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ceres-solver / ceres-solver / faa72ace9abea24877173158bfec451d5b46597e / . / internal / ceres / solver_impl.cc
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// Ceres Solver - A fast non-linear least squares minimizer
// Copyright 2010, 2011, 2012 Google Inc. All rights reserved.
// http://code.google.com/p/ceres-solver/
//
// Redistribution and use in source and binary forms, with or without
// modification, are permitted provided that the following conditions are met:
//
// * Redistributions of source code must retain the above copyright notice,
// this list of conditions and the following disclaimer.
// * Redistributions in binary form must reproduce the above copyright notice,
// this list of conditions and the following disclaimer in the documentation
// and/or other materials provided with the distribution.
// * Neither the name of Google Inc. nor the names of its contributors may be
// used to endorse or promote products derived from this software without
// specific prior written permission.
//
// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
// AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
// IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
// ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
// LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
// CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
// SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
// INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
// CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
// ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
// POSSIBILITY OF SUCH DAMAGE.
//
// Author: keir@google.com (Keir Mierle)
#include "ceres/solver_impl.h"
#include <cstdio>
#include <iostream> // NOLINT
#include <numeric>
#include "ceres/coordinate_descent_minimizer.h"
#include "ceres/evaluator.h"
#include "ceres/gradient_checking_cost_function.h"
#include "ceres/iteration_callback.h"
#include "ceres/levenberg_marquardt_strategy.h"
#include "ceres/linear_solver.h"
#include "ceres/line_search_minimizer.h"
#include "ceres/map_util.h"
#include "ceres/minimizer.h"
#include "ceres/ordered_groups.h"
#include "ceres/parameter_block.h"
#include "ceres/parameter_block_ordering.h"
#include "ceres/problem.h"
#include "ceres/problem_impl.h"
#include "ceres/program.h"
#include "ceres/residual_block.h"
#include "ceres/stringprintf.h"
#include "ceres/trust_region_minimizer.h"
#include "ceres/wall_time.h"
namespace ceres {
namespace internal {
namespace {
// Callback for updating the user's parameter blocks. Updates are only
// done if the step is successful.
class StateUpdatingCallback : public IterationCallback {
public:
StateUpdatingCallback(Program* program, double* parameters)
: program_(program), parameters_(parameters) {}
CallbackReturnType operator()(const IterationSummary& summary) {
if (summary.step_is_successful) {
program_->StateVectorToParameterBlocks(parameters_);
program_->CopyParameterBlockStateToUserState();
}
return SOLVER_CONTINUE;
}
private:
Program* program_;
double* parameters_;
};
void SetSummaryFinalCost(Solver::Summary* summary) {
summary->final_cost = summary->initial_cost;
// We need the loop here, instead of just looking at the last
// iteration because the minimizer maybe making non-monotonic steps.
for (int i = 0; i < summary->iterations.size(); ++i) {
const IterationSummary& iteration_summary = summary->iterations[i];
summary->final_cost = min(iteration_summary.cost, summary->final_cost);
}
}
// Callback for logging the state of the minimizer to STDERR or STDOUT
// depending on the user's preferences and logging level.
class TrustRegionLoggingCallback : public IterationCallback {
public:
explicit TrustRegionLoggingCallback(bool log_to_stdout)
: log_to_stdout_(log_to_stdout) {}
~TrustRegionLoggingCallback() {}
CallbackReturnType operator()(const IterationSummary& summary) {
const char* kReportRowFormat =
"% 4d: f:% 8e d:% 3.2e g:% 3.2e h:% 3.2e "
"rho:% 3.2e mu:% 3.2e li:% 3d it:% 3.2e tt:% 3.2e";
string output = StringPrintf(kReportRowFormat,
summary.iteration,
summary.cost,
summary.cost_change,
summary.gradient_max_norm,
summary.step_norm,
summary.relative_decrease,
summary.trust_region_radius,
summary.linear_solver_iterations,
summary.iteration_time_in_seconds,
summary.cumulative_time_in_seconds);
if (log_to_stdout_) {
cout << output << endl;
} else {
VLOG(1) << output;
}
return SOLVER_CONTINUE;
}
private:
const bool log_to_stdout_;
};
// Callback for logging the state of the minimizer to STDERR or STDOUT
// depending on the user's preferences and logging level.
class LineSearchLoggingCallback : public IterationCallback {
public:
explicit LineSearchLoggingCallback(bool log_to_stdout)
: log_to_stdout_(log_to_stdout) {}
~LineSearchLoggingCallback() {}
CallbackReturnType operator()(const IterationSummary& summary) {
const char* kReportRowFormat =
"% 4d: f:% 8e d:% 3.2e g:% 3.2e h:% 3.2e "
"s:% 3.2e e:% 3d it:% 3.2e tt:% 3.2e";
string output = StringPrintf(kReportRowFormat,
summary.iteration,
summary.cost,
summary.cost_change,
summary.gradient_max_norm,
summary.step_norm,
summary.step_size,
summary.line_search_function_evaluations,
summary.iteration_time_in_seconds,
summary.cumulative_time_in_seconds);
if (log_to_stdout_) {
cout << output << endl;
} else {
VLOG(1) << output;
}
return SOLVER_CONTINUE;
}
private:
const bool log_to_stdout_;
};
// Basic callback to record the execution of the solver to a file for
// offline analysis.
class FileLoggingCallback : public IterationCallback {
public:
explicit FileLoggingCallback(const string& filename)
: fptr_(NULL) {
fptr_ = fopen(filename.c_str(), "w");
CHECK_NOTNULL(fptr_);
}
virtual ~FileLoggingCallback() {
if (fptr_ != NULL) {
fclose(fptr_);
}
}
virtual CallbackReturnType operator()(const IterationSummary& summary) {
fprintf(fptr_,
"%4d %e %e\n",
summary.iteration,
summary.cost,
summary.cumulative_time_in_seconds);
return SOLVER_CONTINUE;
}
private:
FILE* fptr_;
};
// Iterate over each of the groups in order of their priority and fill
// summary with their sizes.
