Ceres¶

class Ceres(*args)¶

Interface to Ceres Solver.

This class exposes the solvers from the non-linear least squares optimization library [ceres2012].

More details about least squares algorithms are available here.

Algorithms are also available for general unconstrained optimization.

Parameters:

problemOptimizationProblem: Optimization problem to solve, either least-squares or general (unconstrained).
algoNamestr: The identifier of the algorithm. Use GetAlgorithmNames() to list available names.

See also

AbdoRackwitz, Cobyla, CMinpack, NLopt, SQP, TNC

Notes

Solvers use first order derivative information.

As for constraint support, only the trust-region solvers allow for bound constraints:

Algorithm	Method type	Problem type support	Constraint support
LEVENBERG_MARQUARDT	trust-region	least-squares	bounds
DOGLEG	trust-region	least-squares	bounds
STEEPEST_DESCENT	line-search	least-squares, general	none
NONLINEAR_CONJUGATE_GRADIENT	line-search	least-squares, general	none
LBFGS	line-search	least-squares, general	none
BFGS	line-search	least-squares, general	none

Ceres least squares solver can be further tweaked thanks to the following ResourceMap parameters, refer to nlls solver options for more details.

Key	Type
Ceres-minimizer_type	str
Ceres-line_search_direction_type	str
Ceres-line_search_type	str
Ceres-nonlinear_conjugate_gradient_type	str
Ceres-max_lbfgs_rank	int
Ceres-use_approximate_eigenvalue_bfgs_scaling	bool
Ceres-line_search_interpolation_type	str
Ceres-min_line_search_step_size	float
Ceres-line_search_sufficient_function_decrease	float
Ceres-max_line_search_step_contraction	float
Ceres-min_line_search_step_contraction	float
Ceres-max_num_line_search_step_size_iterations	int
Ceres-max_num_line_search_direction_restarts	int
Ceres-line_search_sufficient_curvature_decrease	float
Ceres-max_line_search_step_expansion	float
Ceres-trust_region_strategy_type	str
Ceres-dogleg_type	str
Ceres-use_nonmonotonic_steps	bool
Ceres-max_consecutive_nonmonotonic_steps	int
Ceres-max_num_iterations	int
Ceres-max_solver_time_in_seconds	float
Ceres-num_threads	int
Ceres-initial_trust_region_radius	float
Ceres-max_trust_region_radius	float
Ceres-min_trust_region_radius	float
Ceres-min_relative_decrease	float
Ceres-min_lm_diagonal	float
Ceres-max_lm_diagonal	float
Ceres-max_num_consecutive_invalid_steps	int
Ceres-function_tolerance	float
Ceres-gradient_tolerance	float
Ceres-parameter_tolerance	float
Ceres-preconditioner_type	str
Ceres-visibility_clustering_type	str
Ceres-dense_linear_algebra_library_type	str
Ceres-sparse_linear_algebra_library_type	str
Ceres-use_explicit_schur_complement	bool
Ceres-use_postordering	bool
Ceres-dynamic_sparsity	bool
Ceres-min_linear_solver_iterations	int
Ceres-max_linear_solver_iterations	int
Ceres-eta	float
Ceres-jacobi_scaling	bool
Ceres-use_inner_iterations	bool
Ceres-inner_iteration_tolerance	float
Ceres-logging_type	str
Ceres-minimizer_progress_to_stdout	bool
Ceres-trust_region_problem_dump_directory	str
Ceres-trust_region_problem_dump_format_type	str
Ceres-check_gradients	bool
Ceres-gradient_check_relative_precision	float
Ceres-gradient_check_numeric_derivative_relative_step_size	float
Ceres-update_state_every_iteration	bool

Ceres unconstrained solver can be further tweaked using the following ResourceMap parameters, refer to gradient solver options for more details.

