719 lines
26 KiB
C++
719 lines
26 KiB
C++
// Ceres Solver - A fast non-linear least squares minimizer
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// Copyright 2015 Google Inc. All rights reserved.
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// http://ceres-solver.org/
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//
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// Redistribution and use in source and binary forms, with or without
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// modification, are permitted provided that the following conditions are met:
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//
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// * Redistributions of source code must retain the above copyright notice,
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// this list of conditions and the following disclaimer.
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// * Redistributions in binary form must reproduce the above copyright notice,
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// this list of conditions and the following disclaimer in the documentation
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// and/or other materials provided with the distribution.
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// * Neither the name of Google Inc. nor the names of its contributors may be
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// used to endorse or promote products derived from this software without
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// specific prior written permission.
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//
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// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
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// AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
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// IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
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// ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
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// LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
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// CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
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// SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
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// INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
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// CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
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// ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
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// POSSIBILITY OF SUCH DAMAGE.
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//
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// Author: sameeragarwal@google.com (Sameer Agarwal)
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#include "ceres/dogleg_strategy.h"
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#include <cmath>
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#include "Eigen/Dense"
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#include "ceres/array_utils.h"
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#include "ceres/internal/eigen.h"
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#include "ceres/linear_least_squares_problems.h"
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#include "ceres/linear_solver.h"
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#include "ceres/polynomial.h"
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#include "ceres/sparse_matrix.h"
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#include "ceres/trust_region_strategy.h"
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#include "ceres/types.h"
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#include "glog/logging.h"
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namespace ceres {
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namespace internal {
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namespace {
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const double kMaxMu = 1.0;
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const double kMinMu = 1e-8;
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}
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DoglegStrategy::DoglegStrategy(const TrustRegionStrategy::Options& options)
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: linear_solver_(options.linear_solver),
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radius_(options.initial_radius),
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max_radius_(options.max_radius),
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min_diagonal_(options.min_lm_diagonal),
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max_diagonal_(options.max_lm_diagonal),
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mu_(kMinMu),
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min_mu_(kMinMu),
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max_mu_(kMaxMu),
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mu_increase_factor_(10.0),
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increase_threshold_(0.75),
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decrease_threshold_(0.25),
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dogleg_step_norm_(0.0),
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reuse_(false),
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dogleg_type_(options.dogleg_type) {
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CHECK_NOTNULL(linear_solver_);
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CHECK_GT(min_diagonal_, 0.0);
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CHECK_LE(min_diagonal_, max_diagonal_);
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CHECK_GT(max_radius_, 0.0);
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}
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// If the reuse_ flag is not set, then the Cauchy point (scaled
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// gradient) and the new Gauss-Newton step are computed from
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// scratch. The Dogleg step is then computed as interpolation of these
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// two vectors.
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TrustRegionStrategy::Summary DoglegStrategy::ComputeStep(
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const TrustRegionStrategy::PerSolveOptions& per_solve_options,
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SparseMatrix* jacobian,
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const double* residuals,
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double* step) {
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CHECK_NOTNULL(jacobian);
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CHECK_NOTNULL(residuals);
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CHECK_NOTNULL(step);
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const int n = jacobian->num_cols();
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if (reuse_) {
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// Gauss-Newton and gradient vectors are always available, only a
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// new interpolant need to be computed. For the subspace case,
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// the subspace and the two-dimensional model are also still valid.
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switch (dogleg_type_) {
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case TRADITIONAL_DOGLEG:
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ComputeTraditionalDoglegStep(step);
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break;
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case SUBSPACE_DOGLEG:
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ComputeSubspaceDoglegStep(step);
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break;
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}
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TrustRegionStrategy::Summary summary;
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summary.num_iterations = 0;
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summary.termination_type = LINEAR_SOLVER_SUCCESS;
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return summary;
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}
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reuse_ = true;
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// Check that we have the storage needed to hold the various
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// temporary vectors.
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if (diagonal_.rows() != n) {
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diagonal_.resize(n, 1);
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gradient_.resize(n, 1);
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gauss_newton_step_.resize(n, 1);
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}
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// Vector used to form the diagonal matrix that is used to
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// regularize the Gauss-Newton solve and that defines the
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// elliptical trust region
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//
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// || D * step || <= radius_ .
