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// Ceres Solver - A fast non-linear least squares minimizer
// Copyright 2019 Google Inc. All rights reserved.
// 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.
// Author: (Sameer Agarwal)
#include <cmath>
#include <memory>
#include <string>
#include <unordered_set>
#include <vector>
#include "ceres/crs_matrix.h"
#include "ceres/internal/disable_warnings.h"
#include "ceres/internal/port.h"
#include "ceres/iteration_callback.h"
#include "ceres/ordered_groups.h"
#include "ceres/problem.h"
#include "ceres/types.h"
namespace ceres {
// Interface for non-linear least squares solvers.
class CERES_EXPORT Solver {
virtual ~Solver();
// The options structure contains, not surprisingly, options that control how
// the solver operates. The defaults should be suitable for a wide range of
// problems; however, better performance is often obtainable with tweaking.
// The constants are defined inside types.h
struct CERES_EXPORT Options {
// Returns true if the options struct has a valid
// configuration. Returns false otherwise, and fills in *error
// with a message describing the problem.
bool IsValid(std::string* error) const;
// Minimizer options ----------------------------------------
// Ceres supports the two major families of optimization strategies -
// Trust Region and Line Search.
// 1. The line search approach first finds a descent direction
// along which the objective function will be reduced and then
// computes a step size that decides how far should move along
// that direction. The descent direction can be computed by
// various methods, such as gradient descent, Newton's method and
// Quasi-Newton method. The step size can be determined either
// exactly or inexactly.
// 2. The trust region approach approximates the objective
// function using a model function (often a quadratic) over
// a subset of the search space known as the trust region. If the
// model function succeeds in minimizing the true objective
// function the trust region is expanded; conversely, otherwise it
// is contracted and the model optimization problem is solved
// again.
// Trust region methods are in some sense dual to line search methods:
// trust region methods first choose a step size (the size of the
// trust region) and then a step direction while line search methods
// first choose a step direction and then a step size.
MinimizerType minimizer_type = TRUST_REGION;
LineSearchDirectionType line_search_direction_type = LBFGS;
LineSearchType line_search_type = WOLFE;
NonlinearConjugateGradientType nonlinear_conjugate_gradient_type =
// The LBFGS hessian approximation is a low rank approximation to
// the inverse of the Hessian matrix. The rank of the
// approximation determines (linearly) the space and time
// complexity of using the approximation. Higher the rank, the
// better is the quality of the approximation. The increase in
// quality is however is bounded for a number of reasons.
// 1. The method only uses secant information and not actual
// derivatives.
// 2. The Hessian approximation is constrained to be positive
// definite.
// So increasing this rank to a large number will cost time and
// space complexity without the corresponding increase in solution
// quality. There are no hard and fast rules for choosing the
// maximum rank. The best choice usually requires some problem
// specific experimentation.
// For more theoretical and implementation details of the LBFGS
// method, please see:
// Nocedal, J. (1980). "Updating Quasi-Newton Matrices with
// Limited Storage". Mathematics of Computation 35 (151): 773-782.
int max_lbfgs_rank = 20;
// As part of the (L)BFGS update step (BFGS) / right-multiply step (L-BFGS),
// the initial inverse Hessian approximation is taken to be the Identity.
// However, Oren showed that using instead I * \gamma, where \gamma is
// chosen to approximate an eigenvalue of the true inverse Hessian can
// result in improved convergence in a wide variety of cases. Setting
// use_approximate_eigenvalue_bfgs_scaling to true enables this scaling.
// It is important to note that approximate eigenvalue scaling does not
// always improve convergence, and that it can in fact significantly degrade
// performance for certain classes of problem, which is why it is disabled
// by default. In particular it can degrade performance when the
// sensitivity of the problem to different parameters varies significantly,
// as in this case a single scalar factor fails to capture this variation
// and detrimentally downscales parts of the jacobian approximation which
// correspond to low-sensitivity parameters. It can also reduce the
// robustness of the solution to errors in the jacobians.
// Oren S.S., Self-scaling variable metric (SSVM) algorithms
// Part II: Implementation and experiments, Management Science,
// 20(5), 863-874, 1974.
bool use_approximate_eigenvalue_bfgs_scaling = false;
// Degree of the polynomial used to approximate the objective
// function. Valid values are BISECTION, QUADRATIC and CUBIC.
// BISECTION corresponds to pure backtracking search with no
// interpolation.
LineSearchInterpolationType line_search_interpolation_type = CUBIC;
// If during the line search, the step_size falls below this
// value, it is truncated to zero.
double min_line_search_step_size = 1e-9;
// Line search parameters.
