164,9 +165,9 @@
            -
            +

            Installing DFO-LS

            -
            +

            Requirements

            DFO-LS requires the following software to be installed:

            Optional package: DFO-LS versions 1.2 and higher also support the trustregion package for fast trust-region subproblem solutions. To install this, make sure you have a Fortran compiler (e.g. gfortran) and NumPy installed, then run pip install trustregion. You do not have to have trustregion installed for DFO-LS to work, and it is not installed by default.

            -
            -
            +
            +

            Installation using pip

            For easy installation, use pip (http://www.pip-installer.org/) as root:

            .. code-block:: bash
@@ -208,8 +209,8 @@ 
            Installation using pip

            to upgrade DFO-LS to the latest version.

            -
            -
            + +

            Manual installation

            Alternatively, you can download the source code from Github and unpack as follows:

            @@ -238,8 +239,8 @@

            Manual installation +

            +

            Testing

            If you installed DFO-LS manually, you can test your installation by running:

            @@ -248,8 +249,8 @@

            Testing +

            +

            Uninstallation

            If DFO-LS was installed using pip you can uninstall as follows:

            @@ -258,8 +259,8 @@

            Uninstallation - Search — DFO-LS v1.2.3 documentation + Search — DFO-LS v1.3.0 documentation @@ -60,7 +60,7 @@
            - 1.2.3 + 1.3.0
            @@ -84,7 +84,7 @@ -

            Contents:

            +

            Contents:

            • Installing DFO-LS
            • Overview
              • diff --git a/docs/build/html/searchindex.js b/docs/build/html/searchindex.js index fd398c6..7e8fa9d 100644 --- a/docs/build/html/searchindex.js +++ b/docs/build/html/searchindex.js @@ -1 +1 @@ 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No newline at end of file diff --git a/docs/build/html/userguide.html b/docs/build/html/userguide.html index 5b40502..4cfe6d0 100644 --- a/docs/build/html/userguide.html +++ b/docs/build/html/userguide.html @@ -5,10 +5,11 @@ - + + - Using DFO-LS — DFO-LS v1.2.3 documentation + Using DFO-LS — DFO-LS v1.3.0 documentation @@ -61,7 +62,7 @@
              - 1.2.3 + 1.3.0
              @@ -85,7 +86,7 @@ -

              Contents:

              +

              Contents:

            +

            How to use DFO-LS

            The main interface to DFO-LS is via the function solve

            @@ -211,6 +213,7 @@

            How to use DFO-LSsoln.EXIT_MAXFUN_WARNING - maximum allowed objective evaluations reached. This is the most likely return value when using multiple restarts.

          • soln.EXIT_SLOW_WARNING - maximum number of slow iterations reached.

          • soln.EXIT_FALSE_SUCCESS_WARNING - DFO-LS reached the maximum number of restarts which decreased the objective, but to a worse value than was found in a previous run.

            • +

          • soln.EXIT_TR_INCREASE_WARNING - model increase when solving the trust region subproblem with multiple arbitrary constraints.

          • soln.EXIT_INPUT_ERROR - error in the inputs.

          • soln.EXIT_TR_INCREASE_ERROR - error occurred when solving the trust region subproblem.

          • soln.EXIT_LINALG_ERROR - linear algebra error, e.g. the interpolation points produced a singular linear system.

            • @@ -222,8 +225,8 @@

            How to use DFO-LS +

            +

            Optional Arguments

            The solve function has several optional arguments which the user may provide:

            @@ -239,6 +242,7 @@

            Optional Arguments

          • args - a tuple of extra arguments passed to the objective function.

          • bounds - a tuple (lower, upper) with the vectors \(a\) and \(b\) of lower and upper bounds on \(x\) (default is \(a_i=-10^{20}\) and \(b_i=10^{20}\)). To set bounds for either lower or upper, but not both, pass a tuple (lower, None) or (None, upper).

            • +

          • projections - a list [f1,f2,...,fn] of functions that each take as input a point x and return a new point y. The new point y should be given by the projection of x onto a closed convex set. The intersection of all sets corresponding to a function must be non-empty.

          • npt - the number of interpolation points to use (default is len(x0)+1). If using restarts, this is the number of points to use in the first run of the solver, before any restarts (and may be optionally increased via settings in user_params).

          • rhobeg - the initial value of the trust region radius (default is \(0.1\max(\|x_0\|_{\infty}, 1)\), or 0.1 if scaling_within_bounds).

          • rhoend - minimum allowed value of trust region radius, which determines when a successful termination occurs (default is \(10^{-8}\)).

            • @@ -252,8 +256,8 @@

            Optional Argumentsscaling_within_bounds flag is designed to provide an easy way to achieve this, if you have set the bounds lower and upper.

            -

            -
            + +

            A Simple Example

            Suppose we wish to minimize the Rosenbrock test function:

            @@ -299,10 +303,10 @@

            A Simple Exampleexamples directory on the DFO-LS Github page.

            -

            -
            +
            +

            Adding Bounds and More Output

            -

            We can extend the above script to add constraints. To do this, we can add the lines

            +

            We can extend the above script to add constraints. To add bound constraints alone, we can add the lines

            # Define bound constraints (lower <= x <= upper)
 lower = np.array([-10.0, -10.0])
@@ -344,7 +348,7 @@ 
            Adding Bounds and More Outputobjfun:
            
+
            And for the simple bounds example we can now see each evaluation of objfun:
            
 
            
 
            Function eval 1 at point 1 has f = 39.65 at x = [-1.2   0.85]
 Initialising (coordinate directions)
@@ -377,8 +381,97 @@ 
            Adding Bounds and More Output
+
            Handling Arbitrary Convex Constraints
            
+
            DFO-LS can also handle more general constraints where they can be written as the intersection of finitely many convex sets. For example, the below code
+minimizes the Rosenbrock function subject to a constraint set given by the intersection of two convex sets. Note the intersection of the user-provided convex
+sets must be non-empty.
            
+
            
+
            '''
+DFO-LS example: minimize the Rosenbrock function with arbitrary convex constraints
+
+This example defines two functions pball(x) and pbox(x) that project onto ball and
+box constraint sets respectively. It then passes both these functions to the DFO-LS
+solver so that it can find a constrained minimizer to the Rosenbrock function.
+Such a minimizer must lie in the intersection of constraint sets corresponding to
+projection functions pball(x) and pbox(x). The description of the problem is as follows:
+
+    min rosenbrock(x)
+    s.t.
+        -2 <= x[0] <= 1.1,
+        1.1 <= x[1] <= 3,
+        norm(x-c) <= 0.4
+
+where c = [0.7, 1.5] is the centre of the ball.
+'''
+from __future__ import print_function
+import numpy as np
+import dfols
+
+# Define the objective function
+def rosenbrock(x):
+    return np.array([10.0 * (x[1] - x[0] ** 2), 1.0 - x[0]])
+
+# Define the starting point
+x0 = np.array([-1.2, 1])
+
+'''
+Define ball projection function
+Projects the input x onto a ball with
+centre point (0.7,1.5) and radius 0.4.
+'''
+def pball(x):
+    c = np.array([0.7,1.5]) # ball centre
+    r = 0.4 # ball radius
+    return c + (r/np.max([np.linalg.norm(x-c),r]))*(x-c)
+
+'''
+Define box projection function
+Projects the input x onto a box
+such that -2 <= x[0] <= 0.9 and
+1.1 <= x[1] <= 3.
+
+Note: One could equivalently add bound
+constraints as a separate input to the solver
+instead.
+'''
+def pbox(x):
+    l = np.array([-2, 1.1]) # lower bound
+    u = np.array([0.9, 3]) # upper bound
+    return np.minimum(np.maximum(x,l), u)
+
+# For optional extra output details
+import logging
+logging.basicConfig(level=logging.DEBUG, format='%(message)s')
+
+# Call DFO-LS
+soln = dfols.solve(rosenbrock, x0, projections=[pball,pbox])
+
+# Display output
+print(soln)
+
            
+
            
+
            
+
            Note that for bound constraints one can choose to either implement them by defining a projection function as above, or by passing the bounds as input like in the example from the section on adding bound constraints.
            
+
            DFO-LS correctly finds the solution to this constrained problem too. Note that we get a warning because the step computed in the trust region subproblem
+gave an increase in the model. This is common in the case where multiple constraints are active at the optimal point.
            