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//
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jacobian->SquaredColumnNorm(diagonal_.data());
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for (int i = 0; i < n; ++i) {
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diagonal_[i] = std::min(std::max(diagonal_[i], min_diagonal_),
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max_diagonal_);
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}
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diagonal_ = diagonal_.array().sqrt();
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ComputeGradient(jacobian, residuals);
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ComputeCauchyPoint(jacobian);
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LinearSolver::Summary linear_solver_summary =
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ComputeGaussNewtonStep(per_solve_options, jacobian, residuals);
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TrustRegionStrategy::Summary summary;
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summary.residual_norm = linear_solver_summary.residual_norm;
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summary.num_iterations = linear_solver_summary.num_iterations;
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summary.termination_type = linear_solver_summary.termination_type;
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if (linear_solver_summary.termination_type == LINEAR_SOLVER_FATAL_ERROR) {
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return summary;
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}
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if (linear_solver_summary.termination_type != LINEAR_SOLVER_FAILURE) {
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switch (dogleg_type_) {
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// Interpolate the Cauchy point and the Gauss-Newton step.
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case TRADITIONAL_DOGLEG:
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ComputeTraditionalDoglegStep(step);
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break;
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// Find the minimum in the subspace defined by the
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// Cauchy point and the (Gauss-)Newton step.
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case SUBSPACE_DOGLEG:
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if (!ComputeSubspaceModel(jacobian)) {
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summary.termination_type = LINEAR_SOLVER_FAILURE;
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break;
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}
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ComputeSubspaceDoglegStep(step);
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break;
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}
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}
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return summary;
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}
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// The trust region is assumed to be elliptical with the
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// diagonal scaling matrix D defined by sqrt(diagonal_).
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// It is implemented by substituting step' = D * step.
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// The trust region for step' is spherical.
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// The gradient, the Gauss-Newton step, the Cauchy point,
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// and all calculations involving the Jacobian have to
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// be adjusted accordingly.
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void DoglegStrategy::ComputeGradient(
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SparseMatrix* jacobian,
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const double* residuals) {
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gradient_.setZero();
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jacobian->LeftMultiply(residuals, gradient_.data());
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gradient_.array() /= diagonal_.array();
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}
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// The Cauchy point is the global minimizer of the quadratic model
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// along the one-dimensional subspace spanned by the gradient.
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void DoglegStrategy::ComputeCauchyPoint(SparseMatrix* jacobian) {
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// alpha * -gradient is the Cauchy point.
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Vector Jg(jacobian->num_rows());
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Jg.setZero();
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// The Jacobian is scaled implicitly by computing J * (D^-1 * (D^-1 * g))
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// instead of (J * D^-1) * (D^-1 * g).
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Vector scaled_gradient =
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(gradient_.array() / diagonal_.array()).matrix();
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jacobian->RightMultiply(scaled_gradient.data(), Jg.data());
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alpha_ = gradient_.squaredNorm() / Jg.squaredNorm();
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}
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// The dogleg step is defined as the intersection of the trust region
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// boundary with the piecewise linear path from the origin to the Cauchy
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// point and then from there to the Gauss-Newton point (global minimizer
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// of the model function). The Gauss-Newton point is taken if it lies
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// within the trust region.
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void DoglegStrategy::ComputeTraditionalDoglegStep(double* dogleg) {
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VectorRef dogleg_step(dogleg, gradient_.rows());
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// Case 1. The Gauss-Newton step lies inside the trust region, and
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// is therefore the optimal solution to the trust-region problem.
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const double gradient_norm = gradient_.norm();
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const double gauss_newton_norm = gauss_newton_step_.norm();
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if (gauss_newton_norm <= radius_) {
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dogleg_step = gauss_newton_step_;
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dogleg_step_norm_ = gauss_newton_norm;
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dogleg_step.array() /= diagonal_.array();
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VLOG(3) << "GaussNewton step size: " << dogleg_step_norm_
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<< " radius: " << radius_;
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return;
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}
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// Case 2. The Cauchy point and the Gauss-Newton steps lie outside
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// the trust region. Rescale the Cauchy point to the trust region
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// and return.
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if (gradient_norm * alpha_ >= radius_) {
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dogleg_step = -(radius_ / gradient_norm) * gradient_;
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dogleg_step_norm_ = radius_;
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dogleg_step.array() /= diagonal_.array();
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VLOG(3) << "Cauchy step size: " << dogleg_step_norm_
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<< " radius: " << radius_;
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return;
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}
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// Case 3. The Cauchy point is inside the trust region and the
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// Gauss-Newton step is outside. Compute the line joining the two
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// points and the point on it which intersects the trust region
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// boundary.