// Solving the line search problem exactly is computationally
// prohibitive. Fortunately, line search based optimization
// algorithms can still guarantee convergence if instead of an
// exact solution, the line search algorithm returns a solution
// which decreases the value of the objective function
// sufficiently. More precisely, we are looking for a step_size
// s.t.
// f(step_size) <= f(0) + sufficient_decrease * f'(0) * step_size
double line_search_sufficient_function_decrease = 1e-4;
// In each iteration of the line search,
// new_step_size >= max_line_search_step_contraction * step_size
// Note that by definition, for contraction:
// 0 < max_step_contraction < min_step_contraction < 1
double max_line_search_step_contraction = 1e-3;
// In each iteration of the line search,
// new_step_size <= min_line_search_step_contraction * step_size
// Note that by definition, for contraction:
// 0 < max_step_contraction < min_step_contraction < 1
double min_line_search_step_contraction = 0.6;
// Maximum number of trial step size iterations during each line
// search, if a step size satisfying the search conditions cannot
// be found within this number of trials, the line search will
// terminate.
// The minimum allowed value is 0 for trust region minimizer and 1
// otherwise. If 0 is specified for the trust region minimizer,
// then line search will not be used when solving constrained
// optimization problems.
int max_num_line_search_step_size_iterations = 20;
// Maximum number of restarts of the line search direction algorithm before
// terminating the optimization. Restarts of the line search direction
// algorithm occur when the current algorithm fails to produce a new descent
// direction. This typically indicates a numerical failure, or a breakdown
// in the validity of the approximations used.
int max_num_line_search_direction_restarts = 5;
// The strong Wolfe conditions consist of the Armijo sufficient
// decrease condition, and an additional requirement that the
// step-size be chosen s.t. the _magnitude_ ('strong' Wolfe
// conditions) of the gradient along the search direction
// decreases sufficiently. Precisely, this second condition
// is that we seek a step_size s.t.
// |f'(step_size)| <= sufficient_curvature_decrease * |f'(0)|
// Where f() is the line search objective and f'() is the derivative
// of f w.r.t step_size (d f / d step_size).
double line_search_sufficient_curvature_decrease = 0.9;
// During the bracketing phase of the Wolfe search, the step size is
// increased until either a point satisfying the Wolfe conditions is
// found, or an upper bound for a bracket containing a point satisfying
// the conditions is found. Precisely, at each iteration of the
// expansion:
// new_step_size <= max_step_expansion * step_size.
// By definition for expansion, max_step_expansion > 1.0.
double max_line_search_step_expansion = 10.0;
TrustRegionStrategyType trust_region_strategy_type = LEVENBERG_MARQUARDT;
// Type of dogleg strategy to use.
DoglegType dogleg_type = TRADITIONAL_DOGLEG;
// The classical trust region methods are descent methods, in that
// they only accept a point if it strictly reduces the value of
// the objective function.
// Relaxing this requirement allows the algorithm to be more
// efficient in the long term at the cost of some local increase
// in the value of the objective function.
// This is because allowing for non-decreasing objective function
// values in a principled manner allows the algorithm to "jump over
// boulders" as the method is not restricted to move into narrow
// valleys while preserving its convergence properties.
// Setting use_nonmonotonic_steps to true enables the
// non-monotonic trust region algorithm as described by Conn,
// Gould & Toint in "Trust Region Methods", Section 10.1.
// The parameter max_consecutive_nonmonotonic_steps controls the
// window size used by the step selection algorithm to accept
// non-monotonic steps.
// Even though the value of the objective function may be larger
// than the minimum value encountered over the course of the
// optimization, the final parameters returned to the user are the
// ones corresponding to the minimum cost over all iterations.
bool use_nonmonotonic_steps = false;
int max_consecutive_nonmonotonic_steps = 5;
// Maximum number of iterations for the minimizer to run for.
int max_num_iterations = 50;
// Maximum time for which the minimizer should run for.
double max_solver_time_in_seconds = 1e9;
// Number of threads used by Ceres for evaluating the cost and
// jacobians.
int num_threads = 1;
// Trust region minimizer settings.
double initial_trust_region_radius = 1e4;
double max_trust_region_radius = 1e16;
// Minimizer terminates when the trust region radius becomes
// smaller than this value.
double min_trust_region_radius = 1e-32;
// Lower bound for the relative decrease before a step is
// accepted.