Key	Type
Ceres-line_search_direction_type	str
Ceres-line_search_type	str
Ceres-nonlinear_conjugate_gradient_type	str
Ceres-max_lbfgs_rank	int
Ceres-use_approximate_eigenvalue_bfgs_scaling	bool
Ceres-line_search_interpolation_type	str
Ceres-min_line_search_step_size	float
Ceres-line_search_sufficient_function_decrease	float
Ceres-max_line_search_step_contraction	float
Ceres-min_line_search_step_contraction	float
Ceres-max_num_line_search_step_size_iterations	int
Ceres-max_num_line_search_direction_restarts	int
Ceres-line_search_sufficient_curvature_decrease	float
Ceres-max_line_search_step_expansion	float
Ceres-max_num_iterations	int
Ceres-max_solver_time_in_seconds	float
Ceres-function_tolerance	float
Ceres-gradient_tolerance	float
Ceres-parameter_tolerance	float
Ceres-logging_type	str
Ceres-minimizer_progress_to_stdout	bool

Examples

List available algorithms:

>>> import openturns as ot
>>> print(ot.Ceres.GetAlgorithmNames())
[LEVENBERG_MARQUARDT,DOGLEG,...

Solve a least-squares problem:

>>> dim = 2
>>> residualFunction = ot.SymbolicFunction(['x0', 'x1'], ['10*(x1-x0^2)', '1-x0'])
>>> problem = ot.LeastSquaresProblem(residualFunction)
>>> problem.setBounds(ot.Interval([-3.0] * dim, [5.0] * dim))
>>> ot.ResourceMap.AddAsScalar('Ceres-gradient_tolerance', 1e-5)  
>>> algo = ot.Ceres(problem, 'LEVENBERG_MARQUARDT')  
>>> algo.setStartingPoint([0.0] * dim)  
>>> algo.run()  
>>> result = algo.getResult()  
>>> x_star = result.getOptimalPoint()  
>>> y_star = result.getOptimalValue()  

Or, solve a general optimization problem:

>>> dim = 4
>>> linear = ot.SymbolicFunction(['x1', 'x2', 'x3', 'x4'], ['(x1-1)^2+(x2-2)^2+(x3-3)^2+(x4-4)^2'])
>>> problem = ot.OptimizationProblem(linear)
>>> ot.ResourceMap.AddAsScalar('Ceres-gradient_tolerance', 1e-5)  
>>> algo = ot.Ceres(problem, 'BFGS')  
>>> algo.setStartingPoint([0.0] * 4)  
>>> algo.run()  
>>> result = algo.getResult()  
>>> x_star = result.getOptimalPoint()  
>>> y_star = result.getOptimalValue()  

Methods

`GetAlgorithmNames`()	Accessor to the list of algorithms provided, by names.
`getAlgorithmName`()	Accessor to the algorithm name.
`getClassName`()	Accessor to the object's name.
`getId`()	Accessor to the object's id.
`getMaximumAbsoluteError`()	Accessor to maximum allowed absolute error.
`getMaximumConstraintError`()	Accessor to maximum allowed constraint error.
`getMaximumEvaluationNumber`()	Accessor to maximum allowed number of evaluations.
`getMaximumIterationNumber`()	Accessor to maximum allowed number of iterations.
`getMaximumRelativeError`()	Accessor to maximum allowed relative error.
`getMaximumResidualError`()	Accessor to maximum allowed residual error.
`getName`()	Accessor to the object's name.
`getProblem`()	Accessor to optimization problem.
`getResult`()	Accessor to optimization result.
`getShadowedId`()	Accessor to the object's shadowed id.
`getStartingPoint`()	Accessor to starting point.
`getVerbose`()	Accessor to the verbosity flag.
`getVisibility`()	Accessor to the object's visibility state.
`hasName`()	Test if the object is named.
`hasVisibleName`()	Test if the object has a distinguishable name.
`run`()	Launch the optimization.
`setAlgorithmName`(algoName)	Accessor to the algorithm name.
`setMaximumAbsoluteError`(maximumAbsoluteError)	Accessor to maximum allowed absolute error.
`setMaximumConstraintError`(maximumConstraintError)	Accessor to maximum allowed constraint error.
`setMaximumEvaluationNumber`(...)	Accessor to maximum allowed number of evaluations.
`setMaximumIterationNumber`(maximumIterationNumber)	Accessor to maximum allowed number of iterations.
`setMaximumRelativeError`(maximumRelativeError)	Accessor to maximum allowed relative error.
`setMaximumResidualError`(maximumResidualError)	Accessor to maximum allowed residual error.
`setName`(name)	Accessor to the object's name.
`setProblem`(problem)	Accessor to optimization problem.
`setProgressCallback`(*args)	Set up a progress callback.
`setResult`(result)	Accessor to optimization result.
`setShadowedId`(id)	Accessor to the object's shadowed id.
`setStartingPoint`(startingPoint)	Accessor to starting point.
`setStopCallback`(*args)	Set up a stop callback.
`setVerbose`(verbose)	Accessor to the verbosity flag.
`setVisibility`(visible)	Accessor to the object's visibility state.

