From: Denis Davydov Date: Mon, 24 Dec 2018 16:11:08 +0000 (+0100) Subject: add BFGS minimizer X-Git-Tag: v9.1.0-rc1~472^2 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=24bb83d0650b61a5a63f7a90f0c4ead725e9a526;p=dealii.git add BFGS minimizer --- diff --git a/doc/news/changes/minor/20181226DenisDavydov b/doc/news/changes/minor/20181226DenisDavydov new file mode 100644 index 0000000000..e40be830d6 --- /dev/null +++ b/doc/news/changes/minor/20181226DenisDavydov @@ -0,0 +1,3 @@ +New: Add SolverBFGS to minimize a function using the limited memory BFGS approach. +
+(Denis Davydov, 2018/12/26) diff --git a/include/deal.II/optimization/solver_bfgs.h b/include/deal.II/optimization/solver_bfgs.h new file mode 100644 index 0000000000..5f3ed06779 --- /dev/null +++ b/include/deal.II/optimization/solver_bfgs.h @@ -0,0 +1,407 @@ +//----------------------------------------------------------- +// +// Copyright (C) 2018 by the deal.II authors +// +// This file is part of the deal.II library. +// +// The deal.II library is free software; you can use it, redistribute +// it, and/or modify it under the terms of the GNU Lesser General +// Public License as published by the Free Software Foundation; either +// version 2.1 of the License, or (at your option) any later version. +// The full text of the license can be found in the file LICENSE.md at +// the top level directory of deal.II. +// +//--------------------------------------------------------------- + +#ifndef dealii_solver_bfgs_h +#define dealii_solver_bfgs_h + +#include + +#include + +#include + +DEAL_II_NAMESPACE_OPEN + +/** + * Implement the limited memory BFGS minimization method. + * + * This class implements a method to minimize a given function for which only + * the values of the function and its derivatives, but not its second + * derivatives are available. The BFGS method is a variation of the Newton + * method for function minimization in which the Hessian matrix is only + * approximated. In particular, the Hessian is updated using the formula of + * Broyden, Fletcher, Goldfarb, and Shanno (BFGS): + * \f[ + * H^{(k+1)} &= \left[ + * I-\rho_{(k)} s^{(k)} \otimes y^{(k)} + * \right] + * H^{(k)} + * \left[ + * I -\rho^{(k)} y^{(k)} \otimes s^{(k)} + * \right] + * + + * \rho^{(k)} s^{(k)} \otimes s^{(k)} \\ + * y^{(k)} &\dealcoloneq g^{(k+1)} - g^{(k)} \\ + * s^{(k)} &\dealcoloneq x^{(k+1)} - x^{(k)} \\ + * \rho^{(k)} &\dealcoloneq \frac{1}{y^{(k)} \cdot s^{(k)}} + * \f] + * for a symmetric positive definite $H$. Limited memory variant is + * implemented via the two-loop recursion. + * + * @author Denis Davydov, 2018 + */ +template +class SolverBFGS : public Solver +{ +public: + /** + * Number type. + */ + typedef typename VectorType::value_type Number; + + + /** + * Standardized data struct to pipe additional data to the solver. + */ + struct AdditionalData + { + /** + * Constructor. + */ + explicit AdditionalData(const unsigned int max_history_size = 5, + const bool debug_output = false); + + /** + * Maximum history size. + */ + unsigned int max_history_size; + + /** + * Print extra debug output to deallog. + */ + bool debug_output; + }; + + + /** + * Constructor. + */ + explicit SolverBFGS(SolverControl & residual_control, + const AdditionalData &data = AdditionalData()); + + /** + * Solve the unconstrained minimization problem + * \f[ + * \min_{\mathbf x} f(\mathbf x) + * \f] + * starting from initial state @p x. + * + * The function @p compute takes two arguments indicating the values of $x$ + * and of the gradient $g=\nabla f(\mathbf x)=\frac{\partial f}{\partial + * \mathbf x}$. When called, it needs to update the gradient $g$ at the given + * location $x$ and return the value of the function being minimized, i.e., + * $f(\mathbf x)$. + */ + void + solve( + const std::function &compute, + VectorType & x); + + /** + * Connect a slot to perform a custom line-search. + * + * Given the value of function @p f, the current value of unknown @p x, + * the gradient @p g and the search direction @p p, + * return the size $\alpha$ of the step $x \leftarrow x + \alpha p$, + * and update @p x, @p g and @p f accordingly. + */ + boost::signals2::connection + connect_line_search_slot( + const std::function< + Number(Number &f, VectorType &x, VectorType &g, const VectorType &p)> + &slot); + + /** + * Connect a slot to perform a custom preconditioning. + * + * The preconditioner is applied inside the two loop recursion to + * vector `g` using the history of position increments `s` and + * gradient increments `y`. + * + * One possibility is to use the oldest `s,y` pair: + * @code + * const auto preconditioner = [](VectorType & g, + * const FiniteSizeHistory &s, + * const FiniteSizeHistory &y) { + * if (s.size() > 0) + * { + * const unsigned int i = s.size() - 1; + * const auto yy = y[i] * y[i]; + * const auto sy = s[i] * y[i]; + * Assert(yy > 0 && sy > 0, ExcInternalError()); + * g *= sy / yy; + * } + * }; + * @endcode + * + * No preconditioning is performed if the code using this class has not + * attached anything to the signal. + */ + boost::signals2::connection + connect_preconditioner_slot( + const std::function &s, + const FiniteSizeHistory &y)> &slot); + + +protected: + /** + * Additional data to the solver. + */ + const AdditionalData additional_data; + + /** + * Signal used to perform line search. + */ + boost::signals2::signal< + Number(Number &f, VectorType &x, VectorType &g, const VectorType &p)> + line_search_signal; + + /** + * Signal used to perform preconditioning. + */ + boost::signals2::signal &s, + const FiniteSizeHistory &y)> + preconditioner_signal; +}; + + +// ------------------- inline and template functions ---------------- +#ifndef DOXYGEN + +template +SolverBFGS::AdditionalData::AdditionalData( + const unsigned int max_history_size_, + const bool debug_output_) + : max_history_size(max_history_size_) + , debug_output(debug_output_) +{} + + + +template +SolverBFGS::SolverBFGS(SolverControl & solver_control, + const AdditionalData &data) + : Solver(solver_control) + , additional_data(data) +{} + + + +template +boost::signals2::connection +SolverBFGS::connect_line_search_slot( + const std::function< + Number(Number &f, VectorType &x, VectorType &g, const VectorType &p)> &slot) +{ + Assert(line_search_signal.empty(), + ExcMessage("One should not attach more than one line search signal.")); + return line_search_signal.connect(slot); +} + + + +template +boost::signals2::connection +SolverBFGS::connect_preconditioner_slot( + const std::function &s, + const FiniteSizeHistory &y)> &slot) +{ + Assert(preconditioner_signal.empty(), + ExcMessage( + "One should not attach more than one preconditioner signal.")); + return preconditioner_signal.connect(slot); +} + + + +template +void +SolverBFGS::solve( + const std::function &compute, + VectorType & x) +{ + // Also see scipy Fortran implementation + // https://github.com/scipy/scipy/blob/master/scipy/optimize/lbfgsb_src/lbfgsb.f + // and Octave-optim implementation: + // https://sourceforge.net/p/octave/optim/ci/default/tree/src/__bfgsmin.cc + LogStream::Prefix prefix("BFGS"); + + // default line search: + bool first_step = true; + Number f_prev = 0.; + // provide default line search if no signal was attached + VectorType x0; + if (line_search_signal.empty()) + { + x0.reinit(x); + const auto default_line_min = + [&](Number &f, VectorType &x, VectorType &g, const VectorType &p) { + const Number f0 = f; + const Number g0 = g * p; + Assert(g0 < 0, + ExcMessage( + "Function does not decrease along the current direction")); + + // save current solution value (to be used in line_search): + x0 = x; + + // see scipy implementation + // https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.line_search.html#scipy.optimize.line_search + // and Eq. 2.6.8 in Fletcher 2013, Practical methods of optimization + Number df = f_prev - f; + Assert(first_step || df >= 0., + ExcMessage("Function value is not decreasing")); + df = std::max(df, 100. * std::numeric_limits::epsilon()); + // guess a reasonable first step: + const Number a1 = + (first_step ? 1. : std::min(1., -1.01 * 2. * df / g0)); + Assert(a1 > 0., ExcInternalError()); + f_prev = f; + + // 1D line-search function + const auto line_func = + [&](const Number &x_line) -> std::pair { + x = x0; + x.add(x_line, p); + f = compute(x, g); + const Number g_line = g * p; + return std::make_pair(f, g_line); + }; + + // loose line search: + const auto res = LineMinimization::line_search( + line_func, + f0, + g0, + LineMinimization::poly_fit, + a1, + 0.9, + 0.001); + + if (first_step) + first_step = false; + + return res.first; + }; + this->connect_line_search_slot(default_line_min); + } + + // FIXME: Octave has convergence in terms of: + // function change tolerance, default 1e-12 + // parameter change tolerance, default 1e-6 + // gradient tolerance, default 1e-5 + // SolverBase and/or SolverControl need extension + + VectorType g(x), p(x), y_k(x), s_k(x); + + std::vector c1; + c1.reserve(additional_data.max_history_size); + + // limited history + FiniteSizeHistory y(additional_data.max_history_size); + FiniteSizeHistory s(additional_data.max_history_size); + FiniteSizeHistory rho(additional_data.max_history_size); + + unsigned int m = 0; + Number f; + + SolverControl::State conv = SolverControl::iterate; + unsigned int k = 0; + + f = compute(x, g); + + conv = this->iteration_status(k, g.l2_norm(), x); + if (conv != SolverControl::iterate) + return; + + while (conv == SolverControl::iterate) + { + if (additional_data.debug_output) + deallog << "Iteration " << k << " history " << m << std::endl + << "f=" << f << std::endl; + + // 