void SummarizeOrdering(ParameterBlockOrdering* ordering,
vector<int>* summary) {
CHECK_NOTNULL(summary)->clear();
if (ordering == NULL) {
return;
}
const map<int, set<double*> >& group_to_elements =
ordering->group_to_elements();
for (map<int, set<double*> >::const_iterator it = group_to_elements.begin();
it != group_to_elements.end();
++it) {
summary->push_back(it->second.size());
}
}
} // namespace
void SolverImpl::TrustRegionMinimize(
const Solver::Options& options,
Program* program,
CoordinateDescentMinimizer* inner_iteration_minimizer,
Evaluator* evaluator,
LinearSolver* linear_solver,
double* parameters,
Solver::Summary* summary) {
Minimizer::Options minimizer_options(options);
// TODO(sameeragarwal): Add support for logging the configuration
// and more detailed stats.
scoped_ptr<IterationCallback> file_logging_callback;
if (!options.solver_log.empty()) {
file_logging_callback.reset(new FileLoggingCallback(options.solver_log));
minimizer_options.callbacks.insert(minimizer_options.callbacks.begin(),
file_logging_callback.get());
}
TrustRegionLoggingCallback logging_callback(
options.minimizer_progress_to_stdout);
if (options.logging_type != SILENT) {
minimizer_options.callbacks.insert(minimizer_options.callbacks.begin(),
&logging_callback);
}
StateUpdatingCallback updating_callback(program, parameters);
if (options.update_state_every_iteration) {
// This must get pushed to the front of the callbacks so that it is run
// before any of the user callbacks.
minimizer_options.callbacks.insert(minimizer_options.callbacks.begin(),
&updating_callback);
}
minimizer_options.evaluator = evaluator;
scoped_ptr<SparseMatrix> jacobian(evaluator->CreateJacobian());
minimizer_options.jacobian = jacobian.get();
minimizer_options.inner_iteration_minimizer = inner_iteration_minimizer;
TrustRegionStrategy::Options trust_region_strategy_options;
trust_region_strategy_options.linear_solver = linear_solver;
trust_region_strategy_options.initial_radius =
options.initial_trust_region_radius;
trust_region_strategy_options.max_radius = options.max_trust_region_radius;
trust_region_strategy_options.lm_min_diagonal = options.lm_min_diagonal;
trust_region_strategy_options.lm_max_diagonal = options.lm_max_diagonal;
trust_region_strategy_options.trust_region_strategy_type =
options.trust_region_strategy_type;
trust_region_strategy_options.dogleg_type = options.dogleg_type;
scoped_ptr<TrustRegionStrategy> strategy(
TrustRegionStrategy::Create(trust_region_strategy_options));
minimizer_options.trust_region_strategy = strategy.get();
TrustRegionMinimizer minimizer;
double minimizer_start_time = WallTimeInSeconds();
minimizer.Minimize(minimizer_options, parameters, summary);
summary->minimizer_time_in_seconds =
WallTimeInSeconds() - minimizer_start_time;
}
#ifndef CERES_NO_LINE_SEARCH_MINIMIZER
void SolverImpl::LineSearchMinimize(
const Solver::Options& options,
Program* program,
Evaluator* evaluator,
double* parameters,
Solver::Summary* summary) {
Minimizer::Options minimizer_options(options);
// TODO(sameeragarwal): Add support for logging the configuration
// and more detailed stats.
scoped_ptr<IterationCallback> file_logging_callback;
if (!options.solver_log.empty()) {
file_logging_callback.reset(new FileLoggingCallback(options.solver_log));
minimizer_options.callbacks.insert(minimizer_options.callbacks.begin(),
file_logging_callback.get());
}
LineSearchLoggingCallback logging_callback(
options.minimizer_progress_to_stdout);
if (options.logging_type != SILENT) {
minimizer_options.callbacks.insert(minimizer_options.callbacks.begin(),
&logging_callback);
}
StateUpdatingCallback updating_callback(program, parameters);
if (options.update_state_every_iteration) {
// This must get pushed to the front of the callbacks so that it is run
// before any of the user callbacks.
minimizer_options.callbacks.insert(minimizer_options.callbacks.begin(),
&updating_callback);
}
minimizer_options.evaluator = evaluator;
LineSearchMinimizer minimizer;
double minimizer_start_time = WallTimeInSeconds();
minimizer.Minimize(minimizer_options, parameters, summary);
summary->minimizer_time_in_seconds =
WallTimeInSeconds() - minimizer_start_time;
}
#endif // CERES_NO_LINE_SEARCH_MINIMIZER
void SolverImpl::Solve(const Solver::Options& options,
ProblemImpl* problem_impl,
Solver::Summary* summary) {
if (options.minimizer_type == TRUST_REGION) {
TrustRegionSolve(options, problem_impl, summary);
} else {
#ifndef CERES_NO_LINE_SEARCH_MINIMIZER
LineSearchSolve(options, problem_impl, summary);
#else
LOG(FATAL) << "Ceres Solver was compiled with -DLINE_SEARCH_MINIMIZER=OFF";
#endif
}
}
void SolverImpl::TrustRegionSolve(const Solver::Options& original_options,
ProblemImpl* original_problem_impl,
Solver::Summary* summary) {
EventLogger event_logger("TrustRegionSolve");
double solver_start_time = WallTimeInSeconds();
Program* original_program = original_problem_impl->mutable_program();
ProblemImpl* problem_impl = original_problem_impl;
// Reset the summary object to its default values.
*CHECK_NOTNULL(summary) = Solver::Summary();
summary->minimizer_type = TRUST_REGION;
summary->num_parameter_blocks = problem_impl->NumParameterBlocks();
summary->num_parameters = problem_impl->NumParameters();
summary->num_effective_parameters =
original_program->NumEffectiveParameters();
summary->num_residual_blocks = problem_impl->NumResidualBlocks();
summary->num_residuals = problem_impl->NumResiduals();
// Empty programs are usually a user error.
