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 """ Interface to Constrained Optimization By Linear Approximation Functions --------- .. autosummary:: :toctree: generated/ fmin_cobyla """ from __future__ import division, print_function, absolute_import import numpy as np from scipy._lib.six import callable from scipy.optimize import _cobyla from .optimize import OptimizeResult, _check_unknown_options try: from itertools import izip except ImportError: izip = zip __all__ = ['fmin_cobyla'] def fmin_cobyla(func, x0, cons, args=(), consargs=None, rhobeg=1.0, rhoend=1e-4, iprint=1, maxfun=1000, disp=None, catol=2e-4): """ Minimize a function using the Constrained Optimization BY Linear Approximation (COBYLA) method. This method wraps a FORTRAN implementation of the algorithm. Parameters ---------- func : callable Function to minimize. In the form func(x, \\*args). x0 : ndarray Initial guess. cons : sequence Constraint functions; must all be ``>=0`` (a single function if only 1 constraint). Each function takes the parameters `x` as its first argument, and it can return either a single number or an array or list of numbers. args : tuple, optional Extra arguments to pass to function. consargs : tuple, optional Extra arguments to pass to constraint functions (default of None means use same extra arguments as those passed to func). Use ``()`` for no extra arguments. rhobeg : float, optional Reasonable initial changes to the variables. rhoend : float, optional Final accuracy in the optimization (not precisely guaranteed). This is a lower bound on the size of the trust region. iprint : {0, 1, 2, 3}, optional Controls the frequency of output; 0 implies no output. Deprecated. disp : {0, 1, 2, 3}, optional Over-rides the iprint interface. Preferred. maxfun : int, optional Maximum number of function evaluations. catol : float, optional Absolute tolerance for constraint violations. Returns ------- x : ndarray The argument that minimises `f`. See also -------- minimize: Interface to minimization algorithms for multivariate functions. See the 'COBYLA' `method` in particular. Notes ----- This algorithm is based on linear approximations to the objective function and each constraint. We briefly describe the algorithm. Suppose the function is being minimized over k variables. At the jth iteration the algorithm has k+1 points v_1, ..., v_(k+1), an approximate solution x_j, and a radius RHO_j. (i.e. linear plus a constant) approximations to the objective function and constraint functions such that their function values agree with the linear approximation on the k+1 points v_1,.., v_(k+1). This gives a linear program to solve (where the linear approximations of the constraint functions are constrained to be non-negative). However the linear approximations are likely only good approximations near the current simplex, so the linear program is given the further requirement that the solution, which will become x_(j+1), must be within RHO_j from x_j. RHO_j only decreases, never increases. The initial RHO_j is rhobeg and the final RHO_j is rhoend. In this way COBYLA's iterations behave like a trust region algorithm. Additionally, the linear program may be inconsistent, or the approximation may give poor improvement. For details about how these issues are resolved, as well as how the points v_i are updated, refer to the source code or the references below. References ---------- Powell M.J.D. (1994), "A direct search optimization method that models the objective and constraint functions by linear interpolation.", in Advances in Optimization and Numerical Analysis, eds. S. Gomez and J-P Hennart, Kluwer Academic (Dordrecht), pp. 51-67 Powell M.J.D. (1998), "Direct search algorithms for optimization calculations", Acta Numerica 7, 287-336 Powell M.J.D. (2007), "A view of algorithms for optimization without derivatives", Cambridge University Technical Report DAMTP 2007/NA03 Examples -------- Minimize the objective function f(x,y) = x*y subject