/
newton_trust_region.jl
153 lines (125 loc) · 4.58 KB
/
newton_trust_region.jl
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let
#######################################
# First test the subproblem.
srand(42)
n = 5
H = rand(n, n)
H = H' * H + 4 * eye(n)
H_eig = eigfact(H)
U = H_eig[:vectors]
gr = zeros(n)
gr[1] = 1.
s = zeros(Float64, n)
true_s = -H \ gr
s_norm2 = dot(true_s, true_s)
true_m = dot(true_s, gr) + 0.5 * dot(true_s, H * true_s)
# An interior solution
delta = sqrt(s_norm2) + 1.0
m, interior, lambda, hard_case, reached_solution =
Optim.solve_tr_subproblem!(gr, H, delta, s)
@assert interior
@assert !hard_case
@assert reached_solution
@assert abs(m - true_m) < 1e-12
@assert norm(s - true_s) < 1e-12
@assert abs(lambda) < 1e-12
# A boundary solution
delta = 0.5 * sqrt(s_norm2)
m, interior, lambda, hard_case, reached_solution =
Optim.solve_tr_subproblem!(gr, H, delta, s)
@assert !interior
@assert !hard_case
@assert reached_solution
@assert m > true_m
@assert abs(norm(s) - delta) < 1e-12
@assert lambda > 0
# A "hard case" where the gradient is orthogonal to the lowest eigenvector
# Test the checking
hard_case, lambda_1_multiplicity =
Optim.check_hard_case_candidate([-1., 2., 3.], [0., 1., 1.])
@assert hard_case
@assert lambda_1_multiplicity == 1
hard_case, lambda_1_multiplicity =
Optim.check_hard_case_candidate([-1., -1., 3.], [0., 0., 1.])
@assert hard_case
@assert lambda_1_multiplicity == 2
hard_case, lambda_1_multiplicity =
Optim.check_hard_case_candidate([-1., -1., -1.], [0., 0., 0.])
@assert hard_case
@assert lambda_1_multiplicity == 3
hard_case, lambda_1_multiplicity =
Optim.check_hard_case_candidate([1., 2., 3.], [0., 1., 1.])
@assert !hard_case
hard_case, lambda_1_multiplicity =
Optim.check_hard_case_candidate([-1., -1., -1.], [0., 0., 1.])
@assert !hard_case
hard_case, lambda_1_multiplicity =
Optim.check_hard_case_candidate([-1., 2., 3.], [1., 1., 1.])
@assert !hard_case
# Now check an actual had case problem
L = zeros(Float64, n) + 0.1
L[1] = -1.
H = U * diagm(L) * U'
H = 0.5 * (H' + H)
@assert issymmetric(H)
gr = U[:,2][:]
@assert abs(dot(gr, U[:,1][:])) < 1e-12
true_s = -H \ gr
s_norm2 = dot(true_s, true_s)
true_m = dot(true_s, gr) + 0.5 * dot(true_s, H * true_s)
delta = 0.5 * sqrt(s_norm2)
m, interior, lambda, hard_case, reached_solution =
Optim.solve_tr_subproblem!(gr, H, delta, s)
@assert !interior
@assert hard_case
@assert reached_solution
@assert abs(lambda + L[1]) < 1e-4
@assert abs(norm(s) - delta) < 1e-12
#######################################
# Next, test on actual optimization problems.
function f(x::Vector)
(x[1] - 5.0)^4
end
function g!(x::Vector, storage::Vector)
storage[1] = 4.0 * (x[1] - 5.0)^3
end
function h!(x::Vector, storage::Matrix)
storage[1, 1] = 12.0 * (x[1] - 5.0)^2
end
d = TwiceDifferentiableFunction(f, g!, h!)
results = Optim.optimize(d, [0.0], method=NewtonTrustRegion())
@assert length(results.trace) == 0
@assert results.g_converged
@assert norm(results.minimum - [5.0]) < 0.01
eta = 0.9
function f_2(x::Vector)
0.5 * (x[1]^2 + eta * x[2]^2)
end
function g!_2(x::Vector, storage::Vector)
storage[1] = x[1]
storage[2] = eta * x[2]
end
function h!_2(x::Vector, storage::Matrix)
storage[1, 1] = 1.0
storage[1, 2] = 0.0
storage[2, 1] = 0.0
storage[2, 2] = eta
end
d = TwiceDifferentiableFunction(f_2, g!_2, h!_2)
results = Optim.optimize(d, Float64[127, 921], method=NewtonTrustRegion())
@assert results.g_converged
@assert norm(results.minimum - [0.0, 0.0]) < 0.01
# Test Optim.newton for all twice differentiable functions in
# Optim.UnconstrainedProblems.examples
for (name, prob) in Optim.UnconstrainedProblems.examples
if prob.istwicedifferentiable
ddf = DifferentiableFunction(prob.f, prob.g!)
res = Optim.optimize(ddf, prob.initial_x, NewtonTrustRegion(), OptimizationOptions(autodiff = true))
@assert norm(Optim.minimizer(res) - prob.solutions) < 1e-2
res = Optim.optimize(ddf.f, prob.initial_x, NewtonTrustRegion(), OptimizationOptions(autodiff = true))
@assert norm(Optim.minimizer(res) - prob.solutions) < 1e-2
res = Optim.optimize(ddf.f, ddf.g!, prob.initial_x, NewtonTrustRegion(), OptimizationOptions(autodiff = true))
@assert norm(Optim.minimizer(res) - prob.solutions) < 1e-2
end
end
end