1. Two loop recursion to calculate p = - H*g + c1.resize(m); + p = g; + // first loop: + for (unsigned int i = 0; i < m; ++i) + { + c1[i] = rho[i] * (s[i] * p); + p.add(-c1[i], y[i]); + } + // H0 + if (!preconditioner_signal.empty()) + preconditioner_signal(p, s, y); + + // second loop: + for (int i = m - 1; i >= 0; --i) + { + Assert(i >= 0, ExcInternalError()); + const Number c2 = rho[i] * (y[i] * p); + p.add(c1[i] - c2, s[i]); + } + p *= -1.; + + // 2. Line search + s_k = x; + y_k = g; + const Number alpha = line_search_signal(f, x, g, p) + .get(); // <-- signals return boost::optional + s_k.sadd(-1, 1, x); + y_k.sadd(-1, 1, g); + + if (additional_data.debug_output) + deallog << "Line search a=" << alpha << " f=" << f << std::endl; + + // 3. Check convergence + k++; + const Number g_l2 = g.l2_norm(); + conv = this->iteration_status(k, g_l2, x); + if (conv != SolverControl::iterate) + break; + + // 4. Store s, y, rho + const Number curvature = s_k * y_k; + if (additional_data.debug_output) + deallog << "Curvature " << curvature << std::endl; + + if (curvature > 0. && additional_data.max_history_size > 0) + { + s.add(s_k); + y.add(y_k); + rho.add(1. / curvature); + m = s.size(); + + Assert(y.size() == m, ExcInternalError()); + Assert(rho.size() == m, ExcInternalError()); + } + + Assert(m <= additional_data.max_history_size, ExcInternalError()); + } + + // In the case of failure: throw exception. + AssertThrow(conv == SolverControl::success, + SolverControl::NoConvergence(k, g.l2_norm())); +} + +#endif + +DEAL_II_NAMESPACE_CLOSE + +#endif diff --git a/tests/optimization/bfgs_03.cc b/tests/optimization/bfgs_03.cc new file mode 100644 index 0000000000..9853dce836 --- /dev/null +++ b/tests/optimization/bfgs_03.cc @@ -0,0 +1,160 @@ +//----------------------------------------------------------- +// +// Copyright (C) 2018 by the deal.II authors +// +// This file is part of the deal.II library. +// +// The deal.II library is free software; you can use it, redistribute +// it, and/or modify it under the terms of the GNU Lesser General +// Public License as published by the Free Software Foundation; either +// version 2.1 of the License, or (at your option) any later version. +// The full text of the license can be found in the file LICENSE.md at +// the top level directory of deal.II. +// +//--------------------------------------------------------------- + +// test limited memory BFGS with quadratic function +// f(x) = 0.5 x*Lx - x*f +// f'(x) = Lx - f +// where L is 1d FD Laplacian +// +// We compare results to the companion Octave file which uses BFGS +// from optim package. The output of this test is made similar to the +// Octave output. + + +#include + +#include +#include + +#include + +#include +#include + +#include "../tests.h" + +template +void +test() +{ + typedef Vector VectorType; + + // size of the problem + const unsigned int N = 10; + + // parameters: + const unsigned int itmax = 100; + const double tol = 1e-8; + + // 1D Laplace with zero Dirichlet BC on both sides + FullMatrix L(N); + L = 0.; + + for (unsigned int i = 0; i < N; ++i) + { + if (i > 0) + L(i, i - 1) = -1.; + L(i, i) = 2.; + if (i < N - 1) + L(i, i + 1) = -1.; + } + + // L.print_formatted(deallog.get_file_stream(), 6, false, 10); + + // RHS + VectorType b(N); + for (unsigned int i = 0; i < N; ++i) + b(i) = 1.; + + // solution + VectorType x(N); + x = 1.; + + // safety measure to not modify L or b within Lambda. + const FullMatrix &L_const = L; + const VectorType & b_const = b; + const auto func = [&](const VectorType &x, VectorType &g) { + L_const.vmult(g, x); + number res = 0.5 * (g * x) - x * b_const; + g.add(-1, b); + return res; + }; + + // exact line minimization for quadratic function + /* + f(x) := a*x**2 + b*x + c; + g(x) := ''(diff(f(x),x)); + sol : solve([f(0)=f0, f(1)=f1, g(0)=g0],[a,b,c]); + sol2 : solve(diff(f(u),u)=0,u); + subst(sol,sol2); + + u=-g0/(2*(-g0+f1-f0)) + + */ + + bool first_step = true; + + int iteration = 0; + VectorType dx(x); + + const auto line_min = + [&](number &f, VectorType &x, VectorType &g, const VectorType &p) { + deallog << "-------------------" << std::endl + << "Line search " << iteration++ << ":" << std::endl; + + const number g_norm_sqr = g.norm_sqr(); + + deallog << "Gradient:" << std::endl; + g.print(deallog.get_file_stream(), 5, false); + + // directional derivative + const number df = g * p; + Assert(df < 0, ExcInternalError()); + // do the full step + x.add(1., p); + // save old value + const number f_old = f; + // calculate new value + f = func(x, g); + // get the step size + const number denom = -df + f - f_old; + Assert(denom != 0., ExcDivideByZero()); + Assert(denom > 0, ExcInternalError()); + const number step = -df * 0.5 / denom; + // do the step + x.add(step - 1., p); + f = func(x, g); + + first_step = false; + deallog << "Function value: " << f_old << " stepsize: " << step + << " new value: " << f << std::endl; + deallog << "Change:" << std::endl; + dx = p; + dx *= step; + dx.print(deallog.get_file_stream(), 5, false); + + // finally return the step size + return step; + }; + + SolverControl solver_control(itmax, tol, true); + typename SolverBFGS::AdditionalData data(10, false); + SolverBFGS solver(solver_control, data); + solver.connect_line_search_slot(line_min); + solver.solve(func, x); + + deallog << "Limited memory BFGS solution:" << std::endl; + x.print(deallog); +} + +int +main() +{ + std::ofstream logfile("output"); + deallog << std::setprecision(5); + deallog.attach(logfile); + + test(); +} diff --git a/tests/optimization/bfgs_03.output b/tests/optimization/bfgs_03.output new file mode 100644 index 0000000000..202cf2fce4 --- /dev/null +++ b/tests/optimization/bfgs_03.output @@ -0,0 +1,46 @@ + +DEAL:BFGS::Check 0 2.8284 +DEAL:BFGS::Starting value 2.8284 +DEAL:BFGS::------------------- +DEAL:BFGS::Line search 0: +DEAL:BFGS::Gradient: +0.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 0.00000 +DEAL:BFGS::Function value: -9.0000 stepsize: 4.0000 new value: -25.000 +DEAL:BFGS::Change: +0.00000 4.00000 4.00000 4.00000 4.00000 4.00000 4.00000 4.00000 4.00000 0.00000 +DEAL:BFGS::Check 1 7.4833 +DEAL:BFGS::------------------- +DEAL:BFGS::Line search 1: +DEAL:BFGS::Gradient: +-4.00000 3.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 3.00000 -4.00000 +DEAL:BFGS::Function value: -25.000 stepsize: 0.87500 new value: -49.500 +DEAL:BFGS::Change: +3.50000 3.50000 7.00000 7.00000 7.00000 7.00000 7.00000 7.00000 3.50000 3.50000 +DEAL:BFGS::Check 2 4.1833 +DEAL:BFGS::------------------- +DEAL:BFGS::Line search 2: +DEAL:BFGS::Gradient: +-0.50000 -0.50000 2.50000 -1.00000 -1.00000 -1.00000 -1.00000 2.50000 -0.50000 -0.50000 +DEAL:BFGS::Function value: -49.500 stepsize: 0.47619 new value: -53.667 +DEAL:BFGS::Change: +0.83333 0.83333 0.00000 1.66667 1.66667 1.66667 1.66667 0.00000 0.83333 0.83333 +DEAL:BFGS::Check 3 1.8257 +DEAL:BFGS::------------------- +DEAL:BFGS::Line search 3: +DEAL:BFGS::Gradient: +0.33333 0.33333 0.00000 0.66667 -1.00000 -1.00000 0.66667 0.00000 0.33333 0.33333 +DEAL:BFGS::Function value: -53.667 stepsize: 0.60000 new value: -54.667 +DEAL:BFGS::Change: +0.00000 0.00000 0.00000 0.00000 1.00000 1.00000 0.00000 0.00000 0.00000 0.00000 +DEAL:BFGS::Check 4 0.81650 +DEAL:BFGS::------------------- +DEAL:BFGS::Line search 4: +DEAL:BFGS::Gradient: +0.33333 0.33333 0.00000 -0.33333 0.00000 0.00000 -0.33333 0.00000 0.33333 0.33333 +DEAL:BFGS::Function value: -54.667 stepsize: 1.0000 new value: -55.000 +DEAL:BFGS::Change: +-0.33333 -0.33333 0.00000 0.33333 0.33333 0.33333 0.33333 0.00000 -0.33333 -0.33333 +DEAL:BFGS::Check 5 3.5572e-14 +DEAL:BFGS::Convergence step 5 value 3.5572e-14 +DEAL::Limited memory BFGS solution: +DEAL::5.0000 9.0000 12.000 14.000 15.000 15.000 14.000 12.000 9.0000 5.0000 diff --git a/tests/optimization/bfgs_04.cc b/tests/optimization/bfgs_04.cc new file mode 100644 index 0000000000..2052447c08 --- /dev/null +++ b/tests/optimization/bfgs_04.cc @@ -0,0 +1,163 @@ +//----------------------------------------------------------- +// +// Copyright (C) 2018 by the deal.II authors +// +// This file is part of the deal.II library. +// +// The deal.II library is free software; you can use it, redistribute +// it, and/or modify it under the terms of the GNU Lesser General +// Public License as published by the Free Software Foundation; either +// version 2.1 of the License, or (at your option) any later version. +// The full text of the license can be found in the file LICENSE.md at +// the top level directory of deal.II. +// +//--------------------------------------------------------------- + +// test limited memory BFGS with quadratic function +// f(x) = 0.5 x*Lx - x*f +// f'(x) = Lx - f +// where L is 1d FD Laplacian +// Same problem as in bfgs_03, but with +// larger size and lower history. + + +#include + +#include +#include + +#include + +#include +#include + +#include "../tests.h" + +template +void +test() +{ + auto &out = deallog.get_file_stream(); + out << std::setprecision(5) << std::fixed << std::right; + + typedef Vector VectorType; + + // size of the problem + const unsigned int N = 20; + + // parameters: + const unsigned int itmax = 100; + const double tol = 1e-8; + const unsigned int m_max = 4; + + // 1D Laplace with zero Dirichlet BC on both sides + FullMatrix L(N); + L = 0.; + + for (unsigned int i = 0; i < N; ++i) + { + if (i > 0) + L(i, i - 1) = -1.; + L(i, i) = 2.; + if (i < N - 1) + L(i, i + 1) = -1.; + } + + // L.print_formatted(deallog.get_file_stream(), 6, false, 10); + + // RHS + VectorType b(N); + for (unsigned int i = 0; i < N; ++i) + b(i) = 1.; + + // solution + VectorType x(N); + x = 1.; + + // safety measure to not modify L or b within Lambda. + const FullMatrix &L_const = L; + const VectorType & b_const = b; + const auto func = [&](const VectorType &x, VectorType &g) { + L_const.vmult(g, x); + number res = 0.5 * (g * x) - x * b_const; + g.add(-1, b); + return res; + }; + + // exact line minimization for quadratic function + /* + f(x) := a*x**2 + b*x + c; + g(x) := ''(diff(f(x),x)); + sol : solve([f(0)=f0, f(1)=f1, g(0)=g0],[a,b,c]); + sol2 : solve(diff(f(u),u)=0,u); + subst(sol,sol2); + + u=-g0/(2*(-g0+f1-f0)) + + */ + + int iteration = 0; + VectorType dx(x), old_x(x), old_g(x); + + const auto line_min = + [&](number &f, VectorType &x, VectorType &g, const VectorType &p) { + out << "------------------------------------------------" << std::endl + << "Line search " << iteration++ << std::endl + << std::endl; + + const number g_norm_sqr = g.norm_sqr(); + + old_x = x; + old_g = g; + + // directional derivative + const number df = g * p; + Assert(df < 0, ExcInternalError()); + // do the full step + x.add(1., p); + // save old value + const number f_old = f; + // calculate new value + f = func(x, g); + // get the step size + const number denom = -df + f - f_old; + Assert(denom != 0., ExcDivideByZero()); + Assert(denom > 0, ExcInternalError()); + const number step = -df * 0.5 / denom; + // do the step + x.add(step - 1., p); + f = func(x, g); + + out << "function value: " << f_old << " stepsize: " << step << std::endl + << std::endl; + dx = p; + dx *= step; + + const std::string s = " "; + for (unsigned int i = 0; i < N; ++i) + out << s << std::setw(9) << old_x(i) << s << std::setw(9) << old_g(i) + << s << std::setw(9) << dx(i) << std::endl; + + // finally return the step size + return step; + }; + + SolverControl solver_control(itmax, tol, true); + typename SolverBFGS::AdditionalData data(m_max, false); + SolverBFGS solver(solver_control, data); + solver.connect_line_search_slot(line_min); + solver.solve(func, x); + + deallog << "Limited memory BFGS solution:" << std::endl; + x.print(deallog); +} + +int +main() +{ + std::ofstream logfile("output"); + deallog << std::setprecision(5); + deallog.attach(logfile); + + test(); +} diff --git a/tests/optimization/bfgs_04.m b/tests/optimization/bfgs_04.m new file mode 100755 index 0000000000..0a3b1159c9 --- /dev/null +++ b/tests/optimization/bfgs_04.m @@ -0,0 +1,91 @@ +% //----------------------------------------------------------- +% // +% // Copyright (C) 2018 by the deal.II authors +% // +% // This file is part of the deal.II library. +% // +% // The deal.II library is free software; you can use it, redistribute +% // it, and/or modify it under the terms of the GNU Lesser General +% // Public License as published by the Free Software Foundation; either +% // version 2.1 of the License, or (at your option) any later version. +% // The full text of the license can be found in the file LICENSE.md at +% // the top level directory of deal.II. +% // +% //--------------------------------------------------------------- + +% a companion to bfgs_04.cc which minimizes the same function using Octave. + +% bfgs_04.cc matches the Octave. Strangely enough "stepsize" in Octave differs starting from +% iteration 4, however the value, gradient and increment in value (three columns) match perfectly. + +% This file uses bfgsmin() from optim package: https://octave.sourceforge.io/optim/ +% https://octave.sourceforge.io/optim/package_doc/bfgsmin.html#bfgsmin +pkg load optim + +% dimension of Laplace +global N=20; + +% 1D Laplace with zero Dirichlet BC on both sides +function M = get_m() + global N; + for i=1:N + M(i,i)=2; + if i>1 + M(i,i-1)=-1; + end + if i + +#include +#include + +#include +#include + +#include +#include + +#include "../tests.h" + +using namespace LineMinimization; + +template +void +test() +{ + auto &out = deallog.get_file_stream(); + out << std::setprecision(5) << std::fixed << std::right; + + typedef Vector VectorType; + + // size of the problem + const unsigned int N = 21; + + // parameters: + const unsigned int itmax = 150; + const double gtol = 1e-5; // gradient tolerance + const unsigned int m_max = 3; + + // solution + VectorType x(N), x_shifted(x); + x = 0.; + + // shift minimizer to this point + VectorType location(x); + for (unsigned int i = 0; i < N; ++i) + location(i) = double(i) / (N - 1); + + // see + // https://sourceforge.net/p/octave/optim/ci/default/tree/inst/rosenbrock.m#l26 + const auto rosenbrok = [&](VectorType &x, VectorType &g) { + const unsigned int N = x.size(); + double res = 0.; + g = 0; + for (unsigned int i = 0; i < N; ++i) + { + const double xi2 = x(i) * x(i); + + if (i < N - 1) + { + res += 100. * dealii::Utilities::fixed_power<2>(x(i + 1) - xi2) + + dealii::Utilities::fixed_power<2>(1. - x(i)); + + g(i) += -400. * x(i) * (x(i + 1) - xi2) - 2. * (1. - x(i)); + } + + if (i > 0) + g(i) += 200. * (x(i) - x(i - 1) * x(i - 1)); + } + return res; + }; + + + + unsigned int tot_fun_calls = 0; + const auto func = [&](const VectorType &x, VectorType &g) { + tot_fun_calls++; + for (unsigned int i = 0; i < x.size(); ++i) + x_shifted(i) = x(i) - location(i) + 1.; + + return rosenbrok(x_shifted, g); + }; + + VectorType x0(x), dx(x), g_old(x); + + bool first_step = true; + double f_prev = 0.; + + const unsigned int print_n_iterations = 5; + unsigned int iteration = 0; + unsigned int line_search_iterations = 0; + const auto line_min = [&](number & f, + VectorType & x, + VectorType & g, + const VectorType &p) { + if (iteration <= print_n_iterations) + out << "------------------------------------------------" << std::endl + << "Line search " << iteration << std::endl + << std::endl; + + // save current solution value and gradient + x0 = x; + g_old = g; + + const double f0 = f; + const double g0 = g * p; + Assert(g0 < 0, ExcInternalError()); + + // see scipy implementation + // https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.line_search.html#scipy.optimize.line_search + // and Eq. 2.6.8 in Fletcher 2013, Practical methods of optimization + double df = f_prev - f; + Assert(first_step || df >= 0., ExcInternalError()); + df = std::max(df, 100. * std::numeric_limits::epsilon()); + const double a1 = (first_step ? 1. : std::min(1., -1.01 * 2. * df / g0)); + Assert(a1 > 0., ExcInternalError()); + f_prev = f; + + // 1D line-search function + auto line_func = [&](const double &x_line) -> std::pair { + x = x0; + x.add(x_line, p); + const double f_line = func(x, g); + const double g_line = g * p; + f = f_line; + + return std::make_pair(f_line, g_line); + }; + + const auto res = + line_search(line_func, f0, g0, poly_fit, a1, 0.9, 0.001); + + line_search_iterations += res.second; + + if (iteration <= print_n_iterations) + { + out << "function value: " << f0 << " stepsize: " << res.first + << std::endl + << std::endl; + + // change: + dx = p; + dx *= res.first; + + const std::string s = " "; + for (unsigned int i = 0; i < N; ++i) + out << s << std::setw(9) << x0(i) << s << std::setw(9) << g_old(i) + << s << std::setw(9) << dx(i) << std::endl; + } + + if (first_step) + first_step = false; + + iteration++; + return res.first; + }; + + const auto preconditioner = [](VectorType & g, + const FiniteSizeHistory &s, + const FiniteSizeHistory &y) { + if (s.size() > 0) + { + // default preconditioning using the oldest {s,y} pair, see + // lbfgs_recursion() in __bfgsmin.cc of "optim" Octave package. + const unsigned int i = s.size() - 1; + const double yy = y[i] * y[i]; + const double sy = s[i] * y[i]; + Assert(yy > 0 && sy > 0, ExcInternalError()); + g *= sy / yy; + } + }; + + SolverControl solver_control(itmax, gtol, false); + typename SolverBFGS::AdditionalData data(m_max, false); + SolverBFGS solver(solver_control, data); + solver.connect_line_search_slot(line_min); + solver.connect_preconditioner_slot(preconditioner); + solver.solve(func, x); + + Assert(tot_fun_calls == line_search_iterations + 1, ExcInternalError()); + + deallog << "Limited memory BFGS solution:" << std::endl; + x.print(deallog); + + deallog << "Function value: " << func(x, x0) << std::endl; + + x.add(-1, location); + deallog << "Linf error in solution: " << x.linfty_norm() << std::endl; + + deallog << "function calls: " + << (tot_fun_calls - 1) /*one evaluation above*/ << std::endl; +} + +int +main() +{ + std::ofstream logfile("output"); + deallog << std::setprecision(5); + deallog.attach(logfile); + + test(); +} diff --git a/tests/optimization/bfgs_05.m b/tests/optimization/bfgs_05.m new file mode 100755 index 0000000000..9114fdd79b --- /dev/null +++ b/tests/optimization/bfgs_05.m @@ -0,0 +1,48 @@ +% //----------------------------------------------------------- +% // +% // Copyright (C) 2018 by the deal.II authors +% // +% // This file is part of the deal.II library. +% // +% // The deal.II library is free software; you can use it, redistribute +% // it, and/or modify it under the terms of the GNU Lesser General +% // Public License as published by the Free Software Foundation; either +% // version 2.1 of the License, or (at your option) any later version. +% // The full text of the license can be found in the file LICENSE.md at +% // the top level directory of deal.II. +% // +% //--------------------------------------------------------------- + +% a companion to bfgs_05.cc which minimizes the Rosenbrok function using Octave. + +% This file uses bfgsmin() from optim package: https://octave.sourceforge.io/optim/ +% https://octave.sourceforge.io/optim/package_doc/bfgsmin.html#bfgsmin +% +% take Example 3 from bfgsmin_example.m +pkg load optim + +function [obj_value, gradient] = objective(theta, location) + x = theta - location + ones(rows(theta),1); % move minimizer to "location" + [obj_value, gradient] = rosenbrock(x); +endfunction + +% problem parameters +dim = 20; % dimension of Rosenbrock function +theta0 = zeros(dim+1,1); % starting values +location = (0:dim)/dim; % true values +location = location'; + +% solver parameers +mMax = 3; % maximum number of stored residuals +itmax = 120; % maximum allowable number of iterations +ftol = 1; % function change tolerance +xtol = 1; % parameter change tolerance +gtol = 1e-5; % gradient tolerance + +verb = 2; % verbosity [0,3] + +control = {itmax;verb;1;1;mMax;ftol;xtol;gtol}; + +[theta, obj_value, convergence] = bfgsmin("objective", {theta0, location}, control); + +linf_norm = norm(theta-location, 'inf') diff --git a/tests/optimization/bfgs_05.output b/tests/optimization/bfgs_05.output new file mode 100644 index 0000000000..587344381e --- /dev/null +++ b/tests/optimization/bfgs_05.output @@ -0,0 +1,164 @@ + +DEAL:BFGS::Starting value 63.201 +------------------------------------------------ +Line search 0 + +function value: 44.59125 stepsize: 0.00688 + + 0.00000 20.00000 -0.13754 + 0.00000 -9.15000 0.06292 + 0.00000 -15.10000 0.10384 + 0.00000 -18.65000 0.12825 + 0.00000 -20.10000 0.13822 + 0.00000 -19.75000 0.13582 + 0.00000 -17.90000 0.12309 + 0.00000 -14.85000 0.10212 + 0.00000 -10.90000 0.07496 + 0.00000 -6.35000 0.04367 + 0.00000 -1.50000 0.01032 + 0.00000 3.35000 -0.02304 + 0.00000 7.90000 -0.05433 + 0.00000 11.85000 -0.08149 + 0.00000 14.90000 -0.10246 + 0.00000 16.75000 -0.11519 + 0.00000 17.10000 -0.11759 + 0.00000 15.65000 -0.10762 + 0.00000 12.10000 -0.08321 + 0.00000 6.15000 -0.04229 + 0.00000 -0.50000 0.00344 +------------------------------------------------ +Line search 1 + +function value: 31.46982 stepsize: 1.00000 + + -0.13754 -93.10302 0.04478 + 0.06292 62.82505 -0.04890 + 0.10384 7.39497 0.05454 + 0.12825 1.40648 0.07821 + 0.13822 -5.95844 0.09479 + 0.13582 -12.73220 0.10280 + 0.12309 -17.24202 0.10118 + 0.10212 -18.44290 0.08975 + 0.07496 -16.08048 0.06945 + 0.04367 -10.68239 0.04231 + 0.01032 -3.39997 0.01123 + -0.02304 4.25994 -0.02039 + -0.05433 10.76168 -0.04908 + -0.08149 14.86845 -0.07183 + -0.10246 15.91366 -0.08642 + -0.11519 13.96456 -0.09163 + -0.11759 9.83821 -0.08734 + -0.10762 4.94807 -0.07424 + -0.08321 0.98272 -0.05341 + -0.04229 -0.50983 -0.02573 + 0.00344 0.67580 0.00120 +------------------------------------------------ +Line search 2 + +function value: 21.31066 stepsize: 1.00000 + + -0.09276 -51.32933 0.10125 + 0.01402 -21.64466 0.01449 + 0.15838 52.89125 -0.07330 + 0.20646 22.49149 -0.02224 + 0.23302 15.88337 -0.00831 + 0.23862 5.27077 0.01028 + 0.22428 -6.18067 0.02825 + 0.19187 -14.80591 0.03981 + 0.14441 -17.74562 0.04070 + 0.08597 -14.14528 0.03002 + 0.02154 -5.61538 0.01090 + -0.04342 4.37621 -0.01060 + -0.10341 11.95369 -0.02776 + -0.15332 14.43652 -0.03582 + -0.18888 11.33937 -0.03357 + -0.20682 4.37039 -0.02343 + -0.20494 -3.43285 -0.01016 + -0.18186 -8.92041 0.00109 + -0.13662 -9.88283 0.00651 + -0.06802 -5.87687 0.00501 + 0.00464 0.86286 -0.00124 +------------------------------------------------ +Line search 3 + +function value: 14.22694 stepsize: 1.00000 + + 0.00849 15.56485 0.01765 + 0.02851 -18.55220 0.02206 + 0.08508 -19.66673 0.01341 + 0.18422 31.41559 -0.05258 + 0.22471 12.02393 -0.01608 + 0.24890 7.86342 -0.00314 + 0.25253 0.61442 0.01378 + 0.23168 -8.09261 0.03039 + 0.18510 -14.46074 0.03949 + 0.11600 -14.42561 0.03454 + 0.03244 -7.11434 0.01527 + -0.05402 3.54421 -0.01005 + -0.13117 11.41788 -0.02947 + -0.18914 12.71412 -0.03534 + -0.22245 8.06446 -0.02816 + -0.23025 0.90841 -0.01401 + -0.21510 -5.31486 0.00025 + -0.18077 -8.64248 0.00988 + -0.13010 -8.43792 0.01255 + -0.06301 -4.79304 0.00785 + 0.00340 0.64542 -0.00140 +------------------------------------------------ +Line search 4 + +function value: 11.33438 stepsize: 1.00000 + + 0.02614 21.55972 -0.01245 + 0.05057 -9.41840 0.02192 + 0.09849 5.59813 0.00481 + 0.13164 -20.78185 0.00668 + 0.20863 17.72360 -0.03316 + 0.24576 5.72622 -0.00085 + 0.26631 3.26875 0.01611 + 0.26207 -1.85931 0.03631 + 0.22460 -8.59387 0.05269 + 0.15054 -12.44310 0.05270 + 0.04771 -8.34089 0.02750 + -0.06407 2.00137 -0.01337 + -0.16064 10.03092 -0.04540 + -0.22449 10.20601 -0.05269 + -0.25061 4.76104 -0.03931 + -0.24426 -1.29442 -0.01853 + -0.21485 -5.13072 -0.00047 + -0.17089 -6.41353 0.01063 + -0.11756 -5.67239 0.01374 + -0.05517 -3.10137 0.00868 + 0.00199 0.39356 -0.00167 +------------------------------------------------ +Line search 5 + +function value: 9.39817 stepsize: 1.00000 + + 0.01369 2.09034 -0.00803 + 0.07249 16.27957 -0.01397 + 0.10330 -1.08753 0.00313 + 0.13832 -3.17010 0.00613 + 0.17547 -17.22664 0.01211 + 0.24491 11.61622 -0.01544 + 0.28242 5.06870 -0.00275 + 0.29838 4.94126 0.00290 + 0.27729 1.08018 0.01239 + 0.20325 -6.13174 0.02175 + 0.07521 -8.87643 0.01861 + -0.07744 -1.13673 -0.00204 + -0.20604 6.68937 -0.02043 + -0.27718 5.04666 -0.02017 + -0.28992 -0.97195 -0.00884 + -0.26279 -4.68297 0.00148 + -0.21531 -5.19031 0.00692 + -0.16026 -4.13583 0.00845 + -0.10382 -2.78767 0.00745 + -0.04649 -1.29372 0.00407 + 0.00033 0.06337 -0.00053 +DEAL:BFGS::Convergence step 127 value 8.1397e-06 +DEAL::Limited memory BFGS solution: +DEAL::-4.4833e-09 0.050000 0.10000 0.15000 0.20000 0.25000 0.30000 0.35000 0.40000 0.45000 0.50000 0.55000 0.60000 0.65000 0.70000 0.75000 0.80000 0.85000 0.90000 0.95000 1.0000 +DEAL::Function value: 1.3096e-13 +DEAL::Linf error in solution: 1.6564e-07 +DEAL::function calls: 133 diff --git a/tests/optimization/bfgs_05.output.octave b/tests/optimization/bfgs_05.output.octave new file mode 100644 index 0000000000..b83a7e0388 --- /dev/null +++ b/tests/optimization/bfgs_05.output.octave @@ -0,0 +1,3110 @@ +------------------------------------------------ +bfgsmin iteration 0 convergence (f g p): 0 0 0 + +function value: 44.5913 stepsize: 0.00711092 + +used analytic gradient + 0.00000 20.00000 -0.14222 + 0.00000 -9.15000 0.06506 + 0.00000 -15.10000 0.10737 + 0.00000 -18.65000 0.13262 + 0.00000 -20.10000 0.14293 + 0.00000 -19.75000 0.14044 + 0.00000 -17.90000 0.12729 + 0.00000 -14.85000 0.10560 + 0.00000 -10.90000 0.07751 + 0.00000 -6.35000 0.04515 + 0.00000 -1.50000 0.01067 + 0.00000 3.35000 -0.02382 + 0.00000 7.90000 -0.05618 + 0.00000 11.85000 -0.08426 + 0.00000 14.90000 -0.10595 + 0.00000 16.75000 -0.11911 + 0.00000 17.10000 -0.12160 + 0.00000 15.65000 -0.11129 + 0.00000 12.10000 -0.08604 + 0.00000 6.15000 -0.04373 + 0.00000 -0.50000 0.00356 +------------------------------------------------ +bfgsmin iteration 1 convergence (f g p): 0 0 0 + +function value: 31.5771 stepsize: 0.00165409 + +used analytic gradient + -0.14222 -96.10749 0.03592 + 0.06506 65.21654 -0.04736 + 0.10737 8.38754 0.06630 + 0.13262 2.41493 0.09345 + 0.14293 -5.13573 0.11198 + 0.14044 -12.20696 0.12046 + 0.12729 -17.02499 0.11785 + 0.10560 -18.46283 0.10409 + 0.07751 -16.22190 0.08029 + 0.04515 -10.82911 0.04878 + 0.01067 -3.47246 0.01285 + -0.02382 4.28229 -0.02365 + -0.05618 10.83832 -0.05685 + -0.08426 14.91554 -0.08333 + -0.10595 15.83746 -0.10053 + -0.11911 13.70140 -0.10703 + -0.12160 9.38956 -0.10258 + -0.11129 4.39789 -0.08777 + -0.08604 0.48582 -0.06362 + -0.04373 -0.78168 -0.03084 + 0.00356 0.70324 0.00158 +------------------------------------------------ +bfgsmin iteration 2 convergence (f g p): 0 0 0 + +function value: 23.1595 stepsize: 0.00158028 + +used analytic gradient + -0.10630 -60.62528 0.02658 + 0.01770 -19.38494 0.04778 + 0.17367 60.53681 -0.04140 + 0.22607 29.15174 0.00564 + 0.25491 22.52286 0.01687 + 0.26090 10.67866 0.03060 + 0.24514 -2.76216 0.04305 + 0.20968 -13.44473 0.04948 + 0.15780 -17.85744 0.04621 + 0.09393 -14.81813 0.03239 + 0.02352 -6.07835 0.01084 + -0.04747 4.37845 -0.01284 + -0.11303 12.07714 -0.03245 + -0.16759 13.98041 -0.04355 + -0.20648 9.67143 -0.04499 + -0.22614 1.30606 -0.03878 + -0.22417 -7.47705 -0.02876 + -0.19906 -13.04828 -0.01874 + -0.14966 -13.04999 -0.01091 + -0.07457 -7.41192 -0.00461 + 0.00513 0.90554 0.00002 +------------------------------------------------ +bfgsmin iteration 3 convergence (f g p): 1 0 1 + +function value: 19.6471 stepsize: 0.435579 + +used analytic gradient + -0.07972 -62.20698 0.04419 + 0.06548 33.30355 -0.01009 + 0.13228 -6.37137 -0.03239 + 0.23171 45.92197 -0.07369 + 0.27178 26.44966 -0.06852 + 0.29150 18.85397 -0.06567 + 0.28818 7.28696 -0.05562 + 0.25917 -5.43594 -0.04052 + 0.20400 -14.52432 -0.02409 + 0.12632 -15.48456 -0.01014 + 0.03436 -7.80189 -0.00015 + -0.06032 3.60477 0.00766 + -0.14548 11.50157 0.01649 + -0.21114 11.46823 0.02787 + -0.25147 4.28266 0.04032 + -0.26492 -5.80243 0.05044 + -0.25294 -14.23225 0.05492 + -0.21780 -18.02237 0.05169 + -0.16057 -15.95030 0.03999 + -0.07918 -8.57877 0.02017 + 0.00516 0.86076 -0.00145 +------------------------------------------------ +bfgsmin iteration 4 convergence (f g p): 1 0 1 + +function value: 13.7213 stepsize: 1.14318 + +used analytic gradient + -0.03553 -29.07749 0.06636 + 0.05539 19.44095 0.00506 + 0.09989 -5.47804 0.00191 + 0.15802 6.83697 0.00521 + 0.20327 9.76823 0.00791 + 0.22582 1.48244 0.04141 + 0.23257 -3.68271 0.06391 + 0.21865 -9.27686 0.08193 + 0.17991 -13.43998 0.08726 + 0.11618 -13.17660 0.07068 + 0.03421 -6.77333 0.02979 + -0.05265 3.20740 -0.02316 + -0.12899 11.05492 -0.06634 + -0.18327 12.91806 -0.08478 + -0.21114 9.40742 -0.07908 + -0.21448 3.58559 -0.05945 + -0.19802 -1.83960 -0.03651 + -0.16610 -5.38820 -0.01713 + -0.12058 -6.28669 -0.00471 + -0.05901 -3.89232 0.00019 + 0.00370 0.72416 -0.00135 +------------------------------------------------ +bfgsmin iteration 5 convergence (f g p): 1 0 1 + +function value: 10.5274 stepsize: 1.28732 + +used analytic gradient + 0.03083 21.56713 -0.01910 + 0.06046 -2.64280 -0.00410 + 0.10180 -7.69289 -0.00259 + 0.16323 8.21389 -0.02205 + 0.21117 -0.95188 -0.02318 + 0.26723 14.56577 -0.03991 + 0.29648 9.22979 -0.03415 + 0.30059 5.28404 -0.02748 + 0.26717 -2.31301 -0.01392 + 0.18686 -9.74861 0.00152 + 0.06400 -9.31879 0.00860 + -0.07581 0.28861 0.00479 + -0.19533 8.06118 0.00227 + -0.26805 6.07457 0.00981 + -0.29022 -1.27328 0.02182 + -0.27393 -7.18899 0.03007 + -0.23453 -9.56499 0.03214 + -0.18324 -9.30319 0.02930 + -0.12529 -7.42442 0.02274 + -0.05883 -3.90292 0.01171 + 0.00235 0.45489 -0.00116 +------------------------------------------------ +bfgsmin iteration 6 convergence (f g p): 1 0 1 + +function value: 9.16809 stepsize: 0.979367 + +used analytic gradient + 0.01173 7.00016 -0.00461 + 0.05636 2.01369 -0.00228 + 0.09922 0.19366 -0.00537 + 0.14117 -3.67529 0.00929 + 0.18799 0.61248 -0.00447 + 0.22732 -2.60715 0.01560 + 0.26233 2.50117 0.00433 + 0.27311 -1.16069 0.02123 + 0.25325 -3.60103 0.03070 + 0.18838 -6.54700 0.03651 + 0.07261 -7.07714 0.02927 + -0.07103 -0.62232 -0.00292 + -0.19306 7.02795 -0.03688 + -0.25824 7.11943 -0.04102 + -0.26840 2.63406 -0.02492 + -0.24386 -0.95432 -0.00848 + -0.20239 -2.49441 0.00202 + -0.15393 -2.79980 0.00739 + -0.10254 -2.51403 0.00880 + -0.04711 -1.41931 0.00549 + 0.00119 0.23712 -0.00111 +------------------------------------------------ +bfgsmin iteration 7 convergence (f g p): 1 0 1 + +function value: 8.52604 stepsize: 0.439945 + +used analytic gradient + 0.00712 4.12561 -0.00651 + 0.05408 3.72285 -0.00717 + 0.09384 -7.94163 0.00410 + 0.15046 9.51407 -0.00524 + 0.18352 -13.59227 0.00960 + 0.24292 12.74922 -0.00777 + 0.26666 -7.74448 0.00554 + 0.29434 4.82690 -0.00127 + 0.28395 0.49390 0.00632 + 0.22489 -2.18759 0.01219 + 0.10188 -3.83202 0.01390 + -0.07395 -2.79881 0.00351 + -0.22994 2.58590 -0.01323 + -0.29926 2.38129 -0.01366 + -0.29332 -1.15975 -0.00385 + -0.25235 -2.48362 0.00296 + -0.20037 -2.07616 0.00542 + -0.14654 -1.31041 0.00568 + -0.09375 -0.75989 0.00494 + -0.04162 -0.31482 0.00272 + 0.00008 0.00241 -0.00040 +------------------------------------------------ +bfgsmin iteration 8 convergence (f g p): 1 0 1 + +function value: 8.23001 stepsize: 0.906195 + +used analytic gradient + 0.00061 1.72333 -0.00327 + 0.04691 -2.50689 0.00287 + 0.09794 1.08557 -0.00166 + 0.14523 -1.19777 0.00064 + 0.19312 0.96755 0.00019 + 0.23515 -0.91542 -0.00085 + 0.27220 1.09461 -0.00077 + 0.29307 -0.86920 0.00023 + 0.29027 1.92007 -0.00418 + 0.23708 -0.35894 0.00153 + 0.11578 -1.60121 0.00508 + -0.07045 -3.10674 0.00765 + -0.24317 0.08524 -0.00127 + -0.31292 0.54926 -0.00216 + -0.29718 -1.70534 0.00404 + -0.24939 -1.88015 0.00517 + -0.19495 -0.99916 0.00340 + -0.14086 -0.19968 0.00155 + -0.08880 0.19614 0.00048 + -0.03890 0.22007 0.00000 + -0.00032 -0.08854 0.00013 +------------------------------------------------ +bfgsmin iteration 9 convergence (f g p): 1 0 1 + +function value: 8.19127 stepsize: 1.08858 + +used analytic gradient + -0.00267 -2.04125 0.00076 + 0.04978 2.32889 -0.00196 + 0.09628 -1.97322 0.00228 + 0.14587 0.02692 0.00008 + 0.19331 1.23871 -0.00152 + 0.23431 -1.51432 0.00128 + 0.27143 0.60101 -0.00155 + 0.29330 1.21446 -0.00241 + 0.28609 -2.19106 0.00001 + 0.23861 0.59230 -0.00023 + 0.12086 -1.39499 0.00601 + -0.06280 -1.75263 0.00889 + -0.24444 -1.22880 0.00144 + -0.31508 0.13546 -0.00141 + -0.29314 -0.87320 0.00495 + -0.24422 -0.86169 0.00567 + -0.19156 -0.33678 0.00331 + -0.13931 0.09516 0.00107 + -0.08832 0.28502 -0.00011 + -0.03889 0.21816 -0.00038 + -0.00019 -0.06233 0.00021 +------------------------------------------------ +bfgsmin iteration 10 convergence (f g p): 1 0 1 + +function value: 8.16392 stepsize: 5.68739 + +used analytic gradient + -0.00190 -0.65044 0.00426 + 0.04781 -0.84794 0.00237 + 0.09856 1.04814 -0.00243 + 0.14595 -0.19334 0.00131 + 0.19179 -0.80624 0.00111 + 0.23559 0.94901 -0.00222 + 0.26988 -0.44522 -0.00021 + 0.29089 -0.39024 -0.00092 + 0.28611 -1.19107 0.00434 + 0.23838 -1.46836 0.00487 + 0.12688 -0.45894 0.00728 + -0.05391 -0.60443 0.01234 + -0.24300 -2.21862 0.01274 + -0.31649 -0.30383 0.00124 + -0.28819 0.10806 0.00559 + -0.23856 0.23049 0.00512 + -0.18825 0.29277 0.00190 + -0.13824 0.29457 -0.00068 + -0.08843 0.25977 -0.00179 + -0.03927 0.14185 -0.00134 + 0.00002 -0.01826 0.00036 +------------------------------------------------ +bfgsmin iteration 11 convergence (f g p): 1 0 1 + +function value: 8.13367 stepsize: 2.97746 + +used analytic gradient + 0.00235 1.82238 0.00312 + 0.05018 0.78808 0.00196 + 0.09612 -2.84700 0.00398 + 0.14726 1.64514 -0.00244 + 0.19290 0.65006 0.00016 + 0.23336 -1.58261 0.00060 + 0.26967 0.59063 -0.00205 + 0.28996 -2.78055 0.00666 + 0.29045 1.03772 0.00579 + 0.24325 -1.90091 0.01479 + 0.13415 -1.34231 0.01692 + -0.04157 -0.40629 0.02340 + -0.23025 -1.47743 0.02909 + -0.31525 -0.96746 0.00573 + -0.28260 1.15977 0.00647 + -0.23344 1.19655 0.00523 + -0.18635 0.64247 0.00104 + -0.13891 0.15941 -0.00209 + -0.09023 -0.08664 -0.00329 + -0.04060 -0.12238 -0.00221 + 0.00039 0.05985 0.00049 +------------------------------------------------ +bfgsmin iteration 12 convergence (f g p): 1 0 1 + +function value: 8.07513 stepsize: 3.9278 + +used analytic gradient + 0.00547 3.56653 -0.00698 + 0.05214 -0.09011 0.00616 + 0.10010 1.31771 0.00670 + 0.14482 -2.43966 0.00360 + 0.19307 1.54817 -0.00693 + 0.23396 -0.25721 0.00342 + 0.26762 -4.17441 0.01296 + 0.29662 1.90348 0.01198 + 0.29624 -1.89582 0.02679 + 0.25803 1.17141 0.04036 + 0.15107 -3.13403 0.06704 + -0.01817 -1.68458 0.08023 + -0.20116 0.79498 0.08655 + -0.30952 -2.08981 0.02763 + -0.27613 2.29093 0.00920 + -0.22821 2.16400 0.00591 + -0.18531 0.82122 -0.00072 + -0.14100 -0.24853 -0.00537 + -0.09351 -0.72421 -0.00682 + -0.04281 -0.55917 -0.00424 + 0.00088 0.16597 0.00075 +------------------------------------------------ +bfgsmin iteration 13 convergence (f g p): 1 0 1 + +function value: 7.89915 stepsize: 1 + +used analytic gradient + -0.00150 -4.51716 -0.00082 + 0.05830 6.25712 -0.00289 + 0.10680 4.16889 0.00013 + 0.14842 1.23070 0.00119 + 0.18613 -8.05716 0.00371 + 0.23738 0.72482 0.00181 + 0.28058 2.24598 0.00679 + 0.30860 -2.24915 0.01032 + 0.32303 1.99440 0.01537 + 0.29840 -0.45542 0.02660 + 0.21811 0.14207 0.04385 + 0.06207 -6.44758 0.05879 + -0.11461 4.72937 0.05075 + -0.28189 -5.30540 0.02418 + -0.26693 3.40045 0.00139 + -0.22230 3.23105 -0.00055 + -0.18603 0.64944 -0.00157 + -0.14637 -1.30498 -0.00206 + -0.10033 -2.06939 -0.00199 + -0.04705 -1.40579 -0.00106 + 0.00163 0.32470 0.00008 +------------------------------------------------ +bfgsmin iteration 14 convergence (f g p): 1 0 1 + +function value: 7.65629 stepsize: 0.686701 + +used analytic gradient + -0.00233 -4.01674 -0.00177 + 0.05541 3.59409 -0.00268 + 0.10693 4.98945 -0.00269 + 0.14961 0.89269 0.00273 + 0.18984 -5.62038 0.00496 + 0.23919 -1.65027 0.00829 + 0.28737 4.10938 0.00931 + 0.31892 -1.10363 0.01880 + 0.33840 1.89994 0.02975 + 0.32500 -0.08734 0.05163 + 0.26196 1.75424 0.08332 + 0.12086 -5.14685 0.11526 + -0.06386 3.50817 0.10028 + -0.25771 -6.91246 0.05295 + -0.26554 2.89865 0.00202 + -0.22284 3.12119 -0.00248 + -0.18760 0.35061 -0.00291 + -0.14843 -1.71264 -0.00274 + -0.10232 -2.46776 -0.00224 + -0.04811 -1.62010 -0.00104 + 0.00171 0.34168 -0.00003 +------------------------------------------------ +bfgsmin iteration 15 convergence (f g p): 1 0 1 + +function value: 7.60957 stepsize: 0.0878906 + +used analytic gradient + -0.00410 -4.35082 0.00146 + 0.05272 2.67307 -0.00072 + 0.10424 2.23962 -0.00136 + 0.15234 2.73648 -0.00123 + 0.19480 -5.11457 0.00013 + 0.24748 0.88470 -0.00259 + 0.29668 2.58823 -0.00490 + 0.33772 1.78585 -0.00824 + 0.36815 2.58724 -0.01376 + 0.37663 2.66756 -0.02323 + 0.34528 6.61338 -0.03712 + 0.23612 3.11979 -0.04848 + 0.03643 -0.09857 -0.04616 + -0.20476 -11.64869 -0.01839 + -0.26353 0.80826 -0.00333 + -0.22532 2.63068 -0.00157 + -0.19051 -0.19964 0.00073 + -0.15117 -2.25741 0.00226 + -0.10456 -2.92037 0.00266 + -0.04915 -1.83321 0.00159 + 0.00169 0.33700 -0.00024 +------------------------------------------------ +bfgsmin iteration 16 convergence (f g p): 1 0 1 + +function value: 7.41533 stepsize: 0.820359 + +used analytic gradient + -0.00263 -2.90226 0.00588 + 0.05200 1.90826 -0.00407 + 0.10288 1.64688 -0.00250 + 0.15111 1.99299 -0.00448 + 0.19493 -3.46470 0.00690 + 0.24489 0.20784 -0.00134 + 0.29178 2.02354 -0.00408 + 0.32948 1.08093 -0.00291 + 0.35439 1.76406 -0.00498 + 0.35340 1.19254 -0.00440 + 0.30816 3.02983 -0.00877 + 0.18764 0.33923 -0.00536 + -0.00974 -0.50876 -0.00313 + -0.22315 -7.70309 0.01212 + -0.26686 1.18439 -0.00266 + -0.22689 2.35832 -0.00475 + -0.18978 -0.04625 0.00020 + -0.14892 -1.80104 0.00379 + -0.10190 -2.38287 0.00496 + -0.04756 -1.50929 0.00313 + 0.00144 0.28696 -0.00059 +------------------------------------------------ +bfgsmin iteration 17 convergence (f g p): 1 0 1 + +function value: 7.29262 stepsize: 0.355157 + +used analytic gradient + 0.00324 3.44705 -0.00234 + 0.04793 -3.52336 0.00404 + 0.10038 2.55533 -0.00664 + 0.14663 -4.24460 0.00496 + 0.20183 5.76828 -0.00531 + 0.24355 -2.26044 0.00502 + 0.28770 -0.31176 0.00078 + 0.32657 1.82365 -0.00129 + 0.34942 -0.04595 0.00584 + 0.34900 2.48738 0.00813 + 0.29939 -0.00814 0.01998 + 0.18228 1.11516 0.02985 + -0.01287 -2.06709 0.04066 + -0.21103 -4.52312 0.04069 + -0.26952 0.08239 -0.00000 + -0.23165 1.44744 -0.00734 + -0.18958 0.01731 -0.00034 + -0.14513 -1.04350 0.00490 + -0.09694 -1.39413 0.00669 + -0.04443 -0.87916 0.00428 + 0.00085 0.16449 -0.00084 +------------------------------------------------ +bfgsmin