double min_relative_decrease = 1e-3;
// For the Levenberg-Marquadt algorithm, the scaled diagonal of
// the normal equations J'J is used to control the size of the
// trust region. Extremely small and large values along the
// diagonal can make this regularization scheme
// fail. max_lm_diagonal and min_lm_diagonal, clamp the values of
// diag(J'J) from above and below. In the normal course of
// operation, the user should not have to modify these parameters.
double min_lm_diagonal = 1e-6;
double max_lm_diagonal = 1e32;
// Sometimes due to numerical conditioning problems or linear
// solver flakiness, the trust region strategy may return a
// numerically invalid step that can be fixed by reducing the
// trust region size. So the TrustRegionMinimizer allows for a few
// successive invalid steps before it declares NUMERICAL_FAILURE.
int max_num_consecutive_invalid_steps = 5;
// Minimizer terminates when
// (new_cost - old_cost) < function_tolerance * old_cost;
double function_tolerance = 1e-6;
// Minimizer terminates when
// max_i |x - Project(Plus(x, -g(x))| < gradient_tolerance
// This value should typically be 1e-4 * function_tolerance.
double gradient_tolerance = 1e-10;
// Minimizer terminates when
// |step|_2 <= parameter_tolerance * ( |x|_2 + parameter_tolerance)
double parameter_tolerance = 1e-8;
// Linear least squares solver options -------------------------------------
LinearSolverType linear_solver_type =
#if defined(CERES_NO_SPARSE)
// Type of preconditioner to use with the iterative linear solvers.
PreconditionerType preconditioner_type = JACOBI;
// Type of clustering algorithm to use for visibility based
// preconditioning. This option is used only when the
// preconditioner_type is CLUSTER_JACOBI or CLUSTER_TRIDIAGONAL.
VisibilityClusteringType visibility_clustering_type = CANONICAL_VIEWS;
// Subset preconditioner is a general purpose preconditioner for
// linear least squares problems. Given a set of residual blocks,
// it uses the corresponding subset of the rows of the Jacobian to
// construct a preconditioner.
// Suppose the Jacobian J has been horizontally partitioned as
// J = [P]
// [Q]
// Where, Q is the set of rows corresponding to the residual
// blocks in residual_blocks_for_subset_preconditioner.
// The preconditioner is the inverse of the matrix Q'Q.
// Obviously, the efficacy of the preconditioner depends on how
// well the matrix Q approximates J'J, or how well the chosen
// residual blocks approximate the non-linear least squares
// problem.
// If Solver::Options::preconditioner_type == SUBSET, then
// residual_blocks_for_subset_preconditioner must be non-empty.
std::unordered_set<ResidualBlockId> residual_blocks_for_subset_preconditioner;
// Ceres supports using multiple dense linear algebra libraries
// for dense matrix factorizations. Currently EIGEN and LAPACK are
// the valid choices. EIGEN is always available, LAPACK refers to
// the system BLAS + LAPACK library which may or may not be
// available.
// This setting affects the DENSE_QR, DENSE_NORMAL_CHOLESKY and
// DENSE_SCHUR solvers. For small to moderate sized problem EIGEN
// is a fine choice but for large problems, an optimized LAPACK +
// BLAS implementation can make a substantial difference in
// performance.
DenseLinearAlgebraLibraryType dense_linear_algebra_library_type = EIGEN;
// Ceres supports using multiple sparse linear algebra libraries
// for sparse matrix ordering and factorizations. Currently,
// SUITE_SPARSE and CX_SPARSE are the valid choices, depending on
// whether they are linked into Ceres at build time.
SparseLinearAlgebraLibraryType sparse_linear_algebra_library_type =
#elif !defined(CERES_NO_CXSPARSE)
// The order in which variables are eliminated in a linear solver
// can have a significant of impact on the efficiency and accuracy
// of the method. e.g., when doing sparse Cholesky factorization,
// there are matrices for which a good ordering will give a
// Cholesky factor with O(n) storage, where as a bad ordering will
// result in an completely dense factor.
// Ceres allows the user to provide varying amounts of hints to
// the solver about the variable elimination ordering to use. This
// can range from no hints, where the solver is free to decide the
// best possible ordering based on the user's choices like the
// linear solver being used, to an exact order in which the
// variables should be eliminated, and a variety of possibilities
// in between.
// Instances of the ParameterBlockOrdering class are used to
// communicate this information to Ceres.
// Formally an ordering is an ordered partitioning of the
// parameter blocks, i.e, each parameter block belongs to exactly
// one group, and each group has a unique non-negative integer
// associated with it, that determines its order in the set of
// groups.