if (summary->num_parameter_blocks == 0) {
summary->error = "Problem contains no parameter blocks.";
LOG(ERROR) << summary->error;
return;
}
if (summary->num_residual_blocks == 0) {
summary->error = "Problem contains no residual blocks.";
LOG(ERROR) << summary->error;
return;
}
SummarizeOrdering(original_options.linear_solver_ordering,
&(summary->linear_solver_ordering_given));
SummarizeOrdering(original_options.inner_iteration_ordering,
&(summary->inner_iteration_ordering_given));
Solver::Options options(original_options);
options.linear_solver_ordering = NULL;
options.inner_iteration_ordering = NULL;
#ifndef CERES_USE_OPENMP
if (options.num_threads > 1) {
LOG(WARNING)
<< "OpenMP support is not compiled into this binary; "
<< "only options.num_threads=1 is supported. Switching "
<< "to single threaded mode.";
options.num_threads = 1;
}
if (options.num_linear_solver_threads > 1) {
LOG(WARNING)
<< "OpenMP support is not compiled into this binary; "
<< "only options.num_linear_solver_threads=1 is supported. Switching "
<< "to single threaded mode.";
options.num_linear_solver_threads = 1;
}
#endif
summary->num_threads_given = original_options.num_threads;
summary->num_threads_used = options.num_threads;
if (options.lsqp_iterations_to_dump.size() > 0) {
LOG(WARNING) << "Dumping linear least squares problems to disk is"
" currently broken. Ignoring Solver::Options::lsqp_iterations_to_dump";
}
event_logger.AddEvent("Init");
original_program->SetParameterBlockStatePtrsToUserStatePtrs();
event_logger.AddEvent("SetParameterBlockPtrs");
// If the user requests gradient checking, construct a new
// ProblemImpl by wrapping the CostFunctions of problem_impl inside
// GradientCheckingCostFunction and replacing problem_impl with
// gradient_checking_problem_impl.
scoped_ptr<ProblemImpl> gradient_checking_problem_impl;
if (options.check_gradients) {
VLOG(1) << "Checking Gradients";
gradient_checking_problem_impl.reset(
CreateGradientCheckingProblemImpl(
problem_impl,
options.numeric_derivative_relative_step_size,
options.gradient_check_relative_precision));
// From here on, problem_impl will point to the gradient checking
// version.
problem_impl = gradient_checking_problem_impl.get();
}
if (original_options.linear_solver_ordering != NULL) {
if (!IsOrderingValid(original_options, problem_impl, &summary->error)) {
LOG(ERROR) << summary->error;
return;
}
event_logger.AddEvent("CheckOrdering");
options.linear_solver_ordering =
new ParameterBlockOrdering(*original_options.linear_solver_ordering);
event_logger.AddEvent("CopyOrdering");
} else {
options.linear_solver_ordering = new ParameterBlockOrdering;
const ProblemImpl::ParameterMap& parameter_map =
problem_impl->parameter_map();
for (ProblemImpl::ParameterMap::const_iterator it = parameter_map.begin();
it != parameter_map.end();
++it) {
options.linear_solver_ordering->AddElementToGroup(it->first, 0);
}
event_logger.AddEvent("ConstructOrdering");
}
// Create the three objects needed to minimize: the transformed program, the
// evaluator, and the linear solver.
scoped_ptr<Program> reduced_program(CreateReducedProgram(&options,
problem_impl,
&summary->fixed_cost,
&summary->error));
event_logger.AddEvent("CreateReducedProgram");
if (reduced_program == NULL) {
return;
}
SummarizeOrdering(options.linear_solver_ordering,
&(summary->linear_solver_ordering_used));
summary->num_parameter_blocks_reduced = reduced_program->NumParameterBlocks();
summary->num_parameters_reduced = reduced_program->NumParameters();
summary->num_effective_parameters_reduced =
reduced_program->NumEffectiveParameters();
summary->num_residual_blocks_reduced = reduced_program->NumResidualBlocks();
summary->num_residuals_reduced = reduced_program->NumResiduals();
if (summary->num_parameter_blocks_reduced == 0) {
summary->preprocessor_time_in_seconds =
WallTimeInSeconds() - solver_start_time;
double post_process_start_time = WallTimeInSeconds();
LOG(INFO) << "Terminating: FUNCTION_TOLERANCE reached. "
<< "No non-constant parameter blocks found.";
summary->initial_cost = summary->fixed_cost;
summary->final_cost = summary->fixed_cost;
// FUNCTION_TOLERANCE is the right convergence here, as we know
// that the objective function is constant and cannot be changed
// any further.
summary->termination_type = FUNCTION_TOLERANCE;
// Ensure the program state is set to the user parameters on the way out.
original_program->SetParameterBlockStatePtrsToUserStatePtrs();
summary->postprocessor_time_in_seconds =
WallTimeInSeconds() - post_process_start_time;
return;
}
scoped_ptr<LinearSolver>
linear_solver(CreateLinearSolver(&options, &summary->error));
event_logger.AddEvent("CreateLinearSolver");
if (linear_solver == NULL) {
return;
}
summary->linear_solver_type_given = original_options.linear_solver_type;
summary->linear_solver_type_used = options.linear_solver_type;
summary->preconditioner_type = options.preconditioner_type;
summary->num_linear_solver_threads_given =
original_options.num_linear_solver_threads;
summary->num_linear_solver_threads_used = options.num_linear_solver_threads;
summary->sparse_linear_algebra_library =
options.sparse_linear_algebra_library;
summary->trust_region_strategy_type = options.trust_region_strategy_type;
summary->dogleg_type = options.dogleg_type;
// Only Schur types require the lexicographic reordering.
if (IsSchurType(options.linear_solver_type)) {
const int num_eliminate_blocks =
options.linear_solver_ordering
->group_to_elements().begin()
->second.size();
if (!LexicographicallyOrderResidualBlocks(num_eliminate_blocks,
reduced_program.get(),
&summary->error)) {
return;
}
}
scoped_ptr<Evaluator> evaluator(CreateEvaluator(options,
problem_impl->parameter_map(),
reduced_program.get(),
&summary->error));
event_logger.AddEvent("CreateEvaluator");
if (evaluator == NULL) {
return;
}
scoped_ptr<CoordinateDescentMinimizer> inner_iteration_minimizer;
if (options.use_inner_iterations) {
if (reduced_program->parameter_blocks().size() < 2) {
LOG(WARNING) << "Reduced problem only contains one parameter block."
<< "Disabling inner iterations.";
} else {
inner_iteration_minimizer.reset(
CreateInnerIterationMinimizer(original_options,
*reduced_program,
problem_impl->parameter_map(),
summary));
if (inner_iteration_minimizer == NULL) {
LOG(ERROR) << summary->error;
return;
}
}
}
event_logger.AddEvent("CreateIIM");
// The optimizer works on contiguous parameter vectors; allocate some.