to the constraints x**2 + y**2 < 1 and y > 0:: >>> def objective(x): ... return x[0]*x[1] ... >>> def constr1(x): ... return 1 - (x[0]**2 + x[1]**2) ... >>> def constr2(x): ... return x[1] ... >>> from scipy.optimize import fmin_cobyla >>> fmin_cobyla(objective, [0.0, 0.1], [constr1, constr2], rhoend=1e-7) array([-0.70710685, 0.70710671]) The exact solution is (-sqrt(2)/2, sqrt(2)/2). """ err = "cons must be a sequence of callable functions or a single"\ " callable function." try: len(cons) except TypeError: if callable(cons): cons = [cons] else: raise TypeError(err) else: for thisfunc in cons: if not callable(thisfunc): raise TypeError(err) if consargs is None: consargs = args # build constraints con = tuple({'type': 'ineq', 'fun': c, 'args': consargs} for c in cons) # options if disp is not None: iprint = disp opts = {'rhobeg': rhobeg, 'tol': rhoend, 'iprint': iprint, 'disp': iprint != 0, 'maxiter': maxfun, 'catol': catol} sol = _minimize_cobyla(func, x0, args, constraints=con, **opts) if iprint > 0 and not sol['success']: print("COBYLA failed to find a solution: %s" % (sol.message,)) return sol['x'] def _minimize_cobyla(fun, x0, args=(), constraints=(), rhobeg=1.0, tol=1e-4, iprint=1, maxiter=1000, disp=False, catol=2e-4, **unknown_options): """ Minimize a scalar function of one or more variables using the Constrained Optimization BY Linear Approximation (COBYLA) algorithm. Options ------- rhobeg : float Reasonable initial changes to the variables. tol : float Final accuracy in the optimization (not precisely guaranteed). This is a lower bound on the size of the trust region. disp : bool Set to True to print convergence messages. If False, `verbosity` is ignored as set to 0. maxiter : int Maximum number of function evaluations. catol : float Tolerance (absolute) for constraint violations """ _check_unknown_options(unknown_options) maxfun = maxiter rhoend = tol if not disp: iprint = 0 # check constraints if isinstance(constraints, dict): constraints = (constraints, ) for ic, con in enumerate(constraints): # check type try: ctype = con['type'].lower() except KeyError: raise KeyError('Constraint %d has no type defined.' % ic) except TypeError: raise TypeError('Constraints must be defined using a ' 'dictionary.') except AttributeError: raise TypeError("Constraint's type must be a string.") else: if ctype != 'ineq': raise ValueError("Constraints of type '%s' not handled by " "COBYLA." % con['type']) # check function if 'fun' not in con: raise KeyError('Constraint %d has no function defined.' % ic) # check extra arguments if 'args' not in con: con['args'] = () # m is the total number of constraint values # it takes into account that some constraints may be vector-valued cons_lengths = [] for c in constraints: f = c['fun'](x0, *c['args']) try: cons_length = len(f) except TypeError: cons_length = 1 cons_lengths.append(cons_length) m = sum(cons_lengths) def calcfc(x, con): f = fun(x, *args) i = 0 for size, c in izip(cons_lengths, constraints): con[i: i + size] = c['fun'](x, *c['args']) i += size return f info = np.zeros(4, np.float64) xopt, info = _cobyla.minimize(calcfc, m=m, x=np.copy(x0), rhobeg=rhobeg, rhoend=rhoend, iprint=iprint, maxfun=maxfun, dinfo=info) if info[3] > catol: # Check constraint violation info[0] = 4 return OptimizeResult(x=xopt, status=int(info[0]), success=info[0] == 1, message={1: 'Optimization terminated successfully.', 2: 'Maximum number of function evaluations has ' 'been exceeded.', 3: 'Rounding errors are becoming damaging in ' 'COBYLA subroutine.', 4: 'Did not converge to a solution satisfying ' 'the constraints. See `maxcv` for magnitude ' 'of violation.' }.get(info[0], 'Unknown exit status.'), nfev=int(info[1]), fun=info[2], maxcv=info[3]) if __name__ == '__main__': from math import sqrt def fun(x): return x[0] * x[1] def cons(x): return 1 - x[0]**2 - x[1]**2 x = fmin_cobyla(fun, [1., 1.], cons, iprint=3, disp=1) print('\nTheoretical solution: %e, %e' % (1. / sqrt(2.), -1. / sqrt(2.)))