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/* -*- c++ -*- (enables emacs c++ mode) *//*===========================================================================
Copyright (C) 2004-2012 Yves Renard This file is a part of GETFEM++ Getfem++ is free software; you can 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 3 of the License, or (at your option) any later version along with the GCC Runtime Library Exception either version 3.1 or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License and GCC Runtime Library Exception for more details. You should have received a copy of the GNU Lesser General Public License along with this program; if not, write to the Free Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. As a special exception, you may use this file as it is a part of a free software library without restriction. Specifically, if other files instantiate templates or use macros or inline functions from this file, or you compile this file and link it with other files to produce an executable, this file does not by itself cause the resulting executable to be covered by the GNU Lesser General Public License. This exception does not however invalidate any other reasons why the executable file might be covered by the GNU Lesser General Public License. ===========================================================================*/
/**@file gmm_solver_bfgs.h
@author Yves Renard <Yves.Renard@insa-lyon.fr> @date October 14 2004. @brief Implements BFGS (Broyden, Fletcher, Goldfarb, Shanno) algorithm. */#ifndef GMM_BFGS_H
#define GMM_BFGS_H
#include "gmm_kernel.h"
#include "gmm_iter.h"
namespace gmm {
// BFGS algorithm (Broyden, Fletcher, Goldfarb, Shanno)
// Quasi Newton method for optimization problems.
// with Wolfe Line search.
// delta[k] = x[k+1] - x[k]
// gamma[k] = grad f(x[k+1]) - grad f(x[k])
// H[0] = I
// BFGS : zeta[k] = delta[k] - H[k] gamma[k]
// DFP : zeta[k] = H[k] gamma[k]
// tau[k] = gamma[k]^T zeta[k]
// rho[k] = 1 / gamma[k]^T delta[k]
// BFGS : H[k+1] = H[k] + rho[k](zeta[k] delta[k]^T + delta[k] zeta[k]^T)
// - rho[k]^2 tau[k] delta[k] delta[k]^T
// DFP : H[k+1] = H[k] + rho[k] delta[k] delta[k]^T
// - (1/tau[k])zeta[k] zeta[k]^T
// Object representing the inverse of the Hessian
template <typename VECTOR> struct bfgs_invhessian { typedef typename linalg_traits<VECTOR>::value_type T; typedef typename number_traits<T>::magnitude_type R;
std::vector<VECTOR> delta, gamma, zeta; std::vector<T> tau, rho; int version;
template<typename VEC1, typename VEC2> void hmult(const VEC1 &X, VEC2 &Y) { copy(X, Y); for (size_type k = 0 ; k < delta.size(); ++k) { T xdelta = vect_sp(X, delta[k]), xzeta = vect_sp(X, zeta[k]); switch (version) { case 0 : // BFGS
add(scaled(zeta[k], rho[k]*xdelta), Y); add(scaled(delta[k], rho[k]*(xzeta-rho[k]*tau[k]*xdelta)), Y); break; case 1 : // DFP
add(scaled(delta[k], rho[k]*xdelta), Y); add(scaled(zeta[k], -xzeta/tau[k]), Y); break; } } } void restart(void) { delta.resize(0); gamma.resize(0); zeta.resize(0); tau.resize(0); rho.resize(0); } template<typename VECT1, typename VECT2> void update(const VECT1 &deltak, const VECT2 &gammak) { size_type N = vect_size(deltak), k = delta.size(); VECTOR Y(N); hmult(gammak, Y); delta.resize(k+1); gamma.resize(k+1); zeta.resize(k+1); tau.resize(k+1); rho.resize(k+1); resize(delta[k], N); resize(gamma[k], N); resize(zeta[k], N); gmm::copy(deltak, delta[k]); gmm::copy(gammak, gamma[k]); rho[k] = R(1) / vect_sp(deltak, gammak); if (version == 0) add(delta[k], scaled(Y, -1), zeta[k]); else gmm::copy(Y, zeta[k]); tau[k] = vect_sp(gammak, zeta[k]); } bfgs_invhessian(int v = 0) { version = v; } };
template <typename FUNCTION, typename DERIVATIVE, typename VECTOR> void bfgs(FUNCTION f, DERIVATIVE grad, VECTOR &x, int restart, iteration& iter, int version = 0, double lambda_init=0.001, double print_norm=1.0) {
typedef typename linalg_traits<VECTOR>::value_type T; typedef typename number_traits<T>::magnitude_type R; bfgs_invhessian<VECTOR> invhessian(version); VECTOR r(vect_size(x)), d(vect_size(x)), y(vect_size(x)), r2(vect_size(x)); grad(x, r); R lambda = lambda_init, valx = f(x), valy; int nb_restart(0); if (iter.get_noisy() >= 1) cout << "value " << valx / print_norm << " "; while (! iter.finished_vect(r)) {
invhessian.hmult(r, d); gmm::scale(d, T(-1)); // Wolfe Line search
R derivative = gmm::vect_sp(r, d); R lambda_min(0), lambda_max(0), m1 = 0.27, m2 = 0.57; bool unbounded = true, blocked = false, grad_computed = false; for(;;) { add(x, scaled(d, lambda), y); valy = f(y); if (iter.get_noisy() >= 2) { cout.precision(15); cout << "Wolfe line search, lambda = " << lambda << " value = " << valy /print_norm << endl;// << " derivative = " << derivative
// << " lambda min = " << lambda_min << " lambda max = "
// << lambda_max << endl; getchar();
} if (valy <= valx + m1 * lambda * derivative) { grad(y, r2); grad_computed = true; T derivative2 = gmm::vect_sp(r2, d); if (derivative2 >= m2*derivative) break; lambda_min = lambda; } else { lambda_max = lambda; unbounded = false; } if (unbounded) lambda *= R(10); else lambda = (lambda_max + lambda_min) / R(2); if (lambda == lambda_max || lambda == lambda_min) break; // valy <= R(2)*valx replaced by
// valy <= valx + gmm::abs(derivative)*lambda_init
// for compatibility with negative values (08.24.07).
if (valy <= valx + R(2)*gmm::abs(derivative)*lambda && (lambda < R(lambda_init*1E-8) || (!unbounded && lambda_max-lambda_min < R(lambda_init*1E-8)))) { blocked = true; lambda = lambda_init; break; } }
// Rank two update
++iter; if (!grad_computed) grad(y, r2); gmm::add(scaled(r2, -1), r); if (iter.get_iteration() % restart == 0 || blocked) { if (iter.get_noisy() >= 1) cout << "Restart\n"; invhessian.restart(); if (++nb_restart > 10) { if (iter.get_noisy() >= 1) cout << "BFGS is blocked, exiting\n"; return; } } else { invhessian.update(gmm::scaled(d,lambda), gmm::scaled(r,-1)); nb_restart = 0; } copy(r2, r); copy(y, x); valx = valy; if (iter.get_noisy() >= 1) cout << "BFGS value " << valx/print_norm << "\t"; }
}
template <typename FUNCTION, typename DERIVATIVE, typename VECTOR> inline void dfp(FUNCTION f, DERIVATIVE grad, VECTOR &x, int restart, iteration& iter, int version = 1) { bfgs(f, grad, x, restart, iter, version);
}
}
#endif
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