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// a = alpha * -gradient
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// b = gauss_newton_step
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const double b_dot_a = -alpha_ * gradient_.dot(gauss_newton_step_);
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const double a_squared_norm = pow(alpha_ * gradient_norm, 2.0);
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const double b_minus_a_squared_norm =
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a_squared_norm - 2 * b_dot_a + pow(gauss_newton_norm, 2);
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// c = a' (b - a)
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// = alpha * -gradient' gauss_newton_step - alpha^2 |gradient|^2
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const double c = b_dot_a - a_squared_norm;
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const double d = sqrt(c * c + b_minus_a_squared_norm *
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(pow(radius_, 2.0) - a_squared_norm));
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double beta =
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(c <= 0)
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? (d - c) / b_minus_a_squared_norm
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: (radius_ * radius_ - a_squared_norm) / (d + c);
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dogleg_step = (-alpha_ * (1.0 - beta)) * gradient_
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+ beta * gauss_newton_step_;
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dogleg_step_norm_ = dogleg_step.norm();
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dogleg_step.array() /= diagonal_.array();
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VLOG(3) << "Dogleg step size: " << dogleg_step_norm_
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<< " radius: " << radius_;
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}
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// The subspace method finds the minimum of the two-dimensional problem
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//
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// min. 1/2 x' B' H B x + g' B x
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// s.t. || B x ||^2 <= r^2
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//
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// where r is the trust region radius and B is the matrix with unit columns
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// spanning the subspace defined by the steepest descent and Newton direction.
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// This subspace by definition includes the Gauss-Newton point, which is
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// therefore taken if it lies within the trust region.
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void DoglegStrategy::ComputeSubspaceDoglegStep(double* dogleg) {
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VectorRef dogleg_step(dogleg, gradient_.rows());
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// The Gauss-Newton point is inside the trust region if |GN| <= radius_.
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// This test is valid even though radius_ is a length in the two-dimensional
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// subspace while gauss_newton_step_ is expressed in the (scaled)
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// higher dimensional original space. This is because
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//
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// 1. gauss_newton_step_ by definition lies in the subspace, and
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// 2. the subspace basis is orthonormal.
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//
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// As a consequence, the norm of the gauss_newton_step_ in the subspace is
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// the same as its norm in the original space.
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const double gauss_newton_norm = gauss_newton_step_.norm();
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if (gauss_newton_norm <= radius_) {
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dogleg_step = gauss_newton_step_;
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dogleg_step_norm_ = gauss_newton_norm;
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dogleg_step.array() /= diagonal_.array();
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VLOG(3) << "GaussNewton step size: " << dogleg_step_norm_
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<< " radius: " << radius_;
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return;
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}
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// The optimum lies on the boundary of the trust region. The above problem
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// therefore becomes
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//
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// min. 1/2 x^T B^T H B x + g^T B x
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// s.t. || B x ||^2 = r^2
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//
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// Notice the equality in the constraint.
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//
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// This can be solved by forming the Lagrangian, solving for x(y), where
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// y is the Lagrange multiplier, using the gradient of the objective, and
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// putting x(y) back into the constraint. This results in a fourth order
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// polynomial in y, which can be solved using e.g. the companion matrix.
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// See the description of MakePolynomialForBoundaryConstrainedProblem for
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// details. The result is up to four real roots y*, not all of which
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// correspond to feasible points. The feasible points x(y*) have to be
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// tested for optimality.
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if (subspace_is_one_dimensional_) {
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// The subspace is one-dimensional, so both the gradient and
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// the Gauss-Newton step point towards the same direction.
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// In this case, we move along the gradient until we reach the trust
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// region boundary.
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dogleg_step = -(radius_ / gradient_.norm()) * gradient_;
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dogleg_step_norm_ = radius_;
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dogleg_step.array() /= diagonal_.array();
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VLOG(3) << "Dogleg subspace step size (1D): " << dogleg_step_norm_
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<< " radius: " << radius_;
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return;
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}
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Vector2d minimum(0.0, 0.0);
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if (!FindMinimumOnTrustRegionBoundary(&minimum)) {
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// For the positive semi-definite case, a traditional dogleg step
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// is taken in this case.
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LOG(WARNING) << "Failed to compute polynomial roots. "
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<< "Taking traditional dogleg step instead.";
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ComputeTraditionalDoglegStep(dogleg);
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return;
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}
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// Test first order optimality at the minimum.
