Bonmin¶

class Bonmin(*args)¶

Base class for optimization solvers from Bonmin library.

Available constructors:

Bonmin(problem)

Bonmin(problem, algoName)

Parameters

problemOptimizationProblem, optional: Optimization problem to solve. Default is an empty problem.
algoNamestr, optional: Identifier of the optimization method to use. Default is ‘B-BB’

See also

AbdoRackwitz, Cobyla, Dlib, NLopt

Notes

Available algorithms:

Bonmin provides algorithms for the resolution of general optimization only. In particular, least squares problems or nearest point problems are not supported.

Bonmin provides the following algorithms:

Algorithm

Description

B-BB

NLP-based branch-and-bound

B-OA

Outer-approximation decomposition

B-QG

Quesada and Grossmann’s branch-and-cut

B-Hyb

Hybrid outer-approximation based branch-and-cut

B-iFP

Iterated feasibility pump algorithm

Algorithms parameters:

Bonmin algorithms can be adapted using numerous parameters, described here. These parameters can be modified using the ResourceMap. For every option optionName, one simply add a key named Bonmin-optionName with the value to use, as shown below:
>>> import openturns as ot
>>> ot.ResourceMap.AddAsUnsignedInteger('Bonmin-bonmin.iteration_limit', 100)
>>> ot.ResourceMap.AddAsScalar('Bonmin-bonmin.time_limit', 30.0)

Convergence criteria:

To estimate the convergence of the algorithm during the optimization process, Bonmin uses specific tolerance parameters, different from the standard absolute/relative/residual errors used in OpenTURNS. The definition of Bonmin’s parameters can be found in this paper, page 3.

Thus the attributes maximumAbsoluteError, maximumRelativeError, maximumResidualError and maximumConstraintError defined in OptimizationAlgorithm are not used in this case. The tolerances used by Bonmin can be set using specific options (e.g. tol, bonmin.ecp_abs_tol and bonmin.ecp_rel_tol), depending on the algorithm used.

Examples

The code below ensures the optimization of the following problem:

$min \left( - x_0 - x_1 - x_2 \right)$

subject to

$\left(x_1 - \frac{1}{2}\right)^2 + \left(x_2 - \frac{1}{2}\right)^2 \leq \frac{1}{4} x_0 - x_1 \leq 0 x_0 + x_2 + x_3 \leq 2 x_0 \in \{0,1\}^n (x_1, x_2) \in \mathbb{R}^2 x_3 \in \mathbb{N}$

>>> import openturns as ot

>>> # Definition of objective function
>>> objectiveFunction = ot.SymbolicFunction(['x0','x1','x2','x3'], ['-x0 -x1 -x2'])

>>> # Definition of variables bounds
>>> bounds = ot.Interval([0,0,0,0],[1,1e308,1e308,5],[True,True,True,True],[True,False,False,True])

>>> # Definition of variables types
>>> variablesType = [ot.OptimizationProblemImplementation.BINARY,
... ot.OptimizationProblemImplementation.CONTINUOUS,
... ot.OptimizationProblemImplementation.CONTINUOUS,
... ot.OptimizationProblemImplementation.INTEGER]

In OpenTURNS, inequality constraints are defined by a function $h$ such that $h(x) \geq 0$ . The inequality expression above has to be modified to match this formulation.

>>> # Definition of constraints
>>> # Constraints in OpenTURNS are defined as g(x) = 0 and h(x) >= 0
>>> #    No equality constraint -> nothing to do
>>> #    Inequality constraints:
>>> h = ot.SymbolicFunction(['x0','x1','x2','x3'], ['-(x1-1/2)^2 - (x2-1/2)^2 + 1/4', '-x0 + x1', '-x0 - x2 - x3 + 2'])

>>> # Setting up Bonmin problem
>>> problem = ot.OptimizationProblem(objectiveFunction)
>>> problem.setBounds(bounds)
>>> problem.setVariablesType(variablesType)
>>> problem.setInequalityConstraint(h)

>>> bonminAlgorithm = ot.Bonmin(problem,'B-BB')
>>> bonminAlgorithm.setStartingPoint([0,0,0,0])
>>> bonminAlgorithm.setMaximumEvaluationNumber(10000)
>>> bonminAlgorithm.setMaximumIterationNumber(1000)
>>> ot.