Vector parameters(reduced_program->NumParameters());
// Collect the discontiguous parameters into a contiguous state vector.
reduced_program->ParameterBlocksToStateVector(parameters.data());
Vector original_parameters = parameters;
double minimizer_start_time = WallTimeInSeconds();
summary->preprocessor_time_in_seconds =
minimizer_start_time - solver_start_time;
// Run the optimization.
TrustRegionMinimize(options,
reduced_program.get(),
inner_iteration_minimizer.get(),
evaluator.get(),
linear_solver.get(),
parameters.data(),
summary);
event_logger.AddEvent("Minimize");
SetSummaryFinalCost(summary);
// If the user aborted mid-optimization or the optimization
// terminated because of a numerical failure, then return without
// updating user state.
if (summary->termination_type == USER_ABORT ||
summary->termination_type == NUMERICAL_FAILURE) {
return;
}
double post_process_start_time = WallTimeInSeconds();
// Push the contiguous optimized parameters back to the user's
// parameters.
reduced_program->StateVectorToParameterBlocks(parameters.data());
reduced_program->CopyParameterBlockStateToUserState();
// Ensure the program state is set to the user parameters on the way
// out.
original_program->SetParameterBlockStatePtrsToUserStatePtrs();
const map<string, double>& linear_solver_time_statistics =
linear_solver->TimeStatistics();
summary->linear_solver_time_in_seconds =
FindWithDefault(linear_solver_time_statistics,
"LinearSolver::Solve",
0.0);
const map<string, double>& evaluator_time_statistics =
evaluator->TimeStatistics();
summary->residual_evaluation_time_in_seconds =
FindWithDefault(evaluator_time_statistics, "Evaluator::Residual", 0.0);
summary->jacobian_evaluation_time_in_seconds =
FindWithDefault(evaluator_time_statistics, "Evaluator::Jacobian", 0.0);
// Stick a fork in it, we're done.
summary->postprocessor_time_in_seconds =
WallTimeInSeconds() - post_process_start_time;
event_logger.AddEvent("PostProcess");
}
#ifndef CERES_NO_LINE_SEARCH_MINIMIZER
void SolverImpl::LineSearchSolve(const Solver::Options& original_options,
ProblemImpl* original_problem_impl,
Solver::Summary* summary) {
double solver_start_time = WallTimeInSeconds();
Program* original_program = original_problem_impl->mutable_program();
ProblemImpl* problem_impl = original_problem_impl;
// Reset the summary object to its default values.
*CHECK_NOTNULL(summary) = Solver::Summary();
summary->minimizer_type = LINE_SEARCH;
summary->line_search_direction_type =
original_options.line_search_direction_type;
summary->max_lbfgs_rank = original_options.max_lbfgs_rank;
summary->line_search_type = original_options.line_search_type;
summary->num_parameter_blocks = problem_impl->NumParameterBlocks();
summary->num_parameters = problem_impl->NumParameters();
summary->num_residual_blocks = problem_impl->NumResidualBlocks();
summary->num_residuals = problem_impl->NumResiduals();
// Empty programs are usually a user error.
if (summary->num_parameter_blocks == 0) {
summary->error = "Problem contains no parameter blocks.";
LOG(ERROR) << summary->error;
return;
}
if (summary->num_residual_blocks == 0) {
summary->error = "Problem contains no residual blocks.";
LOG(ERROR) << summary->error;
return;
}
Solver::Options options(original_options);
// This ensures that we get a Block Jacobian Evaluator along with
// none of the Schur nonsense. This file will have to be extensively
// refactored to deal with the various bits of cleanups related to
// line search.
options.linear_solver_type = CGNR;
options.linear_solver_ordering = NULL;
options.inner_iteration_ordering = NULL;
#ifndef CERES_USE_OPENMP
if (options.num_threads > 1) {
LOG(WARNING)
<< "OpenMP support is not compiled into this binary; "
<< "only options.num_threads=1 is supported. Switching "
<< "to single threaded mode.";
options.num_threads = 1;
}
#endif // CERES_USE_OPENMP
summary->num_threads_given = original_options.num_threads;
summary->num_threads_used = options.num_threads;
if (original_options.linear_solver_ordering != NULL) {
if (!IsOrderingValid(original_options, problem_impl, &summary->error)) {
LOG(ERROR) << summary->error;
return;
}
options.linear_solver_ordering =
new ParameterBlockOrdering(*original_options.linear_solver_ordering);
} else {
options.linear_solver_ordering = new ParameterBlockOrdering;
const ProblemImpl::ParameterMap& parameter_map =
problem_impl->parameter_map();
for (ProblemImpl::ParameterMap::const_iterator it = parameter_map.begin();
it != parameter_map.end();
++it) {
options.linear_solver_ordering->AddElementToGroup(it->first, 0);
}
}
original_program->SetParameterBlockStatePtrsToUserStatePtrs();
// If the user requests gradient checking, construct a new
// ProblemImpl by wrapping the CostFunctions of problem_impl inside
// GradientCheckingCostFunction and replacing problem_impl with
// gradient_checking_problem_impl.
scoped_ptr<ProblemImpl> gradient_checking_problem_impl;
if (options.check_gradients) {
VLOG(1) << "Checking Gradients";
gradient_checking_problem_impl.reset(
CreateGradientCheckingProblemImpl(
problem_impl,
options.numeric_derivative_relative_step_size,
options.gradient_check_relative_precision));
// From here on, problem_impl will point to the gradient checking
// version.
problem_impl = gradient_checking_problem_impl.get();
}
// Create the three objects needed to minimize: the transformed program, the
// evaluator, and the linear solver.
scoped_ptr<Program> reduced_program(CreateReducedProgram(&options,
problem_impl,
&summary->fixed_cost,
&summary->error));
if (reduced_program == NULL) {
return;
}
summary->num_parameter_blocks_reduced = reduced_program->NumParameterBlocks();
summary->num_parameters_reduced = reduced_program->NumParameters();
summary->num_residual_blocks_reduced = reduced_program->NumResidualBlocks();
summary->num_residuals_reduced = reduced_program->NumResiduals();
if (summary->num_parameter_blocks_reduced == 0) {
summary->preprocessor_time_in_seconds =
WallTimeInSeconds() - solver_start_time;
LOG(INFO) << "Terminating: FUNCTION_TOLERANCE reached. "
<< "No non-constant parameter blocks found.";
// FUNCTION_TOLERANCE is the right convergence here, as we know
// that the objective function is constant and cannot be changed
// any further.