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// The first order KKT conditions state that the minimum x*
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// has to satisfy either || x* ||^2 < r^2 (i.e. has to lie within
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// the trust region), or
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//
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// (B x* + g) + y x* = 0
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//
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// for some positive scalar y.
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// Here, as it is already known that the minimum lies on the boundary, the
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// latter condition is tested. To allow for small imprecisions, we test if
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// the angle between (B x* + g) and -x* is smaller than acos(0.99).
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// The exact value of the cosine is arbitrary but should be close to 1.
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//
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// This condition should not be violated. If it is, the minimum was not
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// correctly determined.
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const double kCosineThreshold = 0.99;
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const Vector2d grad_minimum = subspace_B_ * minimum + subspace_g_;
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const double cosine_angle = -minimum.dot(grad_minimum) /
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(minimum.norm() * grad_minimum.norm());
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if (cosine_angle < kCosineThreshold) {
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LOG(WARNING) << "First order optimality seems to be violated "
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<< "in the subspace method!\n"
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<< "Cosine of angle between x and B x + g is "
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<< cosine_angle << ".\n"
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<< "Taking a regular dogleg step instead.\n"
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<< "Please consider filing a bug report if this "
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<< "happens frequently or consistently.\n";
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ComputeTraditionalDoglegStep(dogleg);
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return;
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}
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// Create the full step from the optimal 2d solution.
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dogleg_step = subspace_basis_ * minimum;
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dogleg_step_norm_ = radius_;
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dogleg_step.array() /= diagonal_.array();
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VLOG(3) << "Dogleg subspace step size: " << dogleg_step_norm_
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<< " radius: " << radius_;
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}
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// Build the polynomial that defines the optimal Lagrange multipliers.
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// Let the Lagrangian be
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//
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// L(x, y) = 0.5 x^T B x + x^T g + y (0.5 x^T x - 0.5 r^2). (1)
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//
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// Stationary points of the Lagrangian are given by
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//
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// 0 = d L(x, y) / dx = Bx + g + y x (2)
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// 0 = d L(x, y) / dy = 0.5 x^T x - 0.5 r^2 (3)
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//
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// For any given y, we can solve (2) for x as
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//
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// x(y) = -(B + y I)^-1 g . (4)
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//
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// As B + y I is 2x2, we form the inverse explicitly:
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//
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// (B + y I)^-1 = (1 / det(B + y I)) adj(B + y I) (5)
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//
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// where adj() denotes adjugation. This should be safe, as B is positive
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// semi-definite and y is necessarily positive, so (B + y I) is indeed
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// invertible.
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// Plugging (5) into (4) and the result into (3), then dividing by 0.5 we
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// obtain
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//
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// 0 = (1 / det(B + y I))^2 g^T adj(B + y I)^T adj(B + y I) g - r^2
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// (6)
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//
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// or
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//
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// det(B + y I)^2 r^2 = g^T adj(B + y I)^T adj(B + y I) g (7a)
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// = g^T adj(B)^T adj(B) g
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// + 2 y g^T adj(B)^T g + y^2 g^T g (7b)
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//
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// as
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//
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// adj(B + y I) = adj(B) + y I = adj(B)^T + y I . (8)
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//
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// The left hand side can be expressed explicitly using
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//
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// det(B + y I) = det(B) + y tr(B) + y^2 . (9)
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//
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// So (7) is a polynomial in y of degree four.