+
            
+
            ****** DFO-LS Results ******
+Solution xmin = [0.9        1.15359245]
+Residual vector = [3.43592448 0.1       ]
+Objective value f(xmin) = 11.81557703
+Needed 10 objective evaluations (at 10 points)
+Approximate Jacobian = [[-1.79826221e+01  1.00004412e+01]
+ [-1.00000000e+00 -1.81976605e-15]]
+Exit flag = 5
+Warning (trust region increase): Either multiple constraints are active or trust region step gave model increase
+****************************
+
            
 
            
-
            
+
            
+
            +

            Example: Noisy Objective Evaluation

            As described in Overview, derivative-free algorithms such as DFO-LS are particularly useful when objfun has noise. Let’s modify the previous example to include random noise in our objective evaluation, and compare it to a derivative-based solver:

            @@ -478,8 +571,8 @@

            Example: Noisy Objective Evaluation +

            +

            Example: Parameter Estimation/Data Fitting

            Next, we show a short example of using DFO-LS to solve a parameter estimation problem (taken from here). Given some observations \((t_i,y_i)\), we wish to calibrate parameters \(x=(x_1,x_2)\) in the exponential decay model

            @@ -562,8 +655,8 @@

            Example: Parameter Estimation/Data Fitting +

            +

            Example: Solving a Nonlinear System of Equations

            Lastly, we give an example of using DFO-LS to solve a nonlinear system of equations (taken from here). We wish to solve the following set of equations

            @@ -615,16 +708,16 @@

            Example: Solving a Nonlinear System of Equations +

            +

            References

            CFMR2018

            Coralia Cartis, Jan Fiala, Benjamin Marteau and Lindon Roberts, Improving the Flexibility and Robustness of Model-Based Derivative-Free Optimization Solvers, ACM Transactions on Mathematical Software, 45:3 (2019), pp. 32:1-32:41 [preprint]