// Given such an ordering, Ceres ensures that the parameter blocks in
// the lowest numbered group are eliminated first, and then the
// parameter blocks in the next lowest numbered group and so on. Within
// each group, Ceres is free to order the parameter blocks as it
// chooses.
// If NULL, then all parameter blocks are assumed to be in the
// same group and the solver is free to decide the best
// ordering.
// e.g. Consider the linear system
// x + y = 3
// 2x + 3y = 7
// There are two ways in which it can be solved. First eliminating x
// from the two equations, solving for y and then back substituting
// for x, or first eliminating y, solving for x and back substituting
// for y. The user can construct three orderings here.
// {0: x}, {1: y} - eliminate x first.
// {0: y}, {1: x} - eliminate y first.
// {0: x, y} - Solver gets to decide the elimination order.
// Thus, to have Ceres determine the ordering automatically using
// heuristics, put all the variables in group 0 and to control the
// ordering for every variable, create groups 0..N-1, one per
// variable, in the desired order.
// Bundle Adjustment
// -----------------
// A particular case of interest is bundle adjustment, where the user
// has two options. The default is to not specify an ordering at all,
// the solver will see that the user wants to use a Schur type solver
// and figure out the right elimination ordering.
// But if the user already knows what parameter blocks are points and
// what are cameras, they can save preprocessing time by partitioning
// the parameter blocks into two groups, one for the points and one
// for the cameras, where the group containing the points has an id
// smaller than the group containing cameras.
std::shared_ptr<ParameterBlockOrdering> linear_solver_ordering;
// Use an explicitly computed Schur complement matrix with
// By default this option is disabled and ITERATIVE_SCHUR
// evaluates matrix-vector products between the Schur
// complement and a vector implicitly by exploiting the algebraic
// expression for the Schur complement.
// The cost of this evaluation scales with the number of non-zeros
// in the Jacobian.
// For small to medium sized problems there is a sweet spot where
// computing the Schur complement is cheap enough that it is much
// more efficient to explicitly compute it and use it for evaluating
// the matrix-vector products.
// Enabling this option tells ITERATIVE_SCHUR to use an explicitly
// computed Schur complement.
// NOTE: This option can only be used with the SCHUR_JACOBI
// preconditioner.
bool use_explicit_schur_complement = false;
// Sparse Cholesky factorization algorithms use a fill-reducing
// ordering to permute the columns of the Jacobian matrix. There
// are two ways of doing this.
// 1. Compute the Jacobian matrix in some order and then have the
// factorization algorithm permute the columns of the Jacobian.
// 2. Compute the Jacobian with its columns already permuted.
// The first option incurs a significant memory penalty. The
// factorization algorithm has to make a copy of the permuted
// Jacobian matrix, thus Ceres pre-permutes the columns of the
// Jacobian matrix and generally speaking, there is no performance
// penalty for doing so.
// In some rare cases, it is worth using a more complicated
// reordering algorithm which has slightly better runtime
// performance at the expense of an extra copy of the Jacobian
// matrix. Setting use_postordering to true enables this tradeoff.
bool use_postordering = false;
// Some non-linear least squares problems are symbolically dense but
// numerically sparse. i.e. at any given state only a small number
// of jacobian entries are non-zero, but the position and number of
// non-zeros is different depending on the state. For these problems
// it can be useful to factorize the sparse jacobian at each solver
// iteration instead of including all of the zero entries in a single
// general factorization.
// If your problem does not have this property (or you do not know),
// then it is probably best to keep this false, otherwise it will
// likely lead to worse performance.
// This settings only affects the SPARSE_NORMAL_CHOLESKY solver.
bool dynamic_sparsity = false;
// TODO(sameeragarwal): Further expand the documentation for the
// following two options.
// If use_mixed_precision_solves is true, the Gauss-Newton matrix
// is computed in double precision, but its factorization is
// computed in single precision. This can result in significant
// time and memory savings at the cost of some accuracy in the
// Gauss-Newton step. Iterative refinement is used to recover some
// of this accuracy back.
// If use_mixed_precision_solves is true, we recommend setting
// max_num_refinement_iterations to 2-3.
// NOTE2: The following two options are currently only applicable
// if sparse_linear_algebra_library_type is EIGEN_SPARSE and
// linear_solver_type is SPARSE_NORMAL_CHOLESKY, or SPARSE_SCHUR.
bool use_mixed_precision_solves = false;
// Number steps of the iterative refinement process to run when
// computing the Gauss-Newton step.
int max_num_refinement_iterations = 0;
// Some non-linear least squares problems have additional
// structure in the way the parameter blocks interact that it is
// beneficial to modify the way the trust region step is computed.