__init__(*args)¶

static GetAlgorithmNames()¶

Accessor to the list of algorithms provided, by names.

Returns:

namesDescription

List of algorithm names provided, according to its naming convention.

The trust region methods are not able to solve general optimization problems, in that case a warning is printed and the default line search method is used instead.

Examples

>>> import openturns as ot
>>> print(ot.Ceres.GetAlgorithmNames())
[LEVENBERG_MARQUARDT,DOGLEG,STEEPEST_DESCENT,NONLINEAR_CONJUGATE_GRADIENT,LBFGS,BFGS]

getAlgorithmName()¶

Accessor to the algorithm name.

Returns:

algoNamestr: The identifier of the algorithm.

getClassName()¶

Accessor to the object’s name.

Returns:

class_namestr: The object class name (object.__class__.__name__).

getId()¶

Accessor to the object’s id.

Returns:

idint: Internal unique identifier.

getMaximumAbsoluteError()¶

Accessor to maximum allowed absolute error.

Returns:

maximumAbsoluteErrorfloat: Maximum allowed absolute error, where the absolute error is defined by $\epsilon^a_n=\|\vect{x}_{n+1}-\vect{x}_n\|_{\infty}$ where $\vect{x}_{n+1}$ and $\vect{x}_n$ are two consecutive approximations of the optimum.

getMaximumConstraintError()¶

Accessor to maximum allowed constraint error.

Returns:

maximumConstraintErrorfloat: Maximum allowed constraint error, where the constraint error is defined by $\gamma_n=\|g(\vect{x}_n)\|_{\infty}$ where $\vect{x}_n$ is the current approximation of the optimum and $g$ is the function that gathers all the equality and inequality constraints (violated values only)

getMaximumEvaluationNumber()¶

Accessor to maximum allowed number of evaluations.

Returns:

Nint: Maximum allowed number of evaluations.

getMaximumIterationNumber()¶

Accessor to maximum allowed number of iterations.

Returns:

Nint: Maximum allowed number of iterations.

getMaximumRelativeError()¶

Accessor to maximum allowed relative error.

Returns:

maximumRelativeErrorfloat: Maximum allowed relative error, where the relative error is defined by $\epsilon^r_n=\epsilon^a_n/\|\vect{x}_{n+1}\|_{\infty}$ if $\|\vect{x}_{n+1}\|_{\infty}\neq 0$ , else $\epsilon^r_n=-1$ .

getMaximumResidualError()¶

Accessor to maximum allowed residual error.

Returns:

maximumResidualErrorfloat: Maximum allowed residual error, where the residual error is defined by $\epsilon^r_n=\frac{\|f(\vect{x}_{n+1})-f(\vect{x}_{n})\|}{\|f(\vect{x}_{n+1})\|}$ if $\|f(\vect{x}_{n+1})\|\neq 0$ , else $\epsilon^r_n=-1$ .

getName()¶

Accessor to the object’s name.

Returns:

namestr: The name of the object.

getProblem()¶

Accessor to optimization problem.

Returns:

problemOptimizationProblem: Optimization problem.

getResult()¶

Accessor to optimization result.

Returns:

resultOptimizationResult: Result class.

getShadowedId()¶

Accessor to the object’s shadowed id.

Returns:

idint: Internal unique identifier.

getStartingPoint()¶

Accessor to starting point.

Returns:

startingPointPoint: Starting point.

getVerbose()¶

Accessor to the verbosity flag.

Returns:

verbosebool: Verbosity flag state.

getVisibility()¶

Accessor to the object’s visibility state.

Returns:

visiblebool: Visibility flag.

hasName()¶

Test if the object is named.

Returns:

hasNamebool: True if the name is not empty.

hasVisibleName()¶

Test if the object has a distinguishable name.

Returns:

hasVisibleNamebool: True if the name is not empty and not the default one.

run()¶: Launch the optimization.

setAlgorithmName(algoName)¶

Accessor to the algorithm name.

Parameters:

algoNamestr: The identifier of the algorithm.

setMaximumAbsoluteError(maximumAbsoluteError)¶

Accessor to maximum allowed absolute error.