            -
            -
            + +
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(decimal 227) - defining Unicode char U+00E4 (decimal 228) - defining Unicode char U+00E5 (decimal 229) - defining Unicode char U+00E6 (decimal 230) - defining Unicode char U+00E7 (decimal 231) - defining Unicode char U+00E8 (decimal 232) - defining Unicode char U+00E9 (decimal 233) - defining Unicode char U+00EA (decimal 234) - defining Unicode char U+00EB (decimal 235) - defining Unicode char U+00EC (decimal 236) - defining Unicode char U+00ED (decimal 237) - defining Unicode char U+00EE (decimal 238) - defining Unicode char U+00EF (decimal 239) - defining Unicode char U+00F0 (decimal 240) - defining Unicode char U+00F1 (decimal 241) - defining Unicode char U+00F2 (decimal 242) - defining Unicode char U+00F3 (decimal 243) - defining Unicode char U+00F4 (decimal 244) - defining Unicode char U+00F5 (decimal 245) - defining Unicode char U+00F6 (decimal 246) - defining Unicode char U+00F8 (decimal 248) - defining Unicode char U+00F9 (decimal 249) - defining Unicode char U+00FA (decimal 250) - defining Unicode char U+00FB (decimal 251) - defining Unicode char U+00FC (decimal 252) - defining Unicode char U+00FD (decimal 253) - defining Unicode char U+00FE (decimal 254) - defining Unicode char U+00FF (decimal 255) - defining Unicode char U+0100 (decimal 256) - defining Unicode char U+0101 (decimal 257) - defining Unicode char U+0102 (decimal 258) - defining Unicode char U+0103 (decimal 259) - defining Unicode char U+0104 (decimal 260) - defining Unicode char U+0105 (decimal 261) - defining Unicode char U+0106 (decimal 262) - defining Unicode char U+0107 (decimal 263) - defining Unicode char U+0108 (decimal 264) - defining Unicode char U+0109 (decimal 265) - defining Unicode char U+010A (decimal 266) - defining Unicode char U+010B (decimal 267) - defining Unicode char U+010C (decimal 268) - defining Unicode char U+010D (decimal 269) - defining Unicode char U+010E (decimal 270) - defining Unicode char U+010F (decimal 271) - defining Unicode char U+0110 (decimal 272) - defining Unicode char U+0111 (decimal 273) - defining Unicode char U+0112 (decimal 274) - defining Unicode char U+0113 (decimal 275) - defining Unicode char U+0114 (decimal 276) - defining Unicode char U+0115 (decimal 277) - defining Unicode char U+0116 (decimal 278) - defining Unicode char U+0117 (decimal 279) - defining Unicode char U+0118 (decimal 280) - defining Unicode char U+0119 (decimal 281) - defining Unicode char U+011A (decimal 282) - defining Unicode char U+011B (decimal 283) - defining Unicode char U+011C (decimal 284) - defining Unicode char U+011D (decimal 285) - defining Unicode char U+011E (decimal 286) - defining Unicode char U+011F (decimal 287) - defining Unicode char U+0120 (decimal 288) - defining Unicode char U+0121 (decimal 289) - defining Unicode char U+0122 (decimal 290) - defining Unicode char U+0123 (decimal 291) - defining Unicode char U+0124 (decimal 292) - defining Unicode char U+0125 (decimal 293) - defining Unicode char U+0128 (decimal 296) - defining Unicode char U+0129 (decimal 297) - defining Unicode char U+012A (decimal 298) - defining Unicode char U+012B (decimal 299) - defining Unicode char U+012C (decimal 300) - defining Unicode char U+012D (decimal 301) - defining Unicode char U+012E (decimal 302) - defining Unicode char U+012F (decimal 303) - defining Unicode char U+0130 (decimal 304) - defining Unicode char U+0131 (decimal 305) - defining Unicode char U+0132 (decimal 306) - defining Unicode char U+0133 (decimal 307) - defining Unicode char U+0134 (decimal 308) - defining Unicode char U+0135 (decimal 309) - defining Unicode char U+0136 (decimal 310) - defining Unicode char U+0137 (decimal 311) - defining Unicode char U+0139 (decimal 313) - defining Unicode char U+013A (decimal 314) - defining Unicode char U+013B (decimal 315) - defining Unicode char U+013C (decimal 316) - defining Unicode char U+013D (decimal 317) - defining Unicode char U+013E (decimal 318) - defining Unicode char U+0141 (decimal 321) - defining Unicode char U+0142 (decimal 322) - defining Unicode char U+0143 (decimal 323) - defining Unicode char U+0144 (decimal 324) - defining Unicode char U+0145 (decimal 325) - defining Unicode char U+0146 (decimal 326) - defining Unicode char U+0147 (decimal 327) - defining Unicode char U+0148 (decimal 328) - defining Unicode char U+014A (decimal 330) - defining Unicode char U+014B (decimal 331) - defining Unicode char U+014C (decimal 332) - defining Unicode char U+014D (decimal 333) - defining Unicode char U+014E (decimal 334) - defining Unicode char U+014F (decimal 335) - defining Unicode char U+0150 (decimal 336) - defining Unicode char U+0151 (decimal 337) - defining Unicode char U+0152 (decimal 338) - defining Unicode char U+0153 (decimal 339) - defining Unicode char U+0154 (decimal 340) - defining Unicode char U+0155 (decimal 341) - defining Unicode char U+0156 (decimal 342) - defining Unicode char U+0157 (decimal 343) - defining Unicode char U+0158 (decimal 344) - defining Unicode char U+0159 (decimal 345) - defining Unicode char U+015A (decimal 346) - defining Unicode char U+015B (decimal 347) - defining Unicode char U+015C (decimal 348) - defining Unicode char U+015D (decimal 349) - defining Unicode char U+015E (decimal 350) - defining Unicode char U+015F (decimal 351) - defining Unicode char U+0160 (decimal 352) - defining Unicode char U+0161 (decimal 353) - defining Unicode char U+0162 (decimal 354) - defining Unicode char U+0163 (decimal 355) - defining Unicode char U+0164 (decimal 356) - defining Unicode char U+0165 (decimal 357) - defining Unicode char U+0168 (decimal 360) - defining Unicode char U+0169 (decimal 361) - defining Unicode char U+016A (decimal 362) - defining Unicode char U+016B (decimal 363) - defining Unicode char U+016C (decimal 364) - defining Unicode char U+016D (decimal 365) - defining Unicode char U+016E (decimal 366) - defining Unicode char U+016F (decimal 367) - defining Unicode char U+0170 (decimal 368) - defining Unicode char U+0171 (decimal 369) - defining Unicode char U+0172 (decimal 370) - defining Unicode char U+0173 (decimal 371) - defining Unicode char U+0174 (decimal 372) - defining Unicode char U+0175 (decimal 373) - defining Unicode char U+0176 (decimal 374) - defining Unicode char U+0177 (decimal 375) - defining Unicode char U+0178 (decimal 376) - defining Unicode char U+0179 (decimal 377) - defining Unicode char U+017A (decimal 378) - defining Unicode char U+017B (decimal 379) - defining Unicode char U+017C (decimal 380) - defining Unicode char U+017D (decimal 381) - defining Unicode char U+017E (decimal 382) - defining Unicode char U+01CD (decimal 461) - defining Unicode char U+01CE (decimal 462) - defining Unicode char U+01CF (decimal 463) - defining Unicode char U+01D0 (decimal 464) - defining Unicode char U+01D1 (decimal 465) - defining Unicode char U+01D2 (decimal 466) - defining Unicode char U+01D3 (decimal 467) - defining Unicode char U+01D4 (decimal 468) - defining Unicode char U+01E2 (decimal 482) - defining Unicode char U+01E3 (decimal 483) - defining Unicode char U+01E6 (decimal 486) - defining Unicode char U+01E7 (decimal 487) - defining Unicode char U+01E8 (decimal 488) - defining Unicode char U+01E9 (decimal 489) - defining Unicode char U+01EA (decimal 490) - defining Unicode char U+01EB (decimal 491) - defining Unicode char U+01F0 (decimal 496) - defining Unicode char U+01F4 (decimal 500) - defining Unicode char U+01F5 (decimal 501) - defining Unicode char U+0218 (decimal 536) - defining Unicode char U+0219 (decimal 537) - defining Unicode char U+021A (decimal 538) - defining Unicode char U+021B (decimal 539) - defining Unicode char U+0232 (decimal 562) - defining Unicode char U+0233 (decimal 563) - defining Unicode char U+1E02 (decimal 