// e.g., consider the following regression problem
// y = a_1 exp(b_1 x) + a_2 exp(b_3 x^2 + c_1)
// Given a set of pairs{(x_i, y_i)}, the user wishes to estimate
// a_1, a_2, b_1, b_2, and c_1.
// Notice here that the expression on the left is linear in a_1
// and a_2, and given any value for b_1, b_2 and c_1, it is
// possible to use linear regression to estimate the optimal
// values of a_1 and a_2. Indeed, its possible to analytically
// eliminate the variables a_1 and a_2 from the problem all
// together. Problems like these are known as separable least
// squares problem and the most famous algorithm for solving them
// is the Variable Projection algorithm invented by Golub &
// Pereyra.
// Similar structure can be found in the matrix factorization with
// missing data problem. There the corresponding algorithm is
// known as Wiberg's algorithm.
// Ruhe & Wedin (Algorithms for Separable Nonlinear Least Squares
// Problems, SIAM Reviews, 22(3), 1980) present an analysis of
// various algorithms for solving separable non-linear least
// squares problems and refer to "Variable Projection" as
// Algorithm I in their paper.
// Implementing Variable Projection is tedious and expensive, and
// they present a simpler algorithm, which they refer to as
// Algorithm II, where once the Newton/Trust Region step has been
// computed for the whole problem (a_1, a_2, b_1, b_2, c_1) and
// additional optimization step is performed to estimate a_1 and
// a_2 exactly.
// This idea can be generalized to cases where the residual is not
// linear in a_1 and a_2, i.e., Solve for the trust region step
// for the full problem, and then use it as the starting point to
// further optimize just a_1 and a_2. For the linear case, this
// amounts to doing a single linear least squares solve. For
// non-linear problems, any method for solving the a_1 and a_2
// optimization problems will do. The only constraint on a_1 and
// a_2 is that they do not co-occur in any residual block.
// This idea can be further generalized, by not just optimizing
// (a_1, a_2), but decomposing the graph corresponding to the
// Hessian matrix's sparsity structure in a collection of
// non-overlapping independent sets and optimizing each of them.
// Setting "use_inner_iterations" to true enables the use of this
// non-linear generalization of Ruhe & Wedin's Algorithm II. This
// version of Ceres has a higher iteration complexity, but also
// displays better convergence behaviour per iteration. Setting
// Solver::Options::num_threads to the maximum number possible is
// highly recommended.
bool use_inner_iterations = false;
// If inner_iterations is true, then the user has two choices.
// 1. Let the solver heuristically decide which parameter blocks
// to optimize in each inner iteration. To do this leave
// Solver::Options::inner_iteration_ordering untouched.
// 2. Specify a collection of of ordered independent sets. Where
// the lower numbered groups are optimized before the higher
// number groups. Each group must be an independent set. Not
// all parameter blocks need to be present in the ordering.
std::shared_ptr<ParameterBlockOrdering> inner_iteration_ordering;
// Generally speaking, inner iterations make significant progress
// in the early stages of the solve and then their contribution
// drops down sharply, at which point the time spent doing inner
// iterations is not worth it.
// Once the relative decrease in the objective function due to
// inner iterations drops below inner_iteration_tolerance, the use
// of inner iterations in subsequent trust region minimizer
// iterations is disabled.
double inner_iteration_tolerance = 1e-3;
// Minimum number of iterations for which the linear solver should
// run, even if the convergence criterion is satisfied.
int min_linear_solver_iterations = 0;
// Maximum number of iterations for which the linear solver should
// run. If the solver does not converge in less than
// max_linear_solver_iterations, then it returns MAX_ITERATIONS,
// as its termination type.
int max_linear_solver_iterations = 500;
// Forcing sequence parameter. The truncated Newton solver uses
// this number to control the relative accuracy with which the
// Newton step is computed.