summary->termination_type = FUNCTION_TOLERANCE;
const double post_process_start_time = WallTimeInSeconds();
SetSummaryFinalCost(summary);
// Ensure the program state is set to the user parameters on the way out.
original_program->SetParameterBlockStatePtrsToUserStatePtrs();
summary->postprocessor_time_in_seconds =
WallTimeInSeconds() - post_process_start_time;
return;
}
scoped_ptr<Evaluator> evaluator(CreateEvaluator(options,
problem_impl->parameter_map(),
reduced_program.get(),
&summary->error));
if (evaluator == NULL) {
return;
}
// The optimizer works on contiguous parameter vectors; allocate some.
Vector parameters(reduced_program->NumParameters());
// Collect the discontiguous parameters into a contiguous state vector.
reduced_program->ParameterBlocksToStateVector(parameters.data());
Vector original_parameters = parameters;
const double minimizer_start_time = WallTimeInSeconds();
summary->preprocessor_time_in_seconds =
minimizer_start_time - solver_start_time;
// Run the optimization.
LineSearchMinimize(options,
reduced_program.get(),
evaluator.get(),
parameters.data(),
summary);
// If the user aborted mid-optimization or the optimization
// terminated because of a numerical failure, then return without
// updating user state.
if (summary->termination_type == USER_ABORT ||
summary->termination_type == NUMERICAL_FAILURE) {
return;
}
const double post_process_start_time = WallTimeInSeconds();
// Push the contiguous optimized parameters back to the user's parameters.
reduced_program->StateVectorToParameterBlocks(parameters.data());
reduced_program->CopyParameterBlockStateToUserState();
SetSummaryFinalCost(summary);
// Ensure the program state is set to the user parameters on the way out.
original_program->SetParameterBlockStatePtrsToUserStatePtrs();
const map<string, double>& evaluator_time_statistics =
evaluator->TimeStatistics();
summary->residual_evaluation_time_in_seconds =
FindWithDefault(evaluator_time_statistics, "Evaluator::Residual", 0.0);
summary->jacobian_evaluation_time_in_seconds =
FindWithDefault(evaluator_time_statistics, "Evaluator::Jacobian", 0.0);
// Stick a fork in it, we're done.
summary->postprocessor_time_in_seconds =
WallTimeInSeconds() - post_process_start_time;
}
#endif // CERES_NO_LINE_SEARCH_MINIMIZER
bool SolverImpl::IsOrderingValid(const Solver::Options& options,
const ProblemImpl* problem_impl,
string* error) {
if (options.linear_solver_ordering->NumElements() !=
problem_impl->NumParameterBlocks()) {
*error = "Number of parameter blocks in user supplied ordering "
"does not match the number of parameter blocks in the problem";
return false;
}
const Program& program = problem_impl->program();
const vector<ParameterBlock*>& parameter_blocks = program.parameter_blocks();
for (vector<ParameterBlock*>::const_iterator it = parameter_blocks.begin();
it != parameter_blocks.end();
++it) {
if (!options.linear_solver_ordering
->IsMember(const_cast<double*>((*it)->user_state()))) {
*error = "Problem contains a parameter block that is not in "
"the user specified ordering.";
return false;
}
}
if (IsSchurType(options.linear_solver_type) &&
options.linear_solver_ordering->NumGroups() > 1) {
const vector<ResidualBlock*>& residual_blocks = program.residual_blocks();
const set<double*>& e_blocks =
options.linear_solver_ordering->group_to_elements().begin()->second;
if (!IsParameterBlockSetIndependent(e_blocks, residual_blocks)) {
*error = "The user requested the use of a Schur type solver. "
"But the first elimination group in the ordering is not an "
"independent set.";
return false;
}
}
return true;
}
bool SolverImpl::IsParameterBlockSetIndependent(
const set<double*>& parameter_block_ptrs,
const vector<ResidualBlock*>& residual_blocks) {
// Loop over each residual block and ensure that no two parameter
// blocks in the same residual block are part of
// parameter_block_ptrs as that would violate the assumption that it
// is an independent set in the Hessian matrix.
for (vector<ResidualBlock*>::const_iterator it = residual_blocks.begin();
it != residual_blocks.end();
++it) {
ParameterBlock* const* parameter_blocks = (*it)->parameter_blocks();
const int num_parameter_blocks = (*it)->NumParameterBlocks();
int count = 0;
for (int i = 0; i < num_parameter_blocks; ++i) {
count += parameter_block_ptrs.count(
parameter_blocks[i]->mutable_user_state());
}
if (count > 1) {
return false;
}
}
return true;
}
// Strips varying parameters and residuals, maintaining order, and updating
// num_eliminate_blocks.
bool SolverImpl::RemoveFixedBlocksFromProgram(Program* program,
ParameterBlockOrdering* ordering,
double* fixed_cost,
string* error) {
vector<ParameterBlock*>* parameter_blocks =
program->mutable_parameter_blocks();
scoped_array<double> residual_block_evaluate_scratch;
if (fixed_cost != NULL) {
residual_block_evaluate_scratch.reset(
new double[program->MaxScratchDoublesNeededForEvaluate()]);
*fixed_cost = 0.0;
}
// Mark all the parameters as unused. Abuse the index member of the parameter
// blocks for the marking.
for (int i = 0; i < parameter_blocks->size(); ++i) {
(*parameter_blocks)[i]->set_index(-1);
}
// Filter out residual that have all-constant parameters, and mark all the
// parameter blocks that appear in residuals.