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// Bringing everything back to the left hand side, the coefficients can
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// be read off as
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//
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// y^4 r^2
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// + y^3 2 r^2 tr(B)
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// + y^2 (r^2 tr(B)^2 + 2 r^2 det(B) - g^T g)
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// + y^1 (2 r^2 det(B) tr(B) - 2 g^T adj(B)^T g)
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// + y^0 (r^2 det(B)^2 - g^T adj(B)^T adj(B) g)
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//
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Vector DoglegStrategy::MakePolynomialForBoundaryConstrainedProblem() const {
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const double detB = subspace_B_.determinant();
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const double trB = subspace_B_.trace();
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const double r2 = radius_ * radius_;
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Matrix2d B_adj;
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B_adj << subspace_B_(1, 1) , -subspace_B_(0, 1),
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-subspace_B_(1, 0) , subspace_B_(0, 0);
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Vector polynomial(5);
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polynomial(0) = r2;
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polynomial(1) = 2.0 * r2 * trB;
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polynomial(2) = r2 * (trB * trB + 2.0 * detB) - subspace_g_.squaredNorm();
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polynomial(3) = -2.0 * (subspace_g_.transpose() * B_adj * subspace_g_
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- r2 * detB * trB);
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polynomial(4) = r2 * detB * detB - (B_adj * subspace_g_).squaredNorm();
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return polynomial;
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}
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// Given a Lagrange multiplier y that corresponds to a stationary point
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// of the Lagrangian L(x, y), compute the corresponding x from the
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// equation
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//
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// 0 = d L(x, y) / dx
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// = B * x + g + y * x
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// = (B + y * I) * x + g
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//
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DoglegStrategy::Vector2d DoglegStrategy::ComputeSubspaceStepFromRoot(
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double y) const {
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const Matrix2d B_i = subspace_B_ + y * Matrix2d::Identity();
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return -B_i.partialPivLu().solve(subspace_g_);
|
|
}
|
|
|
|
// This function evaluates the quadratic model at a point x in the
|
|
// subspace spanned by subspace_basis_.
|
|
double DoglegStrategy::EvaluateSubspaceModel(const Vector2d& x) const {
|
|
return 0.5 * x.dot(subspace_B_ * x) + subspace_g_.dot(x);
|
|
}
|
|
|
|
// This function attempts to solve the boundary-constrained subspace problem
|
|
//
|
|
// min. 1/2 x^T B^T H B x + g^T B x
|
|
// s.t. || B x ||^2 = r^2
|
|
//
|
|
// where B is an orthonormal subspace basis and r is the trust-region radius.
|
|
//
|
|
// This is done by finding the roots of a fourth degree polynomial. If the
|
|
// root finding fails, the function returns false and minimum will be set
|
|
// to (0, 0). If it succeeds, true is returned.
|
|
//
|
|
// In the failure case, another step should be taken, such as the traditional
|
|
// dogleg step.
|
|
bool DoglegStrategy::FindMinimumOnTrustRegionBoundary(Vector2d* minimum) const {
|
|
CHECK_NOTNULL(minimum);
|
|
|
|
// Return (0, 0) in all error cases.
|
|
minimum->setZero();
|
|
|
|
// Create the fourth-degree polynomial that is a necessary condition for
|
|
// optimality.
|
|
const Vector polynomial = MakePolynomialForBoundaryConstrainedProblem();
|
|
|
|
// Find the real parts y_i of its roots (not only the real roots).
|
|
Vector roots_real;
|
|
if (!FindPolynomialRoots(polynomial, &roots_real, NULL)) {
|
|
// Failed to find the roots of the polynomial, i.e. the candidate
|
|
// solutions of the constrained problem. Report this back to the caller.
|
|
return false;
|
|
}
|
|
|
|
// For each root y, compute B x(y) and check for feasibility.
|
|
// Notice that there should always be four roots, as the leading term of
|
|
// the polynomial is r^2 and therefore non-zero. However, as some roots
|
|
// may be complex, the real parts are not necessarily unique.
|
|
double minimum_value = std::numeric_limits<double>::max();
|
|
bool valid_root_found = false;
|
|
for (int i = 0; i < roots_real.size(); ++i) {
|
|
const Vector2d x_i = ComputeSubspaceStepFromRoot(roots_real(i));
|
|
|
|
// Not all roots correspond to points on the trust region boundary.
|
|
// There are at most four candidate solutions. As we are interested
|
|
// in the minimum, it is safe to consider all of them after projecting
|
|
// them onto the trust region boundary.
|
|
if (x_i.norm() > 0) {
|
|
const double f_i = EvaluateSubspaceModel((radius_ / x_i.norm()) * x_i);
|
|
valid_root_found = true;
|
|
if (f_i < minimum_value) {
|
|
minimum_value = f_i;
|
|
*minimum = x_i;
|
|
}
|
|
}
|
|
}
|
|
|
|
return valid_root_found;
|
|
}
|
|
|
|
LinearSolver::Summary DoglegStrategy::ComputeGaussNewtonStep(
|
|
const PerSolveOptions& per_solve_options,
|
|
SparseMatrix* jacobian,
|
|
const double* residuals) {
|
|
const int n = jacobian->num_cols();
|
|
LinearSolver::Summary linear_solver_summary;
|
|
linear_solver_summary.termination_type = LINEAR_SOLVER_FAILURE;
|
|
|
|
// The Jacobian matrix is often quite poorly conditioned. Thus it is
|
|
// necessary to add a diagonal matrix at the bottom to prevent the
|
|
// linear solver from failing.