// This constant is passed to ConjugateGradientsSolver which uses
// it to terminate the iterations when
// (Q_i - Q_{i-1})/Q_i < eta/i
double eta = 1e-1;
// Normalize the jacobian using Jacobi scaling before calling
// the linear least squares solver.
bool jacobi_scaling = true;
// Logging options ---------------------------------------------------------
LoggingType logging_type = PER_MINIMIZER_ITERATION;
// By default the Minimizer progress is logged to VLOG(1), which
// is sent to STDERR depending on the vlog level. If this flag is
// set to true, and logging_type is not SILENT, the logging output
// is sent to STDOUT.
bool minimizer_progress_to_stdout = false;
// List of iterations at which the minimizer should dump the trust
// region problem. Useful for testing and benchmarking. If empty
// (default), no problems are dumped.
std::vector<int> trust_region_minimizer_iterations_to_dump;
// Directory to which the problems should be written to. Should be
// non-empty if trust_region_minimizer_iterations_to_dump is
// non-empty and trust_region_problem_dump_format_type is not
std::string trust_region_problem_dump_directory = "/tmp";
DumpFormatType trust_region_problem_dump_format_type = TEXTFILE;
// Finite differences options ----------------------------------------------
// Check all jacobians computed by each residual block with finite
// differences. This is expensive since it involves computing the
// derivative by normal means (e.g. user specified, autodiff,
// etc), then also computing it using finite differences. The
// results are compared, and if they differ substantially, details
// are printed to the log.
bool check_gradients = false;
// Relative precision to check for in the gradient checker. If the
// relative difference between an element in a jacobian exceeds
// this number, then the jacobian for that cost term is dumped.
double gradient_check_relative_precision = 1e-8;
// WARNING: This option only applies to the to the numeric
// differentiation used for checking the user provided derivatives
// when when Solver::Options::check_gradients is true. If you are
// using NumericDiffCostFunction and are interested in changing
// the step size for numeric differentiation in your cost
// function, please have a look at
// include/ceres/numeric_diff_options.h.
// Relative shift used for taking numeric derivatives when
// Solver::Options::check_gradients is true.
// For finite differencing, each dimension is evaluated at
// slightly shifted values; for the case of central difference,
// this is what gets evaluated:
// delta = gradient_check_numeric_derivative_relative_step_size;
// f_initial = f(x)
// f_forward = f((1 + delta) * x)
// f_backward = f((1 - delta) * x)
// The finite differencing is done along each dimension. The
// reason to use a relative (rather than absolute) step size is
// that this way, numeric differentiation works for functions where
// the arguments are typically large (e.g. 1e9) and when the
// values are small (e.g. 1e-5). It is possible to construct
// "torture cases" which break this finite difference heuristic,
// but they do not come up often in practice.
// TODO(keir): Pick a smarter number than the default above! In
// theory a good choice is sqrt(eps) * x, which for doubles means
// about 1e-8 * x. However, I have found this number too
// optimistic. This number should be exposed for users to change.
double gradient_check_numeric_derivative_relative_step_size = 1e-6;
// If update_state_every_iteration is true, then Ceres Solver will
// guarantee that at the end of every iteration and before any
// user provided IterationCallback is called, the parameter blocks
// are updated to the current best solution found by the
// solver. Thus the IterationCallback can inspect the values of
// the parameter blocks for purposes of computation, visualization
// or termination.
// If update_state_every_iteration is false then there is no such
// guarantee, and user provided IterationCallbacks should not
// expect to look at the parameter blocks and interpret their
// values.
bool update_state_every_iteration = false;
// Callbacks that are executed at the end of each iteration of the
// Minimizer. An iteration may terminate midway, either due to
// numerical failures or because one of the convergence tests has
// been satisfied. In this case none of the callbacks are
// executed.
// Callbacks are executed in the order that they are specified in
// this vector. By default, parameter blocks are updated only at the
// end of the optimization, i.e when the Minimizer terminates. This
// behaviour is controlled by update_state_every_iteration. If the
// user wishes to have access to the updated parameter blocks when
// his/her callbacks are executed, then set
// update_state_every_iteration to true.
// The solver does NOT take ownership of these pointers.
std::vector<IterationCallback*> callbacks;
struct CERES_EXPORT Summary {
// A brief one line description of the state of the solver after
// termination.
std::string BriefReport() const;
// A full multiline description of the state of the solver after
// termination.
std::string FullReport() const;
bool IsSolutionUsable() const;
// Minimizer summary -------------------------------------------------
MinimizerType minimizer_type = TRUST_REGION;
TerminationType termination_type = FAILURE;
// Reason why the solver terminated.
std::string message = "ceres::Solve was not called.";
// Cost of the problem (value of the objective function) before
// the optimization.
double initial_cost = -1.0;
// Cost of the problem (value of the objective function) after the
// optimization.
double final_cost = -1.0;
// The part of the total cost that comes from residual blocks that
// were held fixed by the preprocessor because all the parameter
// blocks that they depend on were fixed.