7682) - defining Unicode char U+1E03 (decimal 7683) - defining Unicode char U+200C (decimal 8204) - defining Unicode char U+2010 (decimal 8208) - defining Unicode char U+2011 (decimal 8209) - defining Unicode char U+2012 (decimal 8210) - defining Unicode char U+2013 (decimal 8211) - defining Unicode char U+2014 (decimal 8212) - defining Unicode char U+2015 (decimal 8213) - defining Unicode char U+2018 (decimal 8216) - defining Unicode char U+2019 (decimal 8217) - defining Unicode char U+201A (decimal 8218) - defining Unicode char U+201C (decimal 8220) - defining Unicode char U+201D (decimal 8221) - defining Unicode char U+201E (decimal 8222) - defining Unicode char U+2030 (decimal 8240) - defining Unicode char U+2031 (decimal 8241) - defining Unicode char U+2039 (decimal 8249) - defining Unicode char U+203A (decimal 8250) - defining Unicode char U+2423 (decimal 9251) - defining Unicode char U+1E20 (decimal 7712) - defining Unicode char U+1E21 (decimal 7713) -) -Now handling font encoding OT1 ... -... processing UTF-8 mapping file for font encoding OT1 - -(/usr/share/texlive/texmf-dist/tex/latex/base/ot1enc.dfu -File: ot1enc.dfu 2017/01/28 v1.1t UTF-8 support for inputenc - defining Unicode char U+00A0 (decimal 160) - defining Unicode char U+00A1 (decimal 161) - defining Unicode char U+00A3 (decimal 163) - defining Unicode char U+00AD (decimal 173) - defining Unicode char U+00B8 (decimal 184) - defining Unicode char U+00BF (decimal 191) - defining Unicode char U+00C5 (decimal 197) - defining Unicode char U+00C6 (decimal 198) - defining Unicode char U+00D8 (decimal 216) - defining Unicode char U+00DF (decimal 223) - defining Unicode char U+00E6 (decimal 230) - defining Unicode char U+00EC (decimal 236) - defining Unicode char U+00ED (decimal 237) - defining Unicode char U+00EE (decimal 238) - defining Unicode char U+00EF (decimal 239) - defining Unicode char U+00F8 (decimal 248) - defining Unicode char U+0131 (decimal 305) - defining Unicode char U+0141 (decimal 321) - defining Unicode char U+0142 (decimal 322) - defining Unicode char U+0152 (decimal 338) - defining Unicode char U+0153 (decimal 339) - defining Unicode char U+0174 (decimal 372) - defining Unicode char U+0175 (decimal 373) - defining Unicode char U+0176 (decimal 374) - defining Unicode char U+0177 (decimal 375) - defining Unicode char U+0218 (decimal 536) - defining Unicode char U+0219 (decimal 537) - defining Unicode char U+021A (decimal 538) - defining Unicode char U+021B (decimal 539) - defining Unicode char U+2013 (decimal 8211) - defining Unicode char U+2014 (decimal 8212) - defining Unicode char U+2018 (decimal 8216) - defining Unicode char U+2019 (decimal 8217) - defining Unicode char U+201C (decimal 8220) - defining Unicode char U+201D (decimal 8221) -) -Now handling font encoding OMS ... -... processing UTF-8 mapping file for font encoding OMS - -(/usr/share/texlive/texmf-dist/tex/latex/base/omsenc.dfu -File: omsenc.dfu 2017/01/28 v1.1t UTF-8 support for inputenc - defining Unicode char U+00A7 (decimal 167) - defining Unicode char U+00B6 (decimal 182) - defining Unicode char U+00B7 (decimal 183) - defining Unicode char U+2020 (decimal 8224) - defining Unicode char U+2021 (decimal 8225) - defining Unicode char U+2022 (decimal 8226) -) -Now handling font encoding OMX ... -... no UTF-8 mapping file for font encoding OMX -Now handling font encoding U ... -... no UTF-8 mapping file for font encoding U - defining Unicode char U+00A9 (decimal 169) - defining Unicode char U+00AA (decimal 170) - defining Unicode char U+00AE (decimal 174) - defining Unicode char U+00BA (decimal 186) - defining Unicode char U+02C6 (decimal 710) - defining Unicode char U+02DC (decimal 732) - defining Unicode char U+200C (decimal 8204) - defining Unicode char U+2026 (decimal 8230) - defining Unicode char U+2122 (decimal 8482) - defining Unicode char U+2423 (decimal 9251) -)) +Package: inputenc 2018/08/11 v1.3c Input encoding file +\inpenc@prehook=\toks15 +\inpenc@posthook=\toks16 +) defining Unicode char U+00A0 (decimal 160) defining Unicode char U+2500 (decimal 9472) defining Unicode char U+2502 (decimal 9474) @@ -357,15 +43,11 @@ Now handling font encoding U ... 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(pdftex.def) Requested size: 352.31625pt x 270.79639pt. - [16] -Overfull \vbox (2.92438pt too high) detected at line 876 - [] - -[17 <./data_fitting.png>] [18] + [18 <./data_fitting.png>] [19] [20] Chapter 4. -[19 +[21 -] [20] -Underfull \hbox (badness 10000) in paragraph at lines 1044--1045 +] [22] +Underfull \hbox (badness 10000) in paragraph at lines 1147--1148 []\T1/pcr/m/n/10 regression.increase_num_extra_steps_with_restart \T1/ptm/m/n/1 0 - The amount to in-crease [] -Underfull \hbox (badness 10000) in paragraph at lines 1079--1080 +Underfull \hbox (badness 10000) in paragraph at lines 1182--1183 []\T1/pcr/m/n/10 restarts.hard.increase_ndirs_initial_amt \T1/ptm/m/n/10 - Amou nt to in-crease \T1/pcr/m/n/10 growing. [] -[21] -Underfull \hbox (badness 8000) in paragraph at lines 1114--1115 +[23] +Underfull \hbox (badness 8000) in paragraph at lines 1217--1218 \T1/ptm/m/n/10 De-fault is \T1/pcr/m/n/10 False \T1/ptm/m/n/10 if $\OML/cmm/m/i t/10 m \OMS/cmsy/m/n/10 ^^U \OML/cmm/m/it/10 n$ \T1/ptm/m/n/10 and \T1/pcr/m/n/ 10 True \T1/ptm/m/n/10 oth-er-wise (op-po-site to \T1/pcr/m/n/10 growing.full_r ank. [] -[22] [23] [24 +[24] [25] [26 ] Chapter 5. -[25] [26] +[27] [28] Chapter 6. -[27 +[29 -] [28] +] [30] Chapter 7. -[29 +[31 -] [30 +] [32 ] (./DFOLS.ind) -Package atveryend Info: Empty hook `BeforeClearDocument' on input line 1397. - [31 +Package atveryend Info: Empty hook `BeforeClearDocument' on input line 1530. + [33 ] -Package atveryend Info: Empty hook `AfterLastShipout' on input line 1397. +Package atveryend Info: Empty hook `AfterLastShipout' on input line 1530. 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b/docs/build/latex/DFOLS.pdf differ diff --git a/docs/build/latex/DFOLS.tex b/docs/build/latex/DFOLS.tex index 2669ee3..e692098 100755 --- a/docs/build/latex/DFOLS.tex +++ b/docs/build/latex/DFOLS.tex @@ -73,8 +73,8 @@ \title{DFO-LS Documentation} -\date{01 June 2021} -\release{1.2.3} +\date{07 November 2021} +\release{1.3.0} \author{Lindon Roberts} \newcommand{\sphinxlogo}{\vbox{}} \renewcommand{\releasename}{Release} @@ -89,20 +89,21 @@ \phantomsection\label{\detokenize{index::doc}} -\sphinxstylestrong{Release:} 1.2.3 +\sphinxstylestrong{Release:} 1.3.0 -\sphinxstylestrong{Date:} 01 June 2021 +\sphinxstylestrong{Date:} 07 November 2021 \sphinxstylestrong{Author:} \sphinxhref{mailto:lindon.roberts@anu.edu.au}{Lindon Roberts} -DFO\sphinxhyphen{}LS is a flexible package for finding local solutions to nonlinear least\sphinxhyphen{}squares minimization problems (with optional bound constraints), without requiring any derivatives of the objective. DFO\sphinxhyphen{}LS stands for Derivative\sphinxhyphen{}Free Optimizer for Least\sphinxhyphen{}Squares. +DFO\sphinxhyphen{}LS is a flexible package for finding local solutions to nonlinear least\sphinxhyphen{}squares minimization problems (with optional constraints), without requiring any derivatives of the objective. DFO\sphinxhyphen{}LS stands for Derivative\sphinxhyphen{}Free Optimizer for Least\sphinxhyphen{}Squares. That is, DFO\sphinxhyphen{}LS solves \begin{equation*} \begin{split}\min_{x\in\mathbb{R}^n} &\quad f(x) := \sum_{i=1}^{m}r_{i}(x)^2 \\ -\text{s.t.} &\quad a \leq x \leq b\end{split} +\text{s.t.