|
|
//
|
|
// We do this by computing the same diagonal matrix as the one used
|
|
// by Levenberg-Marquardt (other choices are possible), and scaling
|
|
// it by a small constant (independent of the trust region radius).
|
|
//
|
|
// If the solve fails, the multiplier to the diagonal is increased
|
|
// up to max_mu_ by a factor of mu_increase_factor_ every time. If
|
|
// the linear solver is still not successful, the strategy returns
|
|
// with LINEAR_SOLVER_FAILURE.
|
|
//
|
|
// Next time when a new Gauss-Newton step is requested, the
|
|
// multiplier starts out from the last successful solve.
|
|
//
|
|
// When a step is declared successful, the multiplier is decreased
|
|
// by half of mu_increase_factor_.
|
|
|
|
while (mu_ < max_mu_) {
|
|
// Dogleg, as far as I (sameeragarwal) understand it, requires a
|
|
// reasonably good estimate of the Gauss-Newton step. This means
|
|
// that we need to solve the normal equations more or less
|
|
// exactly. This is reflected in the values of the tolerances set
|
|
// below.
|
|
//
|
|
// For now, this strategy should only be used with exact
|
|
// factorization based solvers, for which these tolerances are
|
|
// automatically satisfied.
|
|
//
|
|
// The right way to combine inexact solves with trust region
|
|
// methods is to use Stiehaug's method.
|
|
LinearSolver::PerSolveOptions solve_options;
|
|
solve_options.q_tolerance = 0.0;
|
|
solve_options.r_tolerance = 0.0;
|
|
|
|
lm_diagonal_ = diagonal_ * std::sqrt(mu_);
|
|
solve_options.D = lm_diagonal_.data();
|
|
|
|
// As in the LevenbergMarquardtStrategy, solve Jy = r instead
|
|
// of Jx = -r and later set x = -y to avoid having to modify
|
|
// either jacobian or residuals.
|
|
InvalidateArray(n, gauss_newton_step_.data());
|
|
linear_solver_summary = linear_solver_->Solve(jacobian,
|
|
residuals,
|
|
solve_options,
|
|
gauss_newton_step_.data());
|
|
|
|
if (per_solve_options.dump_format_type == CONSOLE ||
|
|
(per_solve_options.dump_format_type != CONSOLE &&
|
|
!per_solve_options.dump_filename_base.empty())) {
|
|
if (!DumpLinearLeastSquaresProblem(per_solve_options.dump_filename_base,
|
|
per_solve_options.dump_format_type,
|
|
jacobian,
|
|
solve_options.D,
|
|
residuals,
|
|
gauss_newton_step_.data(),
|
|
0)) {
|
|
LOG(ERROR) << "Unable to dump trust region problem."
|
|
<< " Filename base: "
|
|
<< per_solve_options.dump_filename_base;
|
|
}
|
|
}
|
|
|
|
if (linear_solver_summary.termination_type == LINEAR_SOLVER_FATAL_ERROR) {
|
|
return linear_solver_summary;
|
|
}
|
|
|
|
if (linear_solver_summary.termination_type == LINEAR_SOLVER_FAILURE ||
|
|
!IsArrayValid(n, gauss_newton_step_.data())) {
|
|
mu_ *= mu_increase_factor_;
|
|
VLOG(2) << "Increasing mu " << mu_;
|
|
linear_solver_summary.termination_type = LINEAR_SOLVER_FAILURE;
|
|
continue;
|
|
}
|
|
break;
|
|
}
|
|
|
|
if (linear_solver_summary.termination_type != LINEAR_SOLVER_FAILURE) {
|
|
// The scaled Gauss-Newton step is D * GN:
|
|
//
|
|
// - (D^-1 J^T J D^-1)^-1 (D^-1 g)
|
|
// = - D (J^T J)^-1 D D^-1 g
|
|
// = D -(J^T J)^-1 g
|
|
//
|
|
gauss_newton_step_.array() *= -diagonal_.array();
|
|
}
|
|
|
|
return linear_solver_summary;
|
|
}
|
|
|
|
void DoglegStrategy::StepAccepted(double step_quality) {
|
|
CHECK_GT(step_quality, 0.0);
|
|
|
|
if (step_quality < decrease_threshold_) {
|
|
radius_ *= 0.5;
|
|
}
|
|
|
|
if (step_quality > increase_threshold_) {
|
|
radius_ = std::max(radius_, 3.0 * dogleg_step_norm_);
|
|
}
|
|
|
|
// Reduce the regularization multiplier, in the hope that whatever
|
|
// was causing the rank deficiency has gone away and we can return
|
|
// to doing a pure Gauss-Newton solve.