double fixed_cost = -1.0;
// IterationSummary for each minimizer iteration in order.
std::vector<IterationSummary> iterations;
// Number of minimizer iterations in which the step was
// accepted. Unless use_non_monotonic_steps is true this is also
// the number of steps in which the objective function value/cost
// went down.
int num_successful_steps = -1.0;
// Number of minimizer iterations in which the step was rejected
// either because it did not reduce the cost enough or the step
// was not numerically valid.
int num_unsuccessful_steps = -1.0;
// Number of times inner iterations were performed.
int num_inner_iteration_steps = -1.0;
// Total number of iterations inside the line search algorithm
// across all invocations. We call these iterations "steps" to
// distinguish them from the outer iterations of the line search
// and trust region minimizer algorithms which call the line
// search algorithm as a subroutine.
int num_line_search_steps = -1.0;
// All times reported below are wall times.
// When the user calls Solve, before the actual optimization
// occurs, Ceres performs a number of preprocessing steps. These
// include error checks, memory allocations, and reorderings. This
// time is accounted for as preprocessing time.
double preprocessor_time_in_seconds = -1.0;
// Time spent in the TrustRegionMinimizer.
double minimizer_time_in_seconds = -1.0;
// After the Minimizer is finished, some time is spent in
// re-evaluating residuals etc. This time is accounted for in the
// postprocessor time.
double postprocessor_time_in_seconds = -1.0;
// Some total of all time spent inside Ceres when Solve is called.
double total_time_in_seconds = -1.0;
// Time (in seconds) spent in the linear solver computing the
// trust region step.
double linear_solver_time_in_seconds = -1.0;
// Number of times the Newton step was computed by solving a
// linear system. This does not include linear solves used by
// inner iterations.
int num_linear_solves = -1;
// Time (in seconds) spent evaluating the residual vector.
double residual_evaluation_time_in_seconds = 1.0;
// Number of residual only evaluations.
int num_residual_evaluations = -1;
// Time (in seconds) spent evaluating the jacobian matrix.
double jacobian_evaluation_time_in_seconds = -1.0;
// Number of Jacobian (and residual) evaluations.
int num_jacobian_evaluations = -1;
// Time (in seconds) spent doing inner iterations.
double inner_iteration_time_in_seconds = -1.0;
// Cumulative timing information for line searches performed as part of the
// solve. Note that in addition to the case when the Line Search minimizer
// is used, the Trust Region minimizer also uses a line search when
// solving a constrained problem.
// Time (in seconds) spent evaluating the univariate cost function as part
// of a line search.
double line_search_cost_evaluation_time_in_seconds = -1.0;
// Time (in seconds) spent evaluating the gradient of the univariate cost
// function as part of a line search.
double line_search_gradient_evaluation_time_in_seconds = -1.0;
// Time (in seconds) spent minimizing the interpolating polynomial
// to compute the next candidate step size as part of a line search.
double line_search_polynomial_minimization_time_in_seconds = -1.0;
// Total time (in seconds) spent performing line searches.
double line_search_total_time_in_seconds = -1.0;
// Number of parameter blocks in the problem.
int num_parameter_blocks = -1;
// Number of parameters in the problem.
int num_parameters = -1;
// Dimension of the tangent space of the problem (or the number of
// columns in the Jacobian for the problem). This is different
// from num_parameters if a parameter block is associated with a
// LocalParameterization
int num_effective_parameters = -1;
// Number of residual blocks in the problem.
int num_residual_blocks = -1;
// Number of residuals in the problem.
int num_residuals = -1;
// Number of parameter blocks in the problem after the inactive
// and constant parameter blocks have been removed. A parameter
// block is inactive if no residual block refers to it.
int num_parameter_blocks_reduced = -1;
// Number of parameters in the reduced problem.
int num_parameters_reduced = -1;
// Dimension of the tangent space of the reduced problem (or the
// number of columns in the Jacobian for the reduced
// problem). This is different from num_parameters_reduced if a
// parameter block in the reduced problem is associated with a
// LocalParameterization.
int num_effective_parameters_reduced = -1;
// Number of residual blocks in the reduced problem.
int num_residual_blocks_reduced = -1;
// Number of residuals in the reduced problem.
int num_residuals_reduced = -1;
// Is the reduced problem bounds constrained.
bool is_constrained = false;
// Number of threads specified by the user for Jacobian and
// residual evaluation.
int num_threads_given = -1;
// Number of threads actually used by the solver for Jacobian and
// residual evaluation. This number is not equal to
// num_threads_given if OpenMP is not available.
int num_threads_used = -1;
// Type of the linear solver requested by the user.