Parameters:

maximumAbsoluteErrorfloat: Maximum allowed absolute error, where the absolute error is defined by $\epsilon^a_n=\|\vect{x}_{n+1}-\vect{x}_n\|_{\infty}$ where $\vect{x}_{n+1}$ and $\vect{x}_n$ are two consecutive approximations of the optimum.

setMaximumConstraintError(maximumConstraintError)¶

Accessor to maximum allowed constraint error.

Parameters:

maximumConstraintErrorfloat: Maximum allowed constraint error, where the constraint error is defined by $\gamma_n=\|g(\vect{x}_n)\|_{\infty}$ where $\vect{x}_n$ is the current approximation of the optimum and $g$ is the function that gathers all the equality and inequality constraints (violated values only)

setMaximumEvaluationNumber(maximumEvaluationNumber)¶

Accessor to maximum allowed number of evaluations.

Parameters:

Nint: Maximum allowed number of evaluations.

setMaximumIterationNumber(maximumIterationNumber)¶

Accessor to maximum allowed number of iterations.

Parameters:

Nint: Maximum allowed number of iterations.

setMaximumRelativeError(maximumRelativeError)¶

Accessor to maximum allowed relative error.

Parameters:

maximumRelativeErrorfloat: Maximum allowed relative error, where the relative error is defined by $\epsilon^r_n=\epsilon^a_n/\|\vect{x}_{n+1}\|_{\infty}$ if $\|\vect{x}_{n+1}\|_{\infty}\neq 0$ , else $\epsilon^r_n=-1$ .

setMaximumResidualError(maximumResidualError)¶

Accessor to maximum allowed residual error.

Parameters:

Maximum allowed residual error, where the residual error is defined by
:math:`epsilon^r_n=frac{|f(vect{x}_{n+1})-f(vect{x}_{n})|}{|f(vect{x}_{n+1})|}`
if :math:`|f(vect{x}_{n+1})|neq 0`, else :math:`epsilon^r_n=-1`.

setName(name)¶

Accessor to the object’s name.

Parameters:

namestr: The name of the object.

setProblem(problem)¶

Accessor to optimization problem.

Parameters:

problemOptimizationProblem: Optimization problem.

setProgressCallback(*args)¶

Set up a progress callback.

Can be used to programmatically report the progress of an optimization.

Parameters:

callbackcallable: Takes a float as argument as percentage of progress.

Examples

>>> import sys
>>> import openturns as ot
>>> rosenbrock = ot.SymbolicFunction(['x1', 'x2'], ['(1-x1)^2+100*(x2-x1^2)^2'])
>>> problem = ot.OptimizationProblem(rosenbrock)
>>> solver = ot.OptimizationAlgorithm(problem)
>>> solver.setStartingPoint([0, 0])
>>> solver.setMaximumResidualError(1.e-3)
>>> solver.setMaximumEvaluationNumber(10000)
>>> def report_progress(progress):
...     sys.stderr.write('-- progress=' + str(progress) + '%\n')
>>> solver.setProgressCallback(report_progress)
>>> solver.run()

setResult(result)¶

Accessor to optimization result.

Parameters:

resultOptimizationResult: Result class.

setShadowedId(id)¶

Accessor to the object’s shadowed id.

Parameters:

idint: Internal unique identifier.

setStartingPoint(startingPoint)¶

Accessor to starting point.

Parameters:

startingPointPoint: Starting point.

setStopCallback(*args)¶

Set up a stop callback.

Can be used to programmatically stop an optimization.

Parameters:

callbackcallable: Returns an int deciding whether to stop or continue.

Examples

>>> import openturns as ot
>>> rosenbrock = ot.SymbolicFunction(['x1', 'x2'], ['(1-x1)^2+100*(x2-x1^2)^2'])
>>> problem = ot.OptimizationProblem(rosenbrock)
>>> solver = ot.OptimizationAlgorithm(problem)
>>> solver.setStartingPoint([0, 0])
>>> solver.setMaximumResidualError(1.e-3)
>>> solver.setMaximumEvaluationNumber(10000)
>>> def ask_stop():
...     return True
>>> solver.setStopCallback(ask_stop)
>>> solver.run()

setVerbose(verbose)¶

Accessor to the verbosity flag.

Parameters:

verbosebool: Verbosity flag state.

setVisibility(visible)¶

Accessor to the object’s visibility state.

Parameters:

visiblebool: Visibility flag.

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