{
vector<ResidualBlock*>* residual_blocks =
program->mutable_residual_blocks();
int j = 0;
for (int i = 0; i < residual_blocks->size(); ++i) {
ResidualBlock* residual_block = (*residual_blocks)[i];
int num_parameter_blocks = residual_block->NumParameterBlocks();
// Determine if the residual block is fixed, and also mark varying
// parameters that appear in the residual block.
bool all_constant = true;
for (int k = 0; k < num_parameter_blocks; k++) {
ParameterBlock* parameter_block = residual_block->parameter_blocks()[k];
if (!parameter_block->IsConstant()) {
all_constant = false;
parameter_block->set_index(1);
}
}
if (!all_constant) {
(*residual_blocks)[j++] = (*residual_blocks)[i];
} else if (fixed_cost != NULL) {
// The residual is constant and will be removed, so its cost is
// added to the variable fixed_cost.
double cost = 0.0;
if (!residual_block->Evaluate(true,
&cost,
NULL,
NULL,
residual_block_evaluate_scratch.get())) {
*error = StringPrintf("Evaluation of the residual %d failed during "
"removal of fixed residual blocks.", i);
return false;
}
*fixed_cost += cost;
}
}
residual_blocks->resize(j);
}
// Filter out unused or fixed parameter blocks, and update
// the ordering.
{
vector<ParameterBlock*>* parameter_blocks =
program->mutable_parameter_blocks();
int j = 0;
for (int i = 0; i < parameter_blocks->size(); ++i) {
ParameterBlock* parameter_block = (*parameter_blocks)[i];
if (parameter_block->index() == 1) {
(*parameter_blocks)[j++] = parameter_block;
} else {
ordering->Remove(parameter_block->mutable_user_state());
}
}
parameter_blocks->resize(j);
}
CHECK(((program->NumResidualBlocks() == 0) &&
(program->NumParameterBlocks() == 0)) ||
((program->NumResidualBlocks() != 0) &&
(program->NumParameterBlocks() != 0)))
<< "Congratulations, you found a bug in Ceres. Please report it.";
return true;
}
Program* SolverImpl::CreateReducedProgram(Solver::Options* options,
ProblemImpl* problem_impl,
double* fixed_cost,
string* error) {
EventLogger event_logger("CreateReducedProgram");
CHECK_NOTNULL(options->linear_solver_ordering);
Program* original_program = problem_impl->mutable_program();
scoped_ptr<Program> transformed_program(new Program(*original_program));
event_logger.AddEvent("TransformedProgram");
ParameterBlockOrdering* linear_solver_ordering =
options->linear_solver_ordering;
const int min_group_id =
linear_solver_ordering->group_to_elements().begin()->first;
const int original_num_groups = linear_solver_ordering->NumGroups();
if (!RemoveFixedBlocksFromProgram(transformed_program.get(),
linear_solver_ordering,
fixed_cost,
error)) {
return NULL;
}
event_logger.AddEvent("RemoveFixedBlocks");
if (transformed_program->NumParameterBlocks() == 0) {
if (transformed_program->NumResidualBlocks() > 0) {
*error = "Zero parameter blocks but non-zero residual blocks"
" in the reduced program. Congratulations, you found a "
"Ceres bug! Please report this error to the developers.";
return NULL;
}
LOG(WARNING) << "No varying parameter blocks to optimize; "
<< "bailing early.";
return transformed_program.release();
}
// If the user supplied an linear_solver_ordering with just one
// group, it is equivalent to the user supplying NULL as
// ordering. Ceres is completely free to choose the parameter block
// ordering as it sees fit. For Schur type solvers, this means that
// the user wishes for Ceres to identify the e_blocks, which we do
// by computing a maximal independent set.
if (original_num_groups == 1 && IsSchurType(options->linear_solver_type)) {
vector<ParameterBlock*> schur_ordering;
const int num_eliminate_blocks = ComputeSchurOrdering(*transformed_program,
&schur_ordering);
CHECK_EQ(schur_ordering.size(), transformed_program->NumParameterBlocks())
<< "Congratulations, you found a Ceres bug! Please report this error "
<< "to the developers.";
for (int i = 0; i < schur_ordering.size(); ++i) {
linear_solver_ordering->AddElementToGroup(
schur_ordering[i]->mutable_user_state(),
(i < num_eliminate_blocks) ? 0 : 1);
}
}
event_logger.AddEvent("SchurOrdering");
if (!ApplyUserOrdering(problem_impl->parameter_map(),
linear_solver_ordering,
transformed_program.get(),
error)) {
return NULL;
}
event_logger.AddEvent("ApplyOrdering");
// If the user requested the use of a Schur type solver, and
// supplied a non-NULL linear_solver_ordering object with more than
// one elimination group, then it can happen that after all the
// parameter blocks which are fixed or unused have been removed from
// the program and the ordering, there are no more parameter blocks
// in the first elimination group.
//
// In such a case, the use of a Schur type solver is not possible,
// as they assume there is at least one e_block. Thus, we
// automatically switch to one of the other solvers, depending on
// the user's indicated preferences.
if (IsSchurType(options->linear_solver_type) &&
original_num_groups > 1 &&
linear_solver_ordering->GroupSize(min_group_id) == 0) {
string msg = "No e_blocks remaining. Switching from ";
if (options->linear_solver_type == SPARSE_SCHUR) {
options->linear_solver_type = SPARSE_NORMAL_CHOLESKY;
msg += "SPARSE_SCHUR to SPARSE_NORMAL_CHOLESKY.";
} else if (options->linear_solver_type == DENSE_SCHUR) {
// TODO(sameeragarwal): This is probably not a great choice.
// Ideally, we should have a DENSE_NORMAL_CHOLESKY, that can
// take a BlockSparseMatrix as input.
options->linear_solver_type = DENSE_QR;
msg += "DENSE_SCHUR to DENSE_QR.";
} else if (options->linear_solver_type == ITERATIVE_SCHUR) {
msg += StringPrintf("ITERATIVE_SCHUR with %s preconditioner "
"to CGNR with JACOBI preconditioner.",
PreconditionerTypeToString(
options->preconditioner_type));
options->linear_solver_type = CGNR;
if (options->preconditioner_type != IDENTITY) {
// CGNR currently only supports the JACOBI preconditioner.
options->preconditioner_type = JACOBI;
}
}
LOG(WARNING) << msg;
}
event_logger.AddEvent("AlternateSolver");
// Since the transformed program is the "active" program, and it is
// mutated, update the parameter offsets and indices.
transformed_program->SetParameterOffsetsAndIndex();
event_logger.AddEvent("SetOffsets");
return transformed_program.release();
}
LinearSolver* SolverImpl::CreateLinearSolver(Solver::Options* options,
string* error) {
CHECK_NOTNULL(options);
CHECK_NOTNULL(options->linear_solver_ordering);
CHECK_NOTNULL(error);
if (options->trust_region_strategy_type == DOGLEG) {
if (options->linear_solver_type == ITERATIVE_SCHUR ||
options->linear_solver_type == CGNR) {
*error = "DOGLEG only supports exact factorization based linear "
"solvers. If you want to use an iterative solver please "
"use LEVENBERG_MARQUARDT as the trust_region_strategy_type";
return NULL;
}
}
#ifdef CERES_NO_SUITESPARSE
if (options->linear_solver_type == SPARSE_NORMAL_CHOLESKY &&
options->sparse_linear_algebra_library == SUITE_SPARSE) {
*error = "Can't use SPARSE_NORMAL_CHOLESKY with SUITESPARSE because "
"SuiteSparse was not enabled when Ceres was built.";
return NULL;
}
if (options->preconditioner_type == CLUSTER_JACOBI) {
*error = "CLUSTER_JACOBI preconditioner not suppored. Please build Ceres "
"with SuiteSparse support.";
return NULL;
}
if (options->preconditioner_type == CLUSTER_TRIDIAGONAL) {
*error = "CLUSTER_TRIDIAGONAL preconditioner not suppored. Please build "
"Ceres with SuiteSparse support.";
return NULL;
}
#endif
#ifdef CERES_NO_CXSPARSE
if (options->linear_solver_type == SPARSE_NORMAL_CHOLESKY &&
options->sparse_linear_algebra_library == CX_SPARSE) {
*error = "Can't use SPARSE_NORMAL_CHOLESKY with CXSPARSE because "
"CXSparse was not enabled when Ceres was built.";
return NULL;
}
#endif
#if defined(CERES_NO_SUITESPARSE) && defined(CERES_NO_CXSPARSE)
if (options->linear_solver_type == SPARSE_SCHUR) {
*error = "Can't use SPARSE_SCHUR because neither SuiteSparse nor"
"CXSparse was enabled when Ceres was compiled.";
return NULL;
}
#endif
if (options->linear_solver_max_num_iterations <= 0) {
*error = "Solver::Options::linear_solver_max_num_iterations is 0.";
return NULL;
}
if (options->linear_solver_min_num_iterations <= 0) {
*error = "Solver::Options::linear_solver_min_num_iterations is 0.";
return NULL;
}
if (options->linear_solver_min_num_iterations >
options->linear_solver_max_num_iterations) {
*error = "Solver::Options::linear_solver_min_num_iterations > "
"Solver::Options::linear_solver_max_num_iterations.";
return NULL;
}
LinearSolver::Options linear_solver_options;
linear_solver_options.min_num_iterations =
options->linear_solver_min_num_iterations;
linear_solver_options.max_num_iterations =
options->linear_solver_max_num_iterations;
linear_solver_options.type = options->linear_solver_type;
linear_solver_options.preconditioner_type = options->preconditioner_type;
linear_solver_options.sparse_linear_algebra_library =
options->sparse_linear_algebra_library;
linear_solver_options.num_threads = options->num_linear_solver_threads;
options->num_linear_solver_threads = linear_solver_options.num_threads;
linear_solver_options.use_block_amd = options->use_block_amd;
const map<int, set<double*> >& groups =
options->linear_solver_ordering->group_to_elements();
for (map<int, set<double*> >::const_iterator it = groups.begin();
it != groups.end();
++it) {
linear_solver_options.elimination_groups.push_back(it->second.size());
}
// Schur type solvers, expect at least two elimination groups. If
// there is only one elimination group, then CreateReducedProgram
// guarantees that this group only contains e_blocks. Thus we add a
// dummy elimination group with zero blocks in it.
if (IsSchurType(linear_solver_options.type) &&
linear_solver_options.elimination_groups.size() == 1) {
linear_solver_options.elimination_groups.push_back(0);
}
return LinearSolver::Create(linear_solver_options);
}
bool SolverImpl::ApplyUserOrdering(
const ProblemImpl::ParameterMap& parameter_map,
const ParameterBlockOrdering* ordering,
Program* program,
string* error) {
if (ordering->NumElements() != program->NumParameterBlocks()) {
*error = StringPrintf("User specified ordering does not have the same "
"number of parameters as the problem. The problem"
"has %d blocks while the ordering has %d blocks.",
program->NumParameterBlocks(),
ordering->NumElements());
return false;
}
vector<ParameterBlock*>* parameter_blocks =
program->mutable_parameter_blocks();
parameter_blocks->clear();
const map<int, set<double*> >& groups =
ordering->group_to_elements();
for (map<int, set<double*> >::const_iterator group_it = groups.begin();
group_it != groups.end();
++group_it) {
const set<double*>& group = group_it->second;
for (set<double*>::const_iterator parameter_block_ptr_it = group.begin();
parameter_block_ptr_it != group.end();
++parameter_block_ptr_it) {
ProblemImpl::ParameterMap::const_iterator parameter_block_it =
parameter_map.find(*parameter_block_ptr_it);
if (parameter_block_it == parameter_map.end()) {
*error = StringPrintf("User specified ordering contains a pointer "
"to a double that is not a parameter block in "
"the problem. The invalid double is in group: %d",
group_it->first);
return false;
}
parameter_blocks->push_back(parameter_block_it->second);
}
}
return true;
}
// Find the minimum index of any parameter block to the given residual.
// Parameter blocks that have indices greater than num_eliminate_blocks are
// considered to have an index equal to num_eliminate_blocks.
static int MinParameterBlock(const ResidualBlock* residual_block,
int num_eliminate_blocks) {
int min_parameter_block_position = num_eliminate_blocks;
for (int i = 0; i < residual_block->NumParameterBlocks(); ++i) {
ParameterBlock* parameter_block = residual_block->parameter_blocks()[i];
if (!parameter_block->IsConstant()) {
CHECK_NE(parameter_block->index(), -1)
<< "Did you forget to call Program::SetParameterOffsetsAndIndex()? "
<< "This is a Ceres bug; please contact the developers!";
min_parameter_block_position = std::min(parameter_block->index(),
min_parameter_block_position);
}
}
return min_parameter_block_position;
}
// Reorder the residuals for program, if necessary, so that the residuals
// involving each E block occur together. This is a necessary condition for the
// Schur eliminator, which works on these "row blocks" in the jacobian.
bool SolverImpl::LexicographicallyOrderResidualBlocks(
const int num_eliminate_blocks,
Program* program,
string* error) {
CHECK_GE(num_eliminate_blocks, 1)
<< "Congratulations, you found a Ceres bug! Please report this error "
<< "to the developers.";
// Create a histogram of the number of residuals for each E block. There is an
// extra bucket at the end to catch all non-eliminated F blocks.
vector<int> residual_blocks_per_e_block(num_eliminate_blocks + 1);
vector<ResidualBlock*>* residual_blocks = program->mutable_residual_blocks();
vector<int> min_position_per_residual(residual_blocks->size());
for (int i = 0; i < residual_blocks->size(); ++i) {
ResidualBlock* residual_block = (*residual_blocks)[i];
int position = MinParameterBlock(residual_block, num_eliminate_blocks);
min_position_per_residual[i] = position;
DCHECK_LE(position, num_eliminate_blocks);
residual_blocks_per_e_block[position]++;
}
// Run a cumulative sum on the histogram, to obtain offsets to the start of
// each histogram bucket (where each bucket is for the residuals for that
// E-block).
vector<int> offsets(num_eliminate_blocks + 1);
std::partial_sum(residual_blocks_per_e_block.begin(),
residual_blocks_per_e_block.end(),
offsets.begin());
CHECK_EQ(offsets.back(), residual_blocks->size())
<< "Congratulations, you found a Ceres bug! Please report this error "
<< "to the developers.";
CHECK(find(residual_blocks_per_e_block.begin(),
residual_blocks_per_e_block.end() - 1, 0) !=
residual_blocks_per_e_block.end())
<< "Congratulations, you found a Ceres bug! Please report this error "
<< "to the developers.";
// Fill in each bucket with the residual blocks for its corresponding E block.
// Each bucket is individually filled from the back of the bucket to the front
// of the bucket. The filling order among the buckets is dictated by the
// residual blocks. This loop uses the offsets as counters; subtracting one
// from each offset as a residual block is placed in the bucket. When the
// filling is finished, the offset pointerts should have shifted down one
// entry (this is verified below).
vector<ResidualBlock*> reordered_residual_blocks(
(*residual_blocks).size(), static_cast<ResidualBlock*>(NULL));
for (int i = 0; i < residual_blocks->size(); ++i) {
int bucket = min_position_per_residual[i];
// Decrement the cursor, which should now point at the next empty position.
offsets[bucket]--;
// Sanity.
CHECK(reordered_residual_blocks[offsets[bucket]] == NULL)
<< "Congratulations, you found a Ceres bug! Please report this error "
<< "to the developers.";
reordered_residual_blocks[offsets[bucket]] = (*residual_blocks)[i];
}
// Sanity check #1: The difference in bucket offsets should match the
// histogram sizes.
for (int i = 0; i < num_eliminate_blocks; ++i) {
CHECK_EQ(residual_blocks_per_e_block[i], offsets[i + 1] - offsets[i])
<< "Congratulations, you found a Ceres bug! Please report this error "
<< "to the developers.";
}
// Sanity check #2: No NULL's left behind.
for (int i = 0; i < reordered_residual_blocks.size(); ++i) {
CHECK(reordered_residual_blocks[i] != NULL)
<< "Congratulations, you found a Ceres bug! Please report this error "
<< "to the developers.";
}
// Now that the residuals are collected by E block, swap them in place.
swap(*program->mutable_residual_blocks(), reordered_residual_blocks);
return true;
}
Evaluator* SolverImpl::CreateEvaluator(
const Solver::Options& options,
const ProblemImpl::ParameterMap& parameter_map,
Program* program,
string* error) {
Evaluator::Options evaluator_options;
evaluator_options.linear_solver_type = options.linear_solver_type;
evaluator_options.num_eliminate_blocks =
(options.linear_solver_ordering->NumGroups() > 0 &&
IsSchurType(options.linear_solver_type))
? (options.linear_solver_ordering
->group_to_elements().begin()
->second.size())
: 0;
evaluator_options.num_threads = options.num_threads;
return Evaluator::Create(evaluator_options, program, error);
}
CoordinateDescentMinimizer* SolverImpl::CreateInnerIterationMinimizer(
const Solver::Options& options,
const Program& program,
const ProblemImpl::ParameterMap& parameter_map,
Solver::Summary* summary) {
scoped_ptr<CoordinateDescentMinimizer> inner_iteration_minimizer(
new CoordinateDescentMinimizer);
scoped_ptr<ParameterBlockOrdering> inner_iteration_ordering;
ParameterBlockOrdering* ordering_ptr = NULL;
if (options.inner_iteration_ordering == NULL) {
// Find a recursive decomposition of the Hessian matrix as a set
// of independent sets of decreasing size and invert it. This
// seems to work better in practice, i.e., Cameras before
// points.
inner_iteration_ordering.reset(new ParameterBlockOrdering);
ComputeRecursiveIndependentSetOrdering(program,
inner_iteration_ordering.get());
inner_iteration_ordering->Reverse();
ordering_ptr = inner_iteration_ordering.get();
} else {
const map<int, set<double*> >& group_to_elements =
options.inner_iteration_ordering->group_to_elements();
// Iterate over each group and verify that it is an independent
// set.
map<int, set<double*> >::const_iterator it = group_to_elements.begin();
for ( ; it != group_to_elements.end(); ++it) {
if (!IsParameterBlockSetIndependent(it->second,
program.residual_blocks())) {
summary->error =
StringPrintf("The user-provided "
"parameter_blocks_for_inner_iterations does not "
"form an independent set. Group Id: %d", it->first);
return NULL;
}
}
ordering_ptr = options.inner_iteration_ordering;
}
if (!inner_iteration_minimizer->Init(program,
parameter_map,
*ordering_ptr,
&summary->error)) {
return NULL;
}
summary->inner_iterations = true;
SummarizeOrdering(ordering_ptr, &(summary->inner_iteration_ordering_used));
return inner_iteration_minimizer.release();
}
} // namespace internal
} // namespace ceres