} &\quad x \in C\\ + &\quad a \leq x \leq b\end{split} \end{equation*} -The upper and lower bounds on the variables are non\sphinxhyphen{}relaxable (i.e. DFO\sphinxhyphen{}LS will never ask to evaluate a point outside the bounds). +The constraint set \(C\) is the intersection of multiple convex sets provided as input by the user. All constraints are non\sphinxhyphen{}relaxable (i.e. DFO\sphinxhyphen{}LS will never ask to evaluate a point that is not feasible). Full details of the DFO\sphinxhyphen{}LS algorithm are given in our paper: C. Cartis, J. Fiala, B. Marteau and L. Roberts, \sphinxhref{https://doi.org/10.1145/3338517}{Improving the Flexibility and Robustness of Model\sphinxhyphen{}Based Derivative\sphinxhyphen{}Free Optimization Solvers}, \sphinxstyleemphasis{ACM Transactions on Mathematical Software}, 45:3 (2019), pp. 32:1\sphinxhyphen{}32:41 {[}\sphinxhref{https://arxiv.org/abs/1804.00154}{preprint}{]} . DFO\sphinxhyphen{}LS is a more flexible version of \sphinxhref{https://github.com/numericalalgorithmsgroup/dfogn}{DFO\sphinxhyphen{}GN}. @@ -248,12 +249,14 @@ \chapter{Overview} \section{When to use DFO\sphinxhyphen{}LS} \label{\detokenize{info:when-to-use-dfo-ls}} -DFO\sphinxhyphen{}LS is designed to solve the nonlinear least\sphinxhyphen{}squares minimization problem (with optional bound constraints) +DFO\sphinxhyphen{}LS is designed to solve the nonlinear least\sphinxhyphen{}squares minimization problem (with optional convex constraints). \begin{equation*} \begin{split}\min_{x\in\mathbb{R}^n} &\quad f(x) := \sum_{i=1}^{m}r_{i}(x)^2 \\ -\text{s.t.} &\quad a \leq x \leq b\end{split} +\text{s.t.} &\quad x \in C\\ + &\quad a \leq x \leq b\end{split} \end{equation*} We call \(f(x)\) the objective function and \(r_i(x)\) the residual functions (or simply residuals). +\(C\) is the intersection of multiple convex sets given as input by the user. DFO\sphinxhyphen{}LS is a \sphinxstyleemphasis{derivative\sphinxhyphen{}free} optimization algorithm, which means it does not require the user to provide the derivatives of \(f(x)\) or \(r_i(x)\), nor does it attempt to estimate them internally (by using finite differencing, for instance). @@ -336,9 +339,9 @@ \section{Nonlinear Least\sphinxhyphen{}Squares Minimization} DFO\sphinxhyphen{}LS is designed to solve the local optimization problem \begin{equation*} \begin{split}\min_{x\in\mathbb{R}^n} &\quad f(x) := \sum_{i=1}^{m}r_{i}(x)^2 \\ -\text{s.t.} &\quad a \leq x \leq b\end{split} +\text{s.t.} &\quad x \in C\end{split} \end{equation*} -where the bound constraints \(a \leq x \leq b\) are optional. The upper and lower bounds on the variables are non\sphinxhyphen{}relaxable (i.e. DFO\sphinxhyphen{}LS will never ask to evaluate a point outside the bounds). +where the set \(C\) is an optional non\sphinxhyphen{}empty, closed and convex constraint set. The constraints are non\sphinxhyphen{}relaxable (i.e. DFO\sphinxhyphen{}LS will never ask to evaluate a point that is not feasible). DFO\sphinxhyphen{}LS iteratively constructs an interpolation\sphinxhyphen{}based model for the objective, and determines a step using a trust\sphinxhyphen{}region framework. For an in\sphinxhyphen{}depth technical description of the algorithm see the paper \sphinxcite{userguide:cfmr2018}. @@ -407,6 +410,9 @@ \section{How to use DFO\sphinxhyphen{}LS} \item {} \sphinxcode{\sphinxupquote{soln.EXIT\_FALSE\_SUCCESS\_WARNING}} \sphinxhyphen{} DFO\sphinxhyphen{}LS reached the maximum number of restarts which decreased the objective, but to a worse value than was found in a previous run. +\item {} +\sphinxcode{\sphinxupquote{soln.EXIT\_TR\_INCREASE\_WARNING}} \sphinxhyphen{} model increase when solving the trust region subproblem with multiple arbitrary constraints. + \item {} \sphinxcode{\sphinxupquote{soln.EXIT\_INPUT\_ERROR}} \sphinxhyphen{} error in the inputs. @@ -450,6 +456,9 @@ \section{Optional Arguments} \item {} \sphinxcode{\sphinxupquote{bounds}} \sphinxhyphen{} a tuple \sphinxcode{\sphinxupquote{(lower, upper)}} with the vectors \(a\) and \(b\) of lower and upper bounds on \(x\) (default is \(a_i=-10^{20}\) and \(b_i=10^{20}\)). To set bounds for either \sphinxcode{\sphinxupquote{lower}} or \sphinxcode{\sphinxupquote{upper}}, but not both, pass a tuple \sphinxcode{\sphinxupquote{(lower, None)}} or \sphinxcode{\sphinxupquote{(None, upper)}}. +\item {} +\sphinxcode{\sphinxupquote{projections}} \sphinxhyphen{} a list \sphinxcode{\sphinxupquote{{[}f1,f2,...,fn{]}}} of functions that each take as input a point \sphinxcode{\sphinxupquote{x}} and return a new point \sphinxcode{\sphinxupquote{y}}. The new point \sphinxcode{\sphinxupquote{y}} should be given by the projection of \sphinxcode{\sphinxupquote{x}} onto a closed convex set. The intersection of all sets corresponding to a function must be non\sphinxhyphen{}empty. + \item {} \sphinxcode{\sphinxupquote{npt}} \sphinxhyphen{} the number of interpolation points to use (default is \sphinxcode{\sphinxupquote{len(x0)+1}}). If using restarts, this is the number of points to use in the first run of the solver, before any restarts (and may be optionally increased via settings in \sphinxcode{\sphinxupquote{user\_params}}). @@ -542,7 +551,7 @@ \section{A Simple Example} \section{Adding Bounds and More Output} \label{\detokenize{userguide:adding-bounds-and-more-output}} -We can extend the above script to add constraints. To do this, we can add the lines +We can extend the above script to add constraints. To add bound constraints alone, we can add the lines \begin{quote} \begin{sphinxVerbatim}[commandchars=\\\{\}] @@ -593,7 +602,7 @@ \section{Adding Bounds and More Output} \end{sphinxVerbatim} \end{quote} -And we can now see each evaluation of \sphinxcode{\sphinxupquote{objfun}}: +And for the simple bounds example we can now see each evaluation of \sphinxcode{\sphinxupquote{objfun}}: \begin{quote} \begin{sphinxVerbatim}[commandchars=\\\{\}] @@ -634,6 +643,100 @@ \section{Adding Bounds and More Output} \end{quote} +\section{Handling Arbitrary Convex Constraints} +\label{\detokenize{userguide:handling-arbitrary-convex-constraints}} +DFO\sphinxhyphen{}LS can also handle more general constraints where they can be written as the intersection of finitely many convex sets. For example, the below code +minimizes the Rosenbrock function subject to a constraint set given by the intersection of two convex sets. Note the intersection of the user\sphinxhyphen{}provided convex +sets must be non\sphinxhyphen{}empty. +\begin{quote} + +\begin{sphinxVerbatim}[commandchars=\\\{\}] +\PYG{l+s+sd}{\PYGZsq{}\PYGZsq{}\PYGZsq{}} +\PYG{l+s+sd}{DFO\PYGZhy{}LS example: minimize the Rosenbrock function with arbitrary convex constraints} + +\PYG{l+s+sd}{This example defines two functions pball(x) and pbox(x) that project onto ball and} +\PYG{l+s+sd}{box constraint sets respectively. It then passes both these functions to the DFO\PYGZhy{}LS} +\PYG{l+s+sd}{solver so that it can find a constrained minimizer to the Rosenbrock function.} +\PYG{l+s+sd}{Such a minimizer must lie in the intersection of constraint sets corresponding to} +\PYG{l+s+sd}{projection functions pball(x) and pbox(x). The description of the problem is as follows:} + +\PYG{l+s+sd}{ min rosenbrock(x)} +\PYG{l+s+sd}{ s.t.} +\PYG{l+s+sd}{ \PYGZhy{}2 \PYGZlt{}= x[0] \PYGZlt{}= 1.1,} +\PYG{l+s+sd}{ 1.1 \PYGZlt{}= x[1] \PYGZlt{}= 3,} +\PYG{l+s+sd}{ norm(x\PYGZhy{}c) \PYGZlt{}= 0.4} + +\PYG{l+s+sd}{where c = [0.7, 1.5] is the centre of the ball.} +\PYG{l+s+sd}{\PYGZsq{}\PYGZsq{}\PYGZsq{}} +\PYG{k+kn}{from} \PYG{n+nn}{\PYGZus{}\PYGZus{}future\PYGZus{}\PYGZus{}} \PYG{k+kn}{import} \PYG{n}{print\PYGZus{}function} +\PYG{k+kn}{import} \PYG{n+nn}{numpy} \PYG{k}{as} \PYG{n+nn}{np} +\PYG{k+kn}{import} \PYG{n+nn}{dfols} + +\PYG{c+c1}{\PYGZsh{} Define the objective function} +\PYG{k}{def} \PYG{n+nf}{rosenbrock}\PYG{p}{(}\PYG{n}{x}\PYG{p}{)}\PYG{p}{:} + \PYG{k}{return} \PYG{n}{np}\PYG{o}{.}\PYG{n}{array}\PYG{p}{(}\PYG{p}{[}\PYG{l+m+mf}{10.0} \PYG{o}{*} \PYG{p}{(}\PYG{n}{x}\PYG{p}{[}\PYG{l+m+mi}{1}\PYG{p}{]} \PYG{o}{\PYGZhy{}} \PYG{n}{x}\PYG{p}{[}\PYG{l+m+mi}{0}\PYG{p}{]} \PYG{o}{*}\PYG{o}{*} \PYG{l+m+mi}{2}\PYG{p}{)}\PYG{p}{,} \PYG{l+m+mf}{1.0} \PYG{o}{\PYGZhy{}} \PYG{n}{x}\PYG{p}{[}\PYG{l+m+mi}{0}\PYG{p}{]}\PYG{p}{]}\PYG{p}{)} + +\PYG{c+c1}{\PYGZsh{} Define the starting point} +\PYG{n}{x0} \PYG{o}{=} \PYG{n}{np}\PYG{o}{.