|
|
mu_ = std::max(min_mu_, 2.0 * mu_ / mu_increase_factor_);
|
|
reuse_ = false;
|
|
}
|
|
|
|
void DoglegStrategy::StepRejected(double step_quality) {
|
|
radius_ *= 0.5;
|
|
reuse_ = true;
|
|
}
|
|
|
|
void DoglegStrategy::StepIsInvalid() {
|
|
mu_ *= mu_increase_factor_;
|
|
reuse_ = false;
|
|
}
|
|
|
|
double DoglegStrategy::Radius() const {
|
|
return radius_;
|
|
}
|
|
|
|
bool DoglegStrategy::ComputeSubspaceModel(SparseMatrix* jacobian) {
|
|
// Compute an orthogonal basis for the subspace using QR decomposition.
|
|
Matrix basis_vectors(jacobian->num_cols(), 2);
|
|
basis_vectors.col(0) = gradient_;
|
|
basis_vectors.col(1) = gauss_newton_step_;
|
|
Eigen::ColPivHouseholderQR<Matrix> basis_qr(basis_vectors);
|
|
|
|
switch (basis_qr.rank()) {
|
|
case 0:
|
|
// This should never happen, as it implies that both the gradient
|
|
// and the Gauss-Newton step are zero. In this case, the minimizer should
|
|
// have stopped due to the gradient being too small.
|
|
LOG(ERROR) << "Rank of subspace basis is 0. "
|
|
<< "This means that the gradient at the current iterate is "
|
|
<< "zero but the optimization has not been terminated. "
|
|
<< "You may have found a bug in Ceres.";
|
|
return false;
|
|
|
|
case 1:
|
|
// Gradient and Gauss-Newton step coincide, so we lie on one of the
|
|
// major axes of the quadratic problem. In this case, we simply move
|
|
// along the gradient until we reach the trust region boundary.
|
|
subspace_is_one_dimensional_ = true;
|
|
return true;
|
|
|
|
case 2:
|
|
subspace_is_one_dimensional_ = false;
|
|
break;
|
|
|
|
default:
|
|
LOG(ERROR) << "Rank of the subspace basis matrix is reported to be "
|
|
<< "greater than 2. As the matrix contains only two "
|
|
<< "columns this cannot be true and is indicative of "
|
|
<< "a bug.";
|
|
return false;
|
|
}
|
|
|
|
// The subspace is two-dimensional, so compute the subspace model.
|
|
// Given the basis U, this is
|
|
//
|
|
// subspace_g_ = g_scaled^T U
|
|
//
|
|
// and
|
|
//
|
|
// subspace_B_ = U^T (J_scaled^T J_scaled) U
|
|
//
|
|
// As J_scaled = J * D^-1, the latter becomes
|
|
//
|
|
// subspace_B_ = ((U^T D^-1) J^T) (J (D^-1 U))
|
|
// = (J (D^-1 U))^T (J (D^-1 U))
|
|
|
|
subspace_basis_ =
|
|
basis_qr.householderQ() * Matrix::Identity(jacobian->num_cols(), 2);
|
|
|
|
subspace_g_ = subspace_basis_.transpose() * gradient_;
|
|
|
|
Eigen::Matrix<double, 2, Eigen::Dynamic, Eigen::RowMajor>
|
|
Jb(2, jacobian->num_rows());
|
|
Jb.setZero();
|
|
|
|
Vector tmp;
|
|
tmp = (subspace_basis_.col(0).array() / diagonal_.array()).matrix();
|
|
jacobian->RightMultiply(tmp.data(), Jb.row(0).data());
|
|
tmp = (subspace_basis_.col(1).array() / diagonal_.array()).matrix();
|
|
jacobian->RightMultiply(tmp.data(), Jb.row(1).data());
|
|
|
|
subspace_B_ = Jb * Jb.transpose();
|
|
|
|
return true;
|
|
}
|
|
|
|
} // namespace internal
|
|
} // namespace ceres
|