iteration 18 convergence (f g p): 1 0 1 + +function value: 7.11897 stepsize: 0.387671 + +used analytic gradient + 0.00091 -0.05722 -0.00675 + 0.05196 4.11842 -0.00153 + 0.09373 -7.64764 0.00966 + 0.15159 5.48544 -0.00309 + 0.19652 -3.53802 -0.00377 + 0.24857 4.55574 -0.00324 + 0.28849 -1.03127 0.00538 + 0.32528 -2.00822 0.00324 + 0.35526 2.78635 0.00268 + 0.35713 0.01456 0.01093 + 0.31936 1.95859 0.02473 + 0.21213 0.29212 0.04011 + 0.02780 -2.69633 0.06367 + -0.17034 -2.18041 0.05835 + -0.26952 -2.42213 0.00717 + -0.23899 -0.00494 -0.00873 + -0.18992 -0.02677 -0.00040 + -0.14023 -0.08385 0.00596 + -0.09025 -0.08842 0.00810 + -0.04016 -0.03002 0.00512 + 0.00001 -0.01653 -0.00095 +------------------------------------------------ +bfgsmin iteration 19 convergence (f g p): 1 0 1 + +function value: 6.94514 stepsize: 2.53419 + +used analytic gradient + -0.00584 -4.81486 0.00521 + 0.05043 1.40122 -0.00186 + 0.10340 3.84808 -0.00582 + 0.14850 0.03207 0.00093 + 0.19275 -4.74812 0.00505 + 0.24533 0.67978 -0.00177 + 0.29386 4.29842 -0.00689 + 0.32853 -2.05982 0.00531 + 0.35794 -0.15275 0.00251 + 0.36806 -0.54721 0.00767 + 0.34409 3.40331 0.01044 + 0.25224 -2.17717 0.03490 + 0.09147 -0.65798 0.05086 + -0.11199 -0.42017 0.04960 + -0.26235 -5.73809 0.01820 + -0.24772 -1.83103 -0.00395 + -0.19032 -0.10645 -0.00004 + -0.13427 1.04851 0.00328 + -0.08216 1.45058 0.00436 + -0.03504 0.96524 0.00264 + -0.00093 -0.23171 -0.00037 +------------------------------------------------ +bfgsmin iteration 20 convergence (f g p): 1 0 1 + +function value: 6.79893 stepsize: 3.10571 + +used analytic gradient + -0.00063 0.06865 0.00367 + 0.04856 -0.21388 -0.00202 + 0.09757 -1.62449 0.00213 + 0.14943 1.27893 -0.00328 + 0.19780 0.60135 0.00326 + 0.24356 -0.35140 -0.00292 + 0.28698 -3.89689 0.00316 + 0.33384 4.85705 -0.00596 + 0.36045 -2.83190 0.00614 + 0.37573 1.40645 -0.00044 + 0.35454 -3.13736 0.00850 + 0.28714 1.16118 0.01609 + 0.14233 0.28976 0.02990 + -0.06239 0.36434 0.02828 + -0.24416 -7.15621 0.03361 + -0.25167 -2.95451 0.00613 + -0.19036 -0.12628 0.00039 + -0.13099 1.65590 -0.00276 + -0.07780 2.26053 -0.00389 + -0.03240 1.46787 -0.00268 + -0.00131 -0.32307 0.00075 +------------------------------------------------ +bfgsmin iteration 21 convergence (f g p): 1 0 1 + +function value: 6.63207 stepsize: 1.66919 + +used analytic gradient + 0.00304 3.83407 -0.00349 + 0.04654 -4.54566 0.00488 + 0.09970 2.62065 0.00043 + 0.14615 -4.14731 0.00224 + 0.20106 6.35500 -0.00883 + 0.24064 -5.79584 0.00471 + 0.29014 2.72574 -0.00056 + 0.32788 -4.58532 0.00465 + 0.36658 5.47580 0.00490 + 0.37529 -4.48961 0.02583 + 0.36303 -1.79995 0.05399 + 0.30323 -0.28409 0.09944 + 0.17223 2.76649 0.15531 + -0.03411 -2.37542 0.15925 + -0.21054 -3.75942 0.10163 + -0.24554 -2.71805 0.00844 + -0.18997 -0.04245 0.00054 + -0.13375 1.14465 -0.00052 + -0.08169 1.53941 -0.00103 + -0.03508 0.95134 -0.00116 + -0.00056 -0.15653 0.00079 +------------------------------------------------ +bfgsmin iteration 22 convergence (f g p): 1 0 1 + +function value: 6.50365 stepsize: 0.307514 + +used analytic gradient + -0.00045 -0.92799 -0.00085 + 0.05142 1.55785 -0.00010 + 0.10013 0.20377 0.00169 + 0.14838 1.43389 -0.00073 + 0.19223 -5.21443 0.00061 + 0.24534 2.60359 -0.00005 + 0.28958 -1.53872 0.00194 + 0.33253 -1.78661 -0.00143 + 0.37148 -1.69099 -0.00278 + 0.40112 -3.34120 -0.00660 + 0.41703 -1.99033 -0.01475 + 0.40267 1.88161 -0.03307 + 0.32755 15.32208 -0.05301 + 0.12514 -5.29606 -0.04861 + -0.10891 -6.74691 -0.01969 + -0.23711 -6.75195 0.00356 + -0.18943 0.03667 -0.00000 + -0.13427 1.04761 -0.00263 + -0.08272 1.34837 -0.00347 + -0.03624 0.71925 -0.00206 + 0.00023 0.00771 0.00023 +------------------------------------------------ +bfgsmin iteration 23 convergence (f g p): 1 0 1 + +function value: 6.24282 stepsize: 3.3732 + +used analytic gradient + -0.00130 -1.56554 0.00672 + 0.05132 1.11771 -0.00200 + 0.10182 2.23793 -0.00484 + 0.14765 -0.22087 -0.00051 + 0.19285 -4.29563 0.01014 + 0.24529 1.53733 -0.00372 + 0.29151 0.95980 -0.00151 + 0.33110 -2.85378 0.00470 + 0.36870 -1.21267 0.00946 + 0.39453 -2.71530 0.01664 + 0.40228 -0.14777 0.02748 + 0.36960 -0.17686 0.05340 + 0.27453 7.58576 0.06945 + 0.07653 -5.04249 0.10220 + -0.12860 -2.87700 0.08866 + -0.23355 -4.62015 0.02548 + -0.18943 0.02662 0.00072 + -0.13690 0.55247 -0.00656 + -0.08619 0.69234 -0.00904 + -0.03830 0.32437 -0.00588 + 0.00046 0.06483 0.00128 +------------------------------------------------ +bfgsmin iteration 24 convergence (f g p): 1 0 1 + +function value: 6.00496 stepsize: 0.477792 + +used analytic gradient + 0.00542 4.65312 -0.00258 + 0.04933 -1.64034 0.00183 + 0.09698 -1.60799 -0.00195 + 0.14714 -2.83790 0.00491 + 0.20299 7.53060 -0.00501 + 0.24157 -5.59615 0.00676 + 0.29001 -0.91182 -0.00123 + 0.33579 -1.40600 0.00441 + 0.37817 -0.47388 0.00155 + 0.41117 -1.47962 0.00356 + 0.42976 -2.13551 0.00012 + 0.42300 7.78783 -0.01225 + 0.34399 3.23500 -0.01249 + 0.17874 -1.77263 0.00677 + -0.03995 -2.70818 0.02349 + -0.20807 -7.21570 0.02210 + -0.18871 -0.09990 0.00047 + -0.14346 -0.71601 -0.00396 + -0.09523 -1.05590 -0.00527 + -0.04418 -0.83308 -0.00309 + 0.00175 0.34233 0.00031 +------------------------------------------------ +bfgsmin iteration 25 convergence (f g p): 1 0 1 + +function value: 5.76673 stepsize: 0.424707 + +used analytic gradient + 0.00284 1.81972 -0.00254 + 0.05116 2.01699 -0.00259 + 0.09503 -6.23380 0.00757 + 0.15206 4.85833 -0.00485 + 0.19798 -2.17693 0.00256 + 0.24833 3.62200 -0.00230 + 0.28877 -6.55644 0.00956 + 0.34020 2.79814 0.00004 + 0.37971 -2.11144 0.00985 + 0.41472 1.21818 0.00998 + 0.42988 1.16392 0.01714 + 0.41074 2.16135 0.02504 + 0.33150 -2.53226 0.05107 + 0.18550 -0.15700 0.07025 + -0.01646 -0.22112 0.06452 + -0.18597 -5.34090 0.03367 + -0.18824 -0.45654 0.00144 + -0.14742 -1.50575 -0.00177 + -0.10050 -2.10134 -0.00234 + -0.04727 -1.45155 -0.00126 + 0.00205 0.40879 -0.00002 +------------------------------------------------ +bfgsmin iteration 26 convergence (f g p): 1 0 1 + +function value: 5.57919 stepsize: 1.2178 + +used analytic gradient + 0.00030 0.81315 -0.00377 + 0.04857 -2.58989 0.00240 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0.03289 + 0.38580 -2.71638 0.06655 + 0.27518 0.43859 0.09068 + 0.09226 -0.06169 0.09995 + -0.10169 -2.54934 0.07361 + -0.18249 -2.87346 0.01005 + -0.14803 -1.66191 0.00277 + -0.10102 -2.20632 0.00387 + -0.04681 -1.35790 0.00296 + 0.00107 0.21221 -0.00113 +------------------------------------------------ +bfgsmin iteration 28 convergence (f g p): 1 0 1 + +function value: 5.1413 stepsize: 0.597099 + +used analytic gradient + -0.00225 -0.82707 0.00101 + 0.04756 -2.80033 0.00103 + 0.10317 4.41238 -0.00220 + 0.14937 -2.44571 0.00060 + 0.20137 1.00216 0.00047 + 0.25155 -2.19739 0.00018 + 0.30801 6.97457 -0.00371 + 0.35125 -2.55732 -0.00022 + 0.40149 9.17125 -0.00712 + 0.42959 -5.53508 -0.00228 + 0.46168 2.58999 -0.00932 + 0.46897 -4.21435 -0.01065 + 0.45235 4.90054 -0.02357 + 0.36586 3.10050 -0.02858 + 0.19222 -1.05583 -0.01939 + -0.02808 -3.70302 -0.00167 + -0.17244 -6.32629 0.00418 + -0.14525 -1.19821 0.00188 + -0.09715 -1.43577 0.00247 + -0.04386 -0.76030 0.00152 + -0.00006 -0.01863 -0.00024 +------------------------------------------------ +bfgsmin iteration 29 convergence (f g p): 1 0 1 + +function value: 4.9937 stepsize: 2.17946 + +used analytic gradient + -0.00124 -0.43215 -0.00001 + 0.04859 -1.29829 0.00166 + 0.10097 1.55042 -0.00203 + 0.14997 -1.15813 0.00319 + 0.20184 1.16622 -0.00096 + 0.25173 -0.72380 0.00526 + 0.30430 3.22475 0.00040 + 0.35103 1.56389 0.00147 + 0.39437 3.01199 0.00304 + 0.42731 -1.25221 0.00889 + 0.45236 -1.16356 0.02116 + 0.45833 -1.26798 0.03706 + 0.42878 0.26780 0.06896 + 0.33729 -0.67125 0.10962 + 0.17283 -0.33471 0.13612 + -0.02975 -0.74485 0.11489 + -0.16826 -5.36274 0.03146 + -0.14337 -0.87641 0.00609 + -0.09468 -0.94998 0.00775 + -0.04234 -0.45672 0.00535 + -0.00029 -0.06993 -0.00153 +------------------------------------------------ +bfgsmin iteration 30 convergence (f g p): 1 0 1 + +function value: 4.9757 stepsize: 0.2557 + +used analytic gradient + -0.00125 -1.10245 0.00086 + 0.05026 1.18300 -0.00034 + 0.09894 -2.42822 0.00039 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+------------------------------------------------ +bfgsmin iteration 33 convergence (f g p): 1 0 1 + +function value: 4.11409 stepsize: 0.657703 + +used analytic gradient + 0.00031 -1.34372 0.00252 + 0.05399 3.93035 -0.00587 + 0.09990 -3.07964 -0.00065 + 0.15345 6.31359 -0.00719 + 0.19303 -5.74279 -0.00055 + 0.24353 1.45164 0.00222 + 0.28716 -4.99659 0.00683 + 0.33710 -3.01059 0.00411 + 0.38832 0.93546 0.00533 + 0.43140 -4.14058 0.00323 + 0.47597 2.83903 0.01022 + 0.50168 -1.22961 0.01309 + 0.50823 1.03823 0.03343 + 0.47291 -2.34812 0.06444 + 0.38205 2.41192 0.07531 + 0.20759 -2.24953 0.11161 + 0.00744 -0.96288 0.09248 + -0.11916 -4.64238 0.02890 + -0.07754 2.19808 -0.00308 + -0.03296 1.34702 -0.00268 + -0.00056 -0.16945 0.00112 +------------------------------------------------ +bfgsmin iteration 34 convergence (f g p): 1 0 1 + +function value: 3.93492 stepsize: 0.583003 + +used analytic gradient + 0.00284 3.03958 -0.00103 + 0.04812 -2.71622 0.00037 + 0.09926 1.49736 -0.00009 + 0.14626 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+------------------------------------------------ +bfgsmin iteration 37 convergence (f g p): 1 0 1 + +function value: 3.59675 stepsize: 1.01761 + +used analytic gradient + 0.00033 1.31232 -0.00302 + 0.04738 -2.91301 0.00461 + 0.10040 2.81763 -0.00508 + 0.14659 -3.25776 0.00638 + 0.19923 1.70235 -0.00230 + 0.24721 -1.75515 0.00432 + 0.29819 0.79090 -0.00002 + 0.34628 -0.93398 0.00346 + 0.39485 0.20803 0.00202 + 0.44032 -1.02082 0.00440 + 0.48356 0.62864 0.00310 + 0.51715 0.05614 0.00586 + 0.53494 -0.33892 0.01057 + 0.52443 2.49842 0.01344 + 0.45695 -0.21019 0.03054 + 0.31699 0.47060 0.04129 + 0.11030 -2.13691 0.04741 + -0.06561 -4.51163 0.03123 + -0.08394 -0.25260 -0.00044 + -0.03853 0.26389 -0.00208 + 0.00049 0.07220 0.00013 +------------------------------------------------ +bfgsmin iteration 38 convergence (f g p): 1 0 1 + +function value: 3.46447 stepsize: 0.693985 + +used analytic gradient + -0.00269 -2.93985 0.00184 + 0.05198 4.94004 -0.00230 + 0.09532 -6.63311 0.00721 + 0.15297 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+bfgsmin iteration 41 convergence (f g p): 1 0 1 + +function value: 2.95985 stepsize: 0.457506 + +used analytic gradient + 0.00001 -0.93732 0.00197 + 0.05237 1.71490 -0.00283 + 0.10165 -1.07599 0.00248 + 0.15445 1.79921 -0.00441 + 0.20503 1.08421 -0.00074 + 0.25549 0.87292 -0.00376 + 0.30658 2.16220 -0.00223 + 0.35566 0.99421 -0.00477 + 0.40516 2.09347 -0.00269 + 0.45208 1.74382 -0.00629 + 0.49571 0.67623 -0.00248 + 0.53547 0.17296 -0.00513 + 0.56760 1.38862 -0.00721 + 0.58067 -1.82298 -0.00851 + 0.56767 1.64060 -0.01829 + 0.49824 1.23723 -0.02348 + 0.35097 1.86798 -0.02400 + 0.13294 -1.14405 -0.00815 + -0.05896 -9.74720 0.01780 + -0.04667 -1.66341 0.00359 + 0.00006 0.00964 -0.00017 +------------------------------------------------ +bfgsmin iteration 42 convergence (f g p): 1 0 1 + +function value: 2.79608 stepsize: 0.612391 + +used analytic gradient + 0.00199 1.78135 -0.00233 + 0.04954 -2.90645 0.00430 + 0.10413 4.32872 -0.00580 + 0.15003 -3.33958 0.00486 + 0.20430 3.62049 -0.00543 + 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convergence (f g p): 1 0 1 + +function value: 2.02508 stepsize: 1.02581 + +used analytic gradient + 0.00065 1.92187 -0.00173 + 0.04650 -3.84373 0.00091 + 0.10022 3.07125 -0.00345 + 0.14637 -1.61790 -0.00096 + 0.19476 -2.21834 -0.00235 + 0.24610 1.62459 -0.00375 + 0.29141 -4.04191 -0.00244 + 0.34265 -0.99620 -0.00303 + 0.39272 0.15895 -0.00391 + 0.43878 -4.39080 -0.00027 + 0.49039 1.11168 -0.00146 + 0.53443 -1.38155 0.00300 + 0.57434 -2.73820 0.00864 + 0.60893 3.89418 0.01462 + 0.61417 -1.46042 0.03090 + 0.58629 2.48231 0.05038 + 0.49133 -1.10263 0.07899 + 0.32387 -1.78111 0.09783 + 0.12411 -0.48834 0.07954 + -0.01263 -4.47346 0.01894 + -0.00014 -0.30800 0.00027 +------------------------------------------------ +bfgsmin iteration 46 convergence (f g p): 1 0 1 + +function value: 1.96678 stepsize: 0.308731 + +used analytic gradient + -0.00109 0.16353 0.00076 + 0.04741 -0.86351 -0.00037 + 0.09677 -0.36168 0.00167 + 0.14542 -0.25677 0.00016 + 0.19241 -2.66869 0.00220 + 0.24235 -0.19589 0.00184 + 0.28898 -3.74271 0.00315 + 0.33961 -1.46600 0.00231 + 0.38881 -2.38173 0.00307 + 0.43851 -2.52539 0.00120 + 0.48893 -1.40748 0.00162 + 0.53744 -1.27372 -0.00066 + 0.58298 -1.31475 -0.00333 + 0.62355 2.46885 -0.00783 + 0.64507 1.79501 -0.01639 + 0.63667 4.10533 -0.02613 + 0.57032 3.04951 -0.03483 + 0.42170 0.10881 -0.03422 + 0.20365 -1.67121 -0.02467 + 0.00631 -8.99724 0.00048 + 0.00013 -0.60894 0.00031 +------------------------------------------------ +bfgsmin iteration 47 convergence (f g p): 1 0 1 + +function value: 1.82628 stepsize: 2.70162 + +used analytic gradient + -0.00032 0.92364 -0.00204 + 0.04704 -2.20471 0.00280 + 0.09844 1.39269 -0.00446 + 0.14557 -1.64313 0.00285 + 0.19461 -1.28472 -0.00160 + 0.24419 -0.50297 0.00282 + 0.29212 -2.29376 0.00115 + 0.34192 -1.65520 0.00436 + 0.39188 -0.75786 0.00020 + 0.43971 -3.20235 0.00559 + 0.49055 -0.03206 0.00110 + 0.53677 -1.24805 0.00497 + 0.57965 -1.20704 0.00556 + 0.61571 2.64735 0.00415 + 0.62868 -0.22331 0.01164 + 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0.35377 1.88930 0.00048 + 0.40378 1.53057 0.00028 + 0.45189 0.02368 0.00205 + 0.50091 2.15787 -0.00129 + 0.54499 0.91489 -0.00109 + 0.58425 -2.77153 0.00192 + 0.62279 -1.24847 0.00163 + 0.65147 -4.43949 0.00685 + 0.66941 3.57880 -0.00182 + 0.64282 4.41432 -0.00781 + 0.54483 -2.02397 -0.00469 + 0.37672 2.48643 0.00192 + 0.15237 -4.59893 0.02744 + 0.01655 -4.88103 0.01176 +------------------------------------------------ +bfgsmin iteration 51 convergence (f g p): 1 0 1 + +function value: 1.09687 stepsize: 3.43706 + +used analytic gradient + -0.00043 -1.03597 0.00309 + 0.05174 1.80754 -0.00286 + 0.10027 -1.66240 0.00571 + 0.15309 1.80237 -0.00129 + 0.20298 0.56505 0.00153 + 0.25298 0.02512 0.00351 + 0.30444 1.56518 -0.00023 + 0.35426 0.87918 0.00198 + 0.40406 0.79307 0.00218 + 0.45394 2.49771 -0.00206 + 0.49962 0.47701 0.00161 + 0.54390 -0.42186 0.00220 + 0.58618 -1.09759 0.00338 + 0.62442 -3.11515 0.01072 + 0.65832 1.97843 0.00423 + 0.66759 2.23758 0.00970 + 0.63501 0.64728 0.01880 + 0.54014 -2.68923 0.03633 + 0.37865 -0.70654 0.05069 + 0.17981 0.85230 0.05453 + 0.02831 -4.89984 0.02402 +------------------------------------------------ +bfgsmin iteration 52 convergence (f g p): 1 0 1 + +function value: 0.985596 stepsize: 1.74591 + +used analytic gradient + 0.00267 2.59792 -0.00060 + 0.04888 -4.58169 0.00427 + 0.10598 5.75976 -0.00379 + 0.15180 -2.40009 0.00159 + 0.20451 1.21391 -0.00218 + 0.25650 3.05780 -0.00212 + 0.30421 -0.87498 -0.00113 + 0.35624 2.09802 -0.00328 + 0.40624 3.03838 -0.00557 + 0.45188 -1.10237 -0.00361 + 0.50123 2.03604 -0.00343 + 0.54610 -0.21695 0.00366 + 0.58956 -2.89428 0.01151 + 0.63514 4.28614 0.00699 + 0.66254 -1.97662 0.00484 + 0.67730 2.26832 0.00070 + 0.65381 -0.64605 0.01814 + 0.57647 -1.51083 0.04318 + 0.42934 -1.49962 0.05781 + 0.23434 2.63836 0.05439 + 0.05233 -5.70348 0.04833 +------------------------------------------------ +bfgsmin iteration 53 convergence (f g p): 1 0 1 + +function value: 0.788855 stepsize: 1.86942 + +used analytic gradient + 0.00206 0.39702 0.00055 + 0.05315 1.46549 -0.00239 + 0.10219 -0.42477 0.00221 + 0.15339 1.59995 -0.00343 + 0.20233 -0.77281 0.00180 + 0.25438 2.23828 -0.00483 + 0.30308 0.15766 -0.00110 + 0.35296 1.47091 -0.00466 + 0.40067 0.17993 -0.00281 + 0.44828 -1.11325 -0.00037 + 0.49779 -1.42142 0.00308 + 0.54977 0.22241 0.00431 + 0.60106 4.31359 -0.00059 + 0.64212 4.70269 -0.00020 + 0.66738 -0.42550 0.00940 + 0.67800 -5.70161 0.02708 + 0.67194 -1.37014 0.03668 + 0.61965 3.01636 0.05767 + 0.48715 -0.45247 0.09436 + 0.28872 -0.62200 0.09495 + 0.10066 -2.81347 0.06705 +------------------------------------------------ +bfgsmin iteration 54 convergence (f g p): 1 0 1 + +function value: 0.648778 stepsize: 0.568461 + +used analytic gradient + 0.00262 1.80337 -0.00139 + 0.05076 -2.05052 0.00049 + 0.10440 4.14478 -0.00263 + 0.14996 -3.45319 0.00082 + 0.20413 4.35393 -0.00233 + 0.24955 -2.89753 0.00001 + 0.30198 2.85200 -0.00168 + 0.34830 -1.63884 -0.00012 + 0.39787 -0.61797 -0.00027 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3.66861 0.05865 + 0.35571 -3.26733 0.07228 + 0.15579 -1.76156 0.05134 +------------------------------------------------ +bfgsmin iteration 56 convergence (f g p): 1 0 1 + +function value: 0.528891 stepsize: 0.598822 + +used analytic gradient + 0.00124 1.06791 -0.00114 + 0.04981 -1.35475 -0.00042 + 0.10169 1.90618 -0.00134 + 0.14965 -0.77693 -0.00140 + 0.19938 -0.23872 0.00044 + 0.24940 0.81234 -0.00249 + 0.29707 -1.52703 0.00133 + 0.34711 -0.75377 -0.00003 + 0.39758 -0.35078 0.00079 + 0.44772 -1.38106 0.00106 + 0.50017 1.14526 -0.00208 + 0.54985 0.82473 -0.00264 + 0.59738 0.04751 -0.00267 + 0.64348 2.31578 -0.00383 + 0.68050 -1.12275 -0.00190 + 0.71083 -0.34677 -0.00344 + 0.72359 0.84317 -0.00846 + 0.70055 4.04183 -0.01727 + 0.60923 3.67045 -0.02331 + 0.42799 -1.96370 -0.01613 + 0.20713 -4.26991 -0.00280 +------------------------------------------------ +bfgsmin iteration 57 convergence (f g p): 1 0 1 + +function value: 0.457551 stepsize: 0.881098 + +used analytic gradient + 0.00010 0.32159 -0.00064 + 0.04939 -0.78770 0.00035 + 0.10035 1.29379 -0.00227 + 0.14825 -1.81836 0.00175 + 0.19982 1.75593 -0.00360 + 0.24690 -2.38165 0.00286 + 0.29840 0.80457 -0.00214 + 0.34708 -1.62784 0.00210 + 0.39837 0.02706 -0.00030 + 0.44878 0.18624 -0.00103 + 0.49810 -0.29801 -0.00029 + 0.54721 0.07936 -0.00061 + 0.59472 -0.02802 0.00040 + 0.63965 0.33915 0.00165 + 0.67860 -0.09969 0.00357 + 0.70739 0.41632 0.00650 + 0.71514 1.07612 0.01133 + 0.68328 1.63975 0.02009 + 0.58592 0.04937 0.03464 + 0.41187 -1.13856 0.04486 + 0.20433 -1.79889 0.03702 +------------------------------------------------ +bfgsmin iteration 58 convergence (f g p): 1 0 1 + +function value: 0.413963 stepsize: 0.818213 + +used analytic gradient + -0.00055 -0.33431 -0.00023 + 0.04974 0.72900 -0.00319 + 0.09808 -1.81832 0.00126 + 0.15000 2.27845 -0.00448 + 0.19622 -3.67847 0.00291 + 0.24976 2.77059 -0.00214 + 0.29626 -3.31201 0.00474 + 0.34918 1.44073 0.00025 + 0.39808 -0.69586 0.00207 + 0.44775 -0.60415 -0.00107 + 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0.31515 -3.23412 0.09360 +------------------------------------------------ +bfgsmin iteration 60 convergence (f g p): 1 0 1 + +function value: 0.243113 stepsize: 0.473312 + +used analytic gradient + -0.00205 -1.21174 0.00072 + 0.04894 1.10347 -0.00023 + 0.09662 -2.27580 0.00157 + 0.14832 -0.07229 0.00006 + 0.19935 0.05454 0.00103 + 0.24992 -0.38823 -0.00019 + 0.30142 1.14335 0.00001 + 0.35078 0.91885 -0.00086 + 0.39823 -2.02547 0.00082 + 0.44987 2.04096 -0.00027 + 0.49633 -2.20147 0.00126 + 0.54647 -0.50754 0.00058 + 0.59611 0.32125 -0.00019 + 0.64300 -0.51049 -0.00067 + 0.68769 0.85143 -0.00232 + 0.72414 -0.96831 -0.00315 + 0.75060 -0.69511 -0.00556 + 0.75606 1.48057 -0.01055 + 0.71768 2.71311 -0.01769 + 0.60720 3.12324 -0.02219 + 0.40875 -4.63313 -0.01346 +------------------------------------------------ +bfgsmin iteration 61 convergence (f g p): 1 0 1 + +function value: 0.20315 stepsize: 1.52488 + +used analytic gradient + -0.00133 -0.54472 0.00027 + 0.04870 -0.04351 -0.00026 + 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+------------------------------------------------ +bfgsmin iteration 64 convergence (f g p): 1 0 1 + +function value: 0.113519 stepsize: 1.08423 + +used analytic gradient + -0.00008 0.34431 0.00141 + 0.04899 -1.28194 0.00263 + 0.10074 1.45197 0.00070 + 0.14925 -1.65936 0.00220 + 0.20152 1.77794 -0.00196 + 0.25013 -1.21475 0.00059 + 0.30184 1.72733 -0.00294 + 0.35018 -1.03959 -0.00003 + 0.40119 1.29150 -0.00089 + 0.44959 -0.79251 0.00089 + 0.49976 0.63795 0.00063 + 0.54822 -0.38905 0.00110 + 0.59675 -0.27771 0.00173 + 0.64435 0.03187 0.00321 + 0.68905 -0.77157 0.00483 + 0.73027 -0.30264 0.00730 + 0.76265 -0.66582 0.01262 + 0.77912 0.27018 0.02041 + 0.76347 1.04609 0.03451 + 0.69187 2.10541 0.06028 + 0.53904 -2.26629 0.10094 +------------------------------------------------ +bfgsmin iteration 65 convergence (f g p): 1 0 1 + +function value: 0.0611149 stepsize: 0.729938 + +used analytic gradient + 0.00133 0.42272 -0.00013 + 0.05162 0.51012 0.00039 + 0.10144 0.22350 -0.00033 + 0.15145 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+bfgsmin iteration 68 convergence (f g p): 1 0 1 + +function value: 0.0312038 stepsize: 0.930256 + +used analytic gradient + 0.00124 0.77664 -0.00123 + 0.05054 -0.63082 0.00015 + 0.10170 1.49934 -0.00184 + 0.14997 -0.89858 0.00031 + 0.20047 0.72381 -0.00048 + 0.24939 -0.70214 0.00022 + 0.29977 0.39727 -0.00001 + 0.34903 -0.92275 0.00059 + 0.40011 0.71902 -0.00082 + 0.44945 -0.85937 0.00085 + 0.50066 0.73732 -0.00093 + 0.55036 -0.07266 0.00006 + 0.60042 0.06720 0.00016 + 0.65052 0.88727 -0.00065 + 0.69867 0.15221 0.00038 + 0.74576 1.02201 -0.00071 + 0.78816 0.58029 0.00041 + 0.82322 0.90270 0.00133 + 0.84304 0.26488 0.00538 + 0.83613 1.76795 0.01221 + 0.77780 -1.48589 0.02949 +------------------------------------------------ +bfgsmin iteration 69 convergence (f g p): 1 0 1 + +function value: 0.0170922 stepsize: 0.822259 + +used analytic gradient + 0.00001 -0.27120 -0.00005 + 0.05069 0.74728 -0.00110 + 0.09986 -0.53017 0.00011 + 0.15028 0.34351 -0.00049 + 0.19999 0.02970 -0.00025 + 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g p): 1 0 1 + +function value: 2.65241e-12 stepsize: 0.871216 + +used analytic gradient + -0.00000 -0.00001 0.00000 + 0.05000 -0.00001 0.00000 + 0.10000 -0.00000 0.00000 + 0.15000 -0.00002 0.00000 + 0.20000 0.00001 -0.00000 + 0.25000 -0.00003 0.00000 + 0.30000 0.00002 -0.00000 + 0.35000 -0.00000 -0.00000 + 0.40000 0.00001 -0.00000 + 0.45000 -0.00000 0.00000 + 0.50000 -0.00001 0.00000 + 0.55000 0.00001 0.00000 + 0.60000 -0.00000 -0.00000 + 0.65000 0.00000 -0.00000 + 0.70000 -0.00000 -0.00000 + 0.75000 0.00001 -0.00000 + 0.80000 0.00000 0.00000 + 0.85000 0.00000 0.00000 + 0.90000 -0.00000 0.00000 + 0.95000 -0.00002 0.00000 + 1.00000 0.00001 0.00000 +------------------------------------------------ +bfgsmin iteration 112 convergence (f g p): 1 0 1 + +function value: 1.37738e-12 stepsize: 1.77508 + +used analytic gradient + -0.00000 0.00000 0.00000 + 0.05000 -0.00001 0.00000 + 0.10000 -0.00001 0.00000 + 0.15000 -0.00001 0.00000 + 0.20000 -0.00001 0.00000 + 0.25000 -0.00000 0.00000 + 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a/tests/optimization/bfgs_05.py b/tests/optimization/bfgs_05.py new file mode 100755 index 0000000000..8eb3f25ea5 --- /dev/null +++ b/tests/optimization/bfgs_05.py @@ -0,0 +1,52 @@ +# //----------------------------------------------------------- +# // +# // Copyright (C) 2018 by the deal.II authors +# // +# // This file is part of the deal.II library. +# // +# // The deal.II library is free software; you can use it, redistribute +# // it, and/or modify it under the terms of the GNU Lesser General +# // Public License as published by the Free Software Foundation; either +# // version 2.1 of the License, or (at your option) any later version. +# // The full text of the license can be found in the file LICENSE.md at +# // the top level directory of deal.II. +# // +# //--------------------------------------------------------------- + +# A companion to bfgs_05.cc which minimizes the Rosenbrok function using numpy + +import numpy as np +from scipy.optimize import fmin_l_bfgs_b +from scipy.optimize import rosen +from scipy.optimize import rosen_der + +# see https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.fmin_l_bfgs_b.html + +# dimension of Rosenbrok function +dim = 20 + +x0 = np.zeros(dim+1) +one = np.ones(dim+1) +location = np.zeros(dim+1) +for i in range(dim+1): + location[i] = (1.*i)/dim + + +def v_rosen(theta): + return rosen(theta - location + one) + + +def g_rosen(theta): + return rosen_der(theta - location + one) + + +x, min_val, info = fmin_l_bfgs_b(func=v_rosen,x0=x0,fprime=g_rosen,m=3, factr=10) +dx = x - location + +print "{0} iterations".format(info['nit']) +print "function value: {0}".format(min_val) +print "linf_norm = {0}".format(np.linalg.norm(dx,ord=np.inf)) +print "Gradient noorm: {0}".format(np.linalg.norm(g_rosen(x))) +print "function calls: {0}".format(info['funcalls']) +print "Solution:" +print x diff --git a/tests/optimization/bfgs_05b.cc b/tests/optimization/bfgs_05b.cc new file mode 100644 index 0000000000..47efae06d1 --- /dev/null +++ b/tests/optimization/bfgs_05b.cc @@ -0,0 +1,138 @@ +//----------------------------------------------------------- +// +// Copyright (C) 2018 by the deal.II authors +// +// This file is part of the deal.II library. +// +// The deal.II library is free software; you can use it, redistribute +// it, and/or modify it under the terms of the GNU Lesser General +// Public License as published by the Free Software Foundation; either +// version 2.1 of the License, or (at your option) any later version. +// The full text of the license can be found in the file LICENSE.md at +// the top level directory of deal.II. +// +//--------------------------------------------------------------- + +// test limited memory BFGS with Rosenbrock function. +// same as bfgs_05 but tests with default line search function which is the +// same asin bfgs_05.cc and therefore the number of iterations until convergence +// is exactly the same. + +#include + +#include +#include + +#include +#include + +#include +#include + +#include "../tests.h" + +using namespace LineMinimization; + +template +void +test() +{ + auto &out = deallog.get_file_stream(); + out << std::setprecision(5) << std::fixed << std::right; + + typedef Vector VectorType; + + // size of the problem + const unsigned int N = 21; + + // parameters: + const unsigned int itmax = 150; + const double gtol = 1e-5; // gradient tolerance + const unsigned int m_max = 3; + + // solution + VectorType x(N), x_shifted(x), x0(x); + x = 0.; + + // shift minimizer to this point + VectorType location(x); + for (unsigned int i = 0; i < N; ++i) + location(i) = double(i) / (N - 1); + + // see + // https://sourceforge.net/p/octave/optim/ci/default/tree/inst/rosenbrock.m#l26 + const auto rosenbrok = [&](VectorType &x, VectorType &g) { + const unsigned int N = x.size(); + double res = 0.; + g = 0; + for (unsigned int i = 0; i < N; ++i) + { + const double xi2 = x(i) * x(i); + + if (i < N - 1) + { + res += 100. * dealii::Utilities::fixed_power<2>(x(i + 1) - xi2) + + dealii::Utilities::fixed_power<2>(1. - x(i)); + + g(i) += -400. * x(i) * (x(i + 1) - xi2) - 2. * (1. - x(i)); + } + + if (i > 0) + g(i) += 200. * (x(i) - x(i - 1) * x(i - 1)); + } + return res; + }; + + + + unsigned int tot_fun_calls = 0; + const auto func = [&](const VectorType &x, VectorType &g) { + tot_fun_calls++; + for (unsigned int i = 0; i < x.size(); ++i) + x_shifted(i) = x(i) - location(i) + 1.; + + return rosenbrok(x_shifted, g); + }; + + const auto preconditioner = [](VectorType & g, + const FiniteSizeHistory &s, + const FiniteSizeHistory &y) { + if (s.size() > 0) + { + // default preconditioning using the oldest {s,y} pair, see + // lbfgs_recursion() in __bfgsmin.cc of "optim" Octave package. + const unsigned int i = s.size() - 1; + const double yy = y[i] * y[i]; + const double sy = s[i] * y[i]; + Assert(yy > 0 && sy > 0, ExcInternalError()); + g *= sy / yy; + } + }; + + SolverControl solver_control(itmax, gtol, false); + typename SolverBFGS::AdditionalData data(m_max, false); + SolverBFGS solver(solver_control, data); + solver.connect_preconditioner_slot(preconditioner); + solver.solve(func, x); + + deallog << "Limited memory BFGS solution:" << std::endl; + x.print(deallog); + + deallog << "Function value: " << func(x, x0) << std::endl; + + x.add(-1, location); + deallog << "Linf error in solution: " << x.linfty_norm() << std::endl; + + deallog << "function calls: " + << (tot_fun_calls - 1) /*one evaluation above*/ << std::endl; +} + +int +main() +{ + std::ofstream logfile("output"); + deallog << std::setprecision(5); + deallog.attach(logfile); + + test(); +} diff --git a/tests/optimization/bfgs_05b.output b/tests/optimization/bfgs_05b.output new file mode 100644 index 0000000000..f76da983f8 --- /dev/null +++ b/tests/optimization/bfgs_05b.output @@ -0,0 +1,8 @@ + +DEAL:BFGS::Starting value 63.201 +DEAL:BFGS::Convergence step 127 value 8.1397e-06 +DEAL::Limited memory BFGS solution: +DEAL::-4.4833e-09 0.050000 0.10000 0.15000 0.20000 0.25000 0.30000 0.35000 0.40000 0.45000 0.50000 0.55000 0.60000 0.65000 0.70000 0.75000 0.80000 0.85000 0.90000 0.95000 1.0000 +DEAL::Function value: 1.3096e-13 +DEAL::Linf error in solution: 1.6564e-07 +DEAL::function calls: 133