}\PYG{n}{array}\PYG{p}{(}\PYG{p}{[}\PYG{o}{\PYGZhy{}}\PYG{l+m+mf}{1.2}\PYG{p}{,} \PYG{l+m+mi}{1}\PYG{p}{]}\PYG{p}{)} + +\PYG{l+s+sd}{\PYGZsq{}\PYGZsq{}\PYGZsq{}} +\PYG{l+s+sd}{Define ball projection function} +\PYG{l+s+sd}{Projects the input x onto a ball with} +\PYG{l+s+sd}{centre point (0.7,1.5) and radius 0.4.} +\PYG{l+s+sd}{\PYGZsq{}\PYGZsq{}\PYGZsq{}} +\PYG{k}{def} \PYG{n+nf}{pball}\PYG{p}{(}\PYG{n}{x}\PYG{p}{)}\PYG{p}{:} + \PYG{n}{c} \PYG{o}{=} \PYG{n}{np}\PYG{o}{.}\PYG{n}{array}\PYG{p}{(}\PYG{p}{[}\PYG{l+m+mf}{0.7}\PYG{p}{,}\PYG{l+m+mf}{1.5}\PYG{p}{]}\PYG{p}{)} \PYG{c+c1}{\PYGZsh{} ball centre} + \PYG{n}{r} \PYG{o}{=} \PYG{l+m+mf}{0.4} \PYG{c+c1}{\PYGZsh{} ball radius} + \PYG{k}{return} \PYG{n}{c} \PYG{o}{+} \PYG{p}{(}\PYG{n}{r}\PYG{o}{/}\PYG{n}{np}\PYG{o}{.}\PYG{n}{max}\PYG{p}{(}\PYG{p}{[}\PYG{n}{np}\PYG{o}{.}\PYG{n}{linalg}\PYG{o}{.}\PYG{n}{norm}\PYG{p}{(}\PYG{n}{x}\PYG{o}{\PYGZhy{}}\PYG{n}{c}\PYG{p}{)}\PYG{p}{,}\PYG{n}{r}\PYG{p}{]}\PYG{p}{)}\PYG{p}{)}\PYG{o}{*}\PYG{p}{(}\PYG{n}{x}\PYG{o}{\PYGZhy{}}\PYG{n}{c}\PYG{p}{)} + +\PYG{l+s+sd}{\PYGZsq{}\PYGZsq{}\PYGZsq{}} +\PYG{l+s+sd}{Define box projection function} +\PYG{l+s+sd}{Projects the input x onto a box} +\PYG{l+s+sd}{such that \PYGZhy{}2 \PYGZlt{}= x[0] \PYGZlt{}= 0.9 and} +\PYG{l+s+sd}{1.1 \PYGZlt{}= x[1] \PYGZlt{}= 3.} + +\PYG{l+s+sd}{Note: One could equivalently add bound} +\PYG{l+s+sd}{constraints as a separate input to the solver} +\PYG{l+s+sd}{instead.} +\PYG{l+s+sd}{\PYGZsq{}\PYGZsq{}\PYGZsq{}} +\PYG{k}{def} \PYG{n+nf}{pbox}\PYG{p}{(}\PYG{n}{x}\PYG{p}{)}\PYG{p}{:} + \PYG{n}{l} \PYG{o}{=} \PYG{n}{np}\PYG{o}{.}\PYG{n}{array}\PYG{p}{(}\PYG{p}{[}\PYG{o}{\PYGZhy{}}\PYG{l+m+mi}{2}\PYG{p}{,} \PYG{l+m+mf}{1.1}\PYG{p}{]}\PYG{p}{)} \PYG{c+c1}{\PYGZsh{} lower bound} + \PYG{n}{u} \PYG{o}{=} \PYG{n}{np}\PYG{o}{.}\PYG{n}{array}\PYG{p}{(}\PYG{p}{[}\PYG{l+m+mf}{0.9}\PYG{p}{,} \PYG{l+m+mi}{3}\PYG{p}{]}\PYG{p}{)} \PYG{c+c1}{\PYGZsh{} upper bound} + \PYG{k}{return} \PYG{n}{np}\PYG{o}{.}\PYG{n}{minimum}\PYG{p}{(}\PYG{n}{np}\PYG{o}{.}\PYG{n}{maximum}\PYG{p}{(}\PYG{n}{x}\PYG{p}{,}\PYG{n}{l}\PYG{p}{)}\PYG{p}{,} \PYG{n}{u}\PYG{p}{)} + +\PYG{c+c1}{\PYGZsh{} For optional extra output details} +\PYG{k+kn}{import} \PYG{n+nn}{logging} +\PYG{n}{logging}\PYG{o}{.}\PYG{n}{basicConfig}\PYG{p}{(}\PYG{n}{level}\PYG{o}{=}\PYG{n}{logging}\PYG{o}{.}\PYG{n}{DEBUG}\PYG{p}{,} \PYG{n+nb}{format}\PYG{o}{=}\PYG{l+s+s1}{\PYGZsq{}}\PYG{l+s+si}{\PYGZpc{}(message)s}\PYG{l+s+s1}{\PYGZsq{}}\PYG{p}{)} + +\PYG{c+c1}{\PYGZsh{} Call DFO\PYGZhy{}LS} +\PYG{n}{soln} \PYG{o}{=} \PYG{n}{dfols}\PYG{o}{.}\PYG{n}{solve}\PYG{p}{(}\PYG{n}{rosenbrock}\PYG{p}{,} \PYG{n}{x0}\PYG{p}{,} \PYG{n}{projections}\PYG{o}{=}\PYG{p}{[}\PYG{n}{pball}\PYG{p}{,}\PYG{n}{pbox}\PYG{p}{]}\PYG{p}{)} + +\PYG{c+c1}{\PYGZsh{} Display output} +\PYG{n+nb}{print}\PYG{p}{(}\PYG{n}{soln}\PYG{p}{)} +\end{sphinxVerbatim} +\end{quote} + +Note that for bound constraints one can choose to either implement them by defining a projection function as above, or by passing the bounds as input like in the example from the section on adding bound constraints. + +DFO\sphinxhyphen{}LS correctly finds the solution to this constrained problem too. Note that we get a warning because the step computed in the trust region subproblem +gave an increase in the model. This is common in the case where multiple constraints are active at the optimal point. +\begin{quote} + +\begin{sphinxVerbatim}[commandchars=\\\{\}] +****** DFO\PYGZhy{}LS Results ****** +Solution xmin = [0.9 1.15359245] +Residual vector = [3.43592448 0.1 ] +Objective value f(xmin) = 11.81557703 +Needed 10 objective evaluations (at 10 points) +Approximate Jacobian = [[\PYGZhy{}1.79826221e+01 1.00004412e+01] + [\PYGZhy{}1.00000000e+00 \PYGZhy{}1.81976605e\PYGZhy{}15]] +Exit flag = 5 +Warning (trust region increase): Either multiple constraints are active or trust region step gave model increase +**************************** +\end{sphinxVerbatim} +\end{quote} + + \section{Example: Noisy Objective Evaluation} \label{\detokenize{userguide:example-noisy-objective-evaluation}} As described in {\hyperref[\detokenize{info::doc}]{\sphinxcrossref{\DUrole{doc}{Overview}}}}, derivative\sphinxhyphen{}free algorithms such as DFO\sphinxhyphen{}LS are particularly useful when \sphinxcode{\sphinxupquote{objfun}} has noise. Let’s modify the previous example to include random noise in our objective evaluation, and compare it to a derivative\sphinxhyphen{}based solver: @@ -1155,6 +1258,25 @@ \section{Dynamically Growing Initial Set} \end{itemize} +\section{Dykstra’s Algorithm} +\label{\detokenize{advanced:dykstra-s-algorithm}}\begin{itemize} +\item {} +\sphinxcode{\sphinxupquote{dykstra.d\_tol}} \sphinxhyphen{} Tolerance on the stopping conditions of Dykstra’s algorithm. Default is \(10^{-10}\). + +\item {} +\sphinxcode{\sphinxupquote{dykstra.max\_iters}} \sphinxhyphen{} The maximum number of iterations Dykstra’s algorithm is allowed to take before stopping. Default is \(100\). + +\end{itemize} + + +\section{Checking Matrix Rank} +\label{\detokenize{advanced:checking-matrix-rank}}\begin{itemize} +\item {} +\sphinxcode{\sphinxupquote{matrix\_rank.r\_tol}} \sphinxhyphen{} Tolerance on what is the smallest posisble diagonal entry value in the QR factorization before being considered zero. Default is \(10^{-18}\). + +\end{itemize} + + \section{References} \label{\detokenize{advanced:references}} @@ -1377,6 +1499,17 @@ \section{Version 1.2.3 (1 Jun 2021)} \end{itemize} +\section{Version 1.3.0 (16 Oct 2021)} +\label{\detokenize{history:version-1-3-0-16-oct-2021}}\begin{itemize} +\item {} +Handle finitely many arbitrary convex constraints in addition to simple bound constraints. + +\item {} +Only new functionality is added, so there is no change to the solver for unconstrained/bound\sphinxhyphen{}constrained problems. + +\end{itemize} + + \chapter{Acknowledgements} \label{\detokenize{index:acknowledgements}} This software was developed under the supervision of \sphinxhref{https://www.maths.ox.ac.uk/people/coralia.cartis}{Coralia Cartis}, and was supported by the EPSRC Centre For Doctoral Training in \sphinxhref{https://www.maths.ox.ac.uk/study-here/postgraduate-study/industrially-focused-mathematical-modelling-epsrc-cdt}{Industrially Focused Mathematical Modelling} (EP/L015803/1) in collaboration with the \sphinxhref{http://www.nag.com/}{Numerical Algorithms Group}. diff --git a/docs/build/latex/DFOLS.toc b/docs/build/latex/DFOLS.toc index 8523942..7108b3f 100755 --- a/docs/build/latex/DFOLS.toc +++ b/docs/build/latex/DFOLS.toc @@ -1,54 +1,58 @@ \babel@toc {english}{} -\contentsline {chapter}{\numberline {1}Installing DFO\sphinxhyphen {}LS}{3}{chapter.1} -\contentsline {section}{\numberline {1.1}Requirements}{3}{section.1.1} -\contentsline {section}{\numberline {1.2}Installation using pip}{3}{section.1.2} -\contentsline {section}{\numberline {1.3}Manual installation}{4}{section.1.3} -\contentsline {section}{\numberline {1.4}Testing}{4}{section.1.4} -\contentsline {section}{\numberline {1.5}Uninstallation}{4}{section.1.5} -\contentsline {chapter}{\numberline {2}Overview}{5}{chapter.2} -\contentsline {section}{\numberline {2.1}When to use DFO\sphinxhyphen {}LS}{5}{section.2.1} -\contentsline {section}{\numberline {2.2}Parameter Fitting}{5}{section.2.2} -\contentsline {section}{\numberline {2.3}Solving Nonlinear Systems of Equations}{6}{section.2.3} -\contentsline {section}{\numberline {2.4}Details of the DFO\sphinxhyphen {}LS Algorithm}{6}{section.2.4} -\contentsline {section}{\numberline {2.5}References}{7}{section.2.5} -\contentsline {chapter}{\numberline {3}Using DFO\sphinxhyphen {}LS}{9}{chapter.3} -\contentsline {section}{\numberline {3.1}Nonlinear Least\sphinxhyphen {}Squares Minimization}{9}{section.3.1} -\contentsline {section}{\numberline {3.2}How to use DFO\sphinxhyphen {}LS}{9}{section.3.2} -\contentsline {section}{\numberline {3.3}Optional Arguments}{10}{section.3.3} -\contentsline {section}{\numberline {3.4}A Simple Example}{11}{section.3.4} -\contentsline {section}{\numberline {3.5}Adding Bounds and More Output}{12}{section.3.5} -\contentsline {section}{\numberline {3.6}Example: Noisy Objective Evaluation}{13}{section.3.6} -\contentsline {section}{\numberline {3.7}Example: Parameter Estimation/Data Fitting}{15}{section.3.7} -\contentsline {section}{\numberline {3.8}Example: Solving a Nonlinear System of Equations}{17}{section.3.8} -\contentsline {section}{\numberline {3.9}References}{18}{section.3.9} -\contentsline {chapter}{\numberline {4}Advanced Usage}{19}{chapter.4} -\contentsline {section}{\numberline {4.1}General Algorithm Parameters}{19}{section.4.1} -\contentsline {section}{\numberline {4.2}Logging and Output}{19}{section.4.2} -\contentsline {section}{\numberline {4.3}Initialization of Points}{20}{section.4.3} -\contentsline {section}{\numberline {4.4}Trust Region Management}{20}{section.4.4} -\contentsline {section}{\numberline {4.5}Termination on Small Objective Value}{20}{section.4.5} -\contentsline {section}{\numberline {4.6}Termination on Slow Progress}{20}{section.4.6} -\contentsline {section}{\numberline {4.7}Stochastic Noise Information}{21}{section.4.7} -\contentsline {section}{\numberline {4.8}Interpolation Management}{21}{section.4.8} -\contentsline {section}{\numberline {4.9}Regression Model Management}{21}{section.4.9} -\contentsline {section}{\numberline {4.10}Multiple Restarts}{21}{section.4.10} -\contentsline {section}{\numberline {4.11}Dynamically Growing Initial Set}{22}{section.4.11} -\contentsline {section}{\numberline {4.12}References}{23}{section.4.12} -\contentsline {chapter}{\numberline {5}Diagnostic Information}{25}{chapter.5} -\contentsline {section}{\numberline {5.1}Current Iterate}{25}{section.5.1} -\contentsline {section}{\numberline {5.2}Trust Region}{25}{section.5.2} -\contentsline {section}{\numberline {5.3}Model Interpolation}{26}{section.5.3} -\contentsline {section}{\numberline {5.4}Iteration Count}{26}{section.5.4} -\contentsline {section}{\numberline {5.5}Algorithm Progress}{26}{section.5.5} -\contentsline {chapter}{\numberline {6}Version History}{27}{chapter.6} -\contentsline {section}{\numberline {6.1}Version 1.0 (6 Feb 2018)}{27}{section.6.1} -\contentsline {section}{\numberline {6.2}Version 1.0.1 (20 Feb 2018)}{27}{section.6.2} -\contentsline {section}{\numberline {6.3}Version 1.0.2 (20 Jun 2018)}{27}{section.6.3} -\contentsline {section}{\numberline {6.4}Version 1.1 (16 Jan 2019)}{27}{section.6.4} -\contentsline {section}{\numberline {6.5}Version 1.1.1 (5 Apr 2019)}{28}{section.6.5} -\contentsline {section}{\numberline {6.6}Version 1.2 (12 Feb 2020)}{28}{section.6.6} -\contentsline {section}{\numberline {6.7}Version 1.2.1 (13 Feb 2020)}{28}{section.6.7} -\contentsline {section}{\numberline {6.8}Version 1.2.2 (26 Feb 2021)}{28}{section.6.8} -\contentsline {section}{\numberline {6.9}Version 1.2.3 (1 Jun 2021)}{28}{section.6.9} -\contentsline {chapter}{\numberline {7}Acknowledgements}{29}{chapter.7} -\contentsline {chapter}{Bibliography}{31}{chapter*.3} +\contentsline {chapter}{\numberline {1}Installing DFO\sphinxhyphen {}LS}{3}{chapter.1}% +\contentsline {section}{\numberline {1.1}Requirements}{3}{section.1.1}% +\contentsline {section}{\numberline {1.2}Installation using pip}{3}{section.1.2}% +\contentsline {section}{\numberline {1.3}Manual installation}{4}{section.1.3}% +\contentsline {section}{\numberline {1.4}Testing}{4}{section.1.4}% +\contentsline {section}{\numberline {1.5}Uninstallation}{4}{section.1.5}% +\contentsline {chapter}{\numberline {2}Overview}{5}{chapter.2}% +\contentsline {section}{\numberline {2.1}When to use DFO\sphinxhyphen {}LS}{5}{section.2.1}% +\contentsline {section}{\numberline {2.2}Parameter Fitting}{5}{section.2.2}% +\contentsline {section}{\numberline {2.3}Solving Nonlinear Systems of Equations}{6}{section.2.3}% +\contentsline {section}{\numberline {2.4}Details of the DFO\sphinxhyphen {}LS Algorithm}{7}{section.2.4}% +\contentsline {section}{\numberline {2.5}References}{7}{section.2.5}% +\contentsline {chapter}{\numberline {3}Using DFO\sphinxhyphen {}LS}{9}{chapter.3}% +\contentsline {section}{\numberline {3.1}Nonlinear Least\sphinxhyphen {}Squares Minimization}{9}{section.3.1}% +\contentsline {section}{\numberline {3.2}How to use DFO\sphinxhyphen {}LS}{9}{section.3.2}% +\contentsline {section}{\numberline {3.3}Optional Arguments}{10}{section.3.3}% +\contentsline {section}{\numberline {3.4}A Simple Example}{11}{section.3.4}% +\contentsline {section}{\numberline {3.5}Adding Bounds and More Output}{12}{section.3.5}% +\contentsline {section}{\numberline {3.6}Handling Arbitrary Convex Constraints}{14}{section.3.6}% +\contentsline {section}{\numberline {3.7}Example: Noisy Objective Evaluation}{15}{section.3.7}% +\contentsline {section}{\numberline {3.8}Example: Parameter Estimation/Data Fitting}{17}{section.3.8}% +\contentsline {section}{\numberline {3.9}Example: Solving a Nonlinear System of Equations}{19}{section.3.9}% +\contentsline {section}{\numberline {3.10}References}{20}{section.3.10}% +\contentsline {chapter}{\numberline {4}Advanced Usage}{21}{chapter.4}% +\contentsline {section}{\numberline {4.1}General Algorithm Parameters}{21}{section.4.1}% +\contentsline {section}{\numberline {4.2}Logging and Output}{21}{section.4.2}% +\contentsline {section}{\numberline {4.3}Initialization of Points}{22}{section.4.3}% +\contentsline {section}{\numberline {4.4}Trust Region Management}{22}{section.4.4}% +\contentsline {section}{\numberline {4.5}Termination on Small Objective Value}{22}{section.4.5}% +\contentsline {section}{\numberline {4.6}Termination on Slow Progress}{22}{section.4.6}% +\contentsline {section}{\numberline {4.7}Stochastic Noise Information}{23}{section.4.7}% +\contentsline {section}{\numberline {4.8}Interpolation Management}{23}{section.4.8}% +\contentsline {section}{\numberline {4.9}Regression Model Management}{23}{section.4.9}% +\contentsline {section}{\numberline {4.10}Multiple Restarts}{23}{section.4.10}% +\contentsline {section}{\numberline {4.11}Dynamically Growing Initial Set}{24}{section.4.11}% +\contentsline {section}{\numberline {4.12}Dykstra’s Algorithm}{25}{section.4.12}% +\contentsline {section}{\numberline {4.13}Checking Matrix Rank}{25}{section.4.13}% +\contentsline {section}{\numberline {4.14}References}{25}{section.4.14}% +\contentsline {chapter}{\numberline {5}Diagnostic Information}{27}{chapter.5}% +\contentsline {section}{\numberline {5.1}Current Iterate}{27}{section.5.1}% +\contentsline {section}{\numberline {5.2}Trust Region}{27}{section.5.2}% +\contentsline {section}{\numberline {5.3}Model Interpolation}{28}{section.5.3}% +\contentsline {section}{\numberline {5.4}Iteration Count}{28}{section.5.4}% +\contentsline {section}{\numberline {5.5}Algorithm Progress}{28}{section.5.5}% +\contentsline {chapter}{\numberline {6}Version History}{29}{chapter.6}% +\contentsline {section}{\numberline {6.1}Version 1.0 (6 Feb 2018)}{29}{section.6.1}% +\contentsline {section}{\numberline {6.2}Version 1.0.1 (20 Feb 2018)}{29}{section.6.2}% +\contentsline {section}{\numberline {6.3}Version 1.0.2 (20 Jun 2018)}{29}{section.6.3}% +\contentsline {section}{\numberline {6.4}Version 1.1 (16 Jan 2019)}{29}{section.6.4}% +\contentsline {section}{\numberline {6.5}Version 1.1.1 (5 Apr 2019)}{30}{section.6.5}% +\contentsline {section}{\numberline {6.6}Version 1.2 (12 Feb 2020)}{30}{section.6.6}% +\contentsline {section}{\numberline {6.7}Version 1.2.1 (13 Feb 2020)}{30}{section.6.7}% +\contentsline {section}{\numberline {6.8}Version 1.2.2 (26 Feb 2021)}{30}{section.6.8}% +\contentsline {section}{\numberline {6.9}Version 1.2.3 (1 Jun 2021)}{30}{section.6.9}% +\contentsline {section}{\numberline {6.10}Version 1.3.0 (16 Oct 2021)}{30}{section.6.10}% +\contentsline {chapter}{\numberline {7}Acknowledgements}{31}{chapter.7}% +\contentsline {chapter}{Bibliography}{33}{chapter*.3}% diff --git a/docs/build/latex/sphinxhighlight.sty b/docs/build/latex/sphinxhighlight.sty index 9be977b..fe63e4f 100755 --- a/docs/build/latex/sphinxhighlight.sty +++ b/docs/build/latex/sphinxhighlight.sty @@ -52,7 +52,7 @@ \@namedef{PYG@tok@gp}{\let\PYG@bf=\textbf\def\PYG@tc##1{\textcolor[rgb]{0.78,0.36,0.04}{##1}}} \@namedef{PYG@tok@go}{\def\PYG@tc##1{\textcolor[rgb]{0.20,0.20,0.20}{##1}}} \@namedef{PYG@tok@gt}{\def\PYG@tc##1{\textcolor[rgb]{0.00,0.27,0.87}{##1}}} -\@namedef{PYG@tok@err}{\def\PYG@bc##1{{\setlength{\fboxsep}{-\fboxrule}\fcolorbox[rgb]{1.00,0.00,0.00}{1,1,1}{\strut ##1}}}} +\@namedef{PYG@tok@err}{\def\PYG@bc##1{{\setlength{\fboxsep}{\string -\fboxrule}\fcolorbox[rgb]{1.00,0.00,0.00}{1,1,1}{\strut ##1}}}} \@namedef{PYG@tok@kc}{\let\PYG@bf=\textbf\def\PYG@tc##1{\textcolor[rgb]{0.00,0.44,0.13}{##1}}} \@namedef{PYG@tok@kd}{\let\PYG@bf=\textbf\def\PYG@tc##1{\textcolor[rgb]{0.00,0.44,0.13}{##1}}} \@namedef{PYG@tok@kn}{\let\PYG@bf=\textbf\def\PYG@tc##1{\textcolor[rgb]{0.00,0.44,0.13}{##1}}} diff --git a/docs/history.rst b/docs/history.rst index fd4d8ef..0d1938c 100644 --- a/docs/history.rst +++ b/docs/history.rst @@ -45,3 +45,8 @@ Version 1.2.2 (26 Feb 2021) Version 1.2.3 (1 Jun 2021) --------------------------- * Minor update to customise handling of NaNs in objective evaluations - no changes to the DFO-LS algorithm. + +Version 1.3.0 (16 Oct 2021) +--------------------------- +* Handle finitely many arbitrary convex constraints in addition to simple bound constraints. +* Only new functionality is added, so there is no change to the solver for unconstrained/bound-constrained problems. diff --git a/docs/index.rst b/docs/index.rst index a0e467e..031d6c9 100644 --- a/docs/index.rst +++ b/docs/index.rst @@ -11,16 +11,17 @@ DFO-LS: Derivative-Free Optimizer for Least-Squares Minimization **Author:** `Lindon Roberts `_ -DFO-LS is a flexible package for finding local solutions to nonlinear least-squares minimization problems (with optional bound constraints), without requiring any derivatives of the objective. DFO-LS stands for Derivative-Free Optimizer for Least-Squares. +DFO-LS is a flexible package for finding local solutions to nonlinear least-squares minimization problems (with optional constraints), without requiring any derivatives of the objective. DFO-LS stands for Derivative-Free Optimizer for Least-Squares. That is, DFO-LS solves .. math:: \min_{x\in\mathbb{R}^n} &\quad f(x) := \sum_{i=1}^{m}r_{i}(x)^2 \\ - \text{s.t.} &\quad a \leq x \leq b + \text{s.t.} &\quad x \in C\\ + &\quad a \leq x \leq b -The upper and lower bounds on the variables are non-relaxable (i.e. DFO-LS will never ask to evaluate a point outside the bounds). +The constraint set :math:`C` is the intersection of multiple convex sets provided as input by the user. All constraints are non-relaxable (i.e. DFO-LS will never ask to evaluate a point that is not feasible). Full details of the DFO-LS algorithm are given in our paper: C. Cartis, J. Fiala, B. Marteau and L. Roberts, `Improving the Flexibility and Robustness of Model-Based Derivative-Free Optimization Solvers `_, *ACM Transactions on Mathematical Software*, 45:3 (2019), pp. 32:1-32:41 [`preprint `_] . DFO-LS is a more flexible version of `DFO-GN `_. diff --git a/docs/info.rst b/docs/info.rst index 824a232..e4e0b29 100644 --- a/docs/info.rst +++ b/docs/info.rst @@ -3,14 +3,16 @@ Overview When to use DFO-LS ------------------ -DFO-LS is designed to solve the nonlinear least-squares minimization problem (with optional bound constraints) +DFO-LS is designed to solve the nonlinear least-squares minimization problem (with optional convex constraints). .. math:: \min_{x\in\mathbb{R}^n} &\quad f(x) := \sum_{i=1}^{m}r_{i}(x)^2 \\ - \text{s.t.} &\quad a \leq x \leq b + \text{s.t.} &\quad x \in C\\ + &\quad a \leq x \leq b We call :math:`f(x)` the objective function and :math:`r_i(x)` the residual functions (or simply residuals). +:math:`C` is the intersection of multiple convex sets given as input by the user. DFO-LS is a *derivative-free* optimization algorithm, which means it does not require the user to provide the derivatives of :math:`f(x)` or :math:`r_i(x)`, nor does it attempt to estimate them internally (by using finite differencing, for instance). diff --git a/docs/userguide.rst b/docs/userguide.rst index 6804892..e8a7b54 100644 --- a/docs/userguide.rst +++ b/docs/userguide.rst @@ -9,9 +9,9 @@ DFO-LS is designed to solve the local optimization problem .. math:: \min_{x\in\mathbb{R}^n} &\quad f(x) := \sum_{i=1}^{m}r_{i}(x)^2 \\ - \text{s.t.} &\quad a \leq x \leq b + \text{s.t.} &\quad x \in C -where the bound constraints :math:`a \leq x \leq b` are optional. The upper and lower bounds on the variables are non-relaxable (i.e. DFO-LS will never ask to evaluate a point outside the bounds). +where the set :math:`C` is an optional non-empty, closed and convex constraint set. The constraints are non-relaxable (i.e. DFO-LS will never ask to evaluate a point that is not feasible). DFO-LS iteratively constructs an interpolation-based model for the objective, and determines a step using a trust-region framework. For an in-depth technical description of the algorithm see the paper [CFMR2018]_. @@ -48,6 +48,7 @@ The possible values of :code:`soln.flag` are defined by the following variables: * :code:`soln.EXIT_MAXFUN_WARNING` - maximum allowed objective evaluations reached. This is the most likely return value when using multiple restarts. * :code:`soln.EXIT_SLOW_WARNING` - maximum number of slow iterations reached. * :code:`soln.EXIT_FALSE_SUCCESS_WARNING` - DFO-LS reached the maximum number of restarts which decreased the objective, but to a worse value than was found in a previous run. +* :code:`soln.EXIT_TR_INCREASE_WARNING` - model increase when solving the trust region subproblem with multiple arbitrary constraints. * :code:`soln.EXIT_INPUT_ERROR` - error in the inputs. * :code:`soln.EXIT_TR_INCREASE_ERROR` - error occurred when solving the trust region subproblem. * :code:`soln.EXIT_LINALG_ERROR` - linear algebra error, e.g. the interpolation points produced a singular linear system. @@ -75,6 +76,7 @@ These arguments are: * :code:`args` - a tuple of extra arguments passed to the objective function. * :code:`bounds` - a tuple :code:`(lower, upper)` with the vectors :math:`a` and :math:`b` of lower and upper bounds on :math:`x` (default is :math:`a_i=-10^{20}` and :math:`b_i=10^{20}`). To set bounds for either :code:`lower` or :code:`upper`, but not both, pass a tuple :code:`(lower, None)` or :code:`(None, upper)`. +* :code:`projections` - a list :code:`[f1,f2,...,fn]` of functions that each take as input a point :code:`x` and return a new point :code:`y`. The new point :code:`y` should be given by the projection of :code:`x` onto a closed convex set. The intersection of all sets corresponding to a function must be non-empty. * :code:`npt` - the number of interpolation points to use (default is :code:`len(x0)+1`). If using restarts, this is the number of points to use in the first run of the solver, before any restarts (and may be optionally increased via settings in :code:`user_params`). * :code:`rhobeg` - the initial value of the trust region radius (default is :math:`0.1\max(\|x_0\|_{\infty}, 1)`, or 0.1 if :code:`scaling_within_bounds`). * :code:`rhoend` - minimum allowed value of trust region radius, which determines when a successful termination occurs (default is :math:`10^{-8}`). @@ -144,7 +146,7 @@ This and all following problems can be found in the `examples