LinearSolverType linear_solver_type_given =
#if defined(CERES_NO_SPARSE)
// Type of the linear solver actually used. This may be different
// from linear_solver_type_given if Ceres determines that the
// problem structure is not compatible with the linear solver
// requested or if the linear solver requested by the user is not
// available, e.g. The user requested SPARSE_NORMAL_CHOLESKY but
// no sparse linear algebra library was available.
LinearSolverType linear_solver_type_used =
#if defined(CERES_NO_SPARSE)
// Size of the elimination groups given by the user as hints to
// the linear solver.
std::vector<int> linear_solver_ordering_given;
// Size of the parameter groups used by the solver when ordering
// the columns of the Jacobian. This maybe different from
// linear_solver_ordering_given if the user left
// linear_solver_ordering_given blank and asked for an automatic
// ordering, or if the problem contains some constant or inactive
// parameter blocks.
std::vector<int> linear_solver_ordering_used;
// For Schur type linear solvers, this string describes the
// template specialization which was detected in the problem and
// should be used.
std::string schur_structure_given;
// This is the Schur template specialization that was actually
// instantiated and used. The reason this will be different from
// schur_structure_given is because the corresponding template
// specialization does not exist.
// Template specializations can be added to ceres by editing
// internal/ceres/
std::string schur_structure_used;
// True if the user asked for inner iterations to be used as part
// of the optimization.
bool inner_iterations_given = false;
// True if the user asked for inner iterations to be used as part
// of the optimization and the problem structure was such that
// they were actually performed. e.g., in a problem with just one
// parameter block, inner iterations are not performed.
bool inner_iterations_used = false;
// Size of the parameter groups given by the user for performing
// inner iterations.
std::vector<int> inner_iteration_ordering_given;
// Size of the parameter groups given used by the solver for
// performing inner iterations. This maybe different from
// inner_iteration_ordering_given if the user left
// inner_iteration_ordering_given blank and asked for an automatic
// ordering, or if the problem contains some constant or inactive
// parameter blocks.
std::vector<int> inner_iteration_ordering_used;
// Type of the preconditioner requested by the user.
PreconditionerType preconditioner_type_given = IDENTITY;
// Type of the preconditioner actually used. This may be different
// from linear_solver_type_given if Ceres determines that the
// problem structure is not compatible with the linear solver
// requested or if the linear solver requested by the user is not
// available.
PreconditionerType preconditioner_type_used = IDENTITY;
// Type of clustering algorithm used for visibility based
// preconditioning. Only meaningful when the preconditioner_type
VisibilityClusteringType visibility_clustering_type = CANONICAL_VIEWS;
// Type of trust region strategy.
TrustRegionStrategyType trust_region_strategy_type = LEVENBERG_MARQUARDT;
// Type of dogleg strategy used for solving the trust region
// problem.
DoglegType dogleg_type = TRADITIONAL_DOGLEG;
// Type of the dense linear algebra library used.
DenseLinearAlgebraLibraryType dense_linear_algebra_library_type = EIGEN;
// Type of the sparse linear algebra library used.
SparseLinearAlgebraLibraryType sparse_linear_algebra_library_type =
// Type of line search direction used.
LineSearchDirectionType line_search_direction_type = LBFGS;
// Type of the line search algorithm used.
LineSearchType line_search_type = WOLFE;
// When performing line search, the degree of the polynomial used
// to approximate the objective function.
LineSearchInterpolationType line_search_interpolation_type = CUBIC;
// If the line search direction is NONLINEAR_CONJUGATE_GRADIENT,
// then this indicates the particular variant of non-linear
// conjugate gradient used.
NonlinearConjugateGradientType nonlinear_conjugate_gradient_type =
// If the type of the line search direction is LBFGS, then this
// indicates the rank of the Hessian approximation.
int max_lbfgs_rank = -1;
// Once a least squares problem has been built, this function takes
// the problem and optimizes it based on the values of the options
// parameters. Upon return, a detailed summary of the work performed
// by the preprocessor, the non-linear minimizer and the linear
// solver are reported in the summary object.
virtual void Solve(const Options& options,
Problem* problem,
Solver::Summary* summary);
// Helper function which avoids going through the interface.
CERES_EXPORT void Solve(const Solver::Options& options,
Problem* problem,
Solver::Summary* summary);
} // namespace ceres
#include "ceres/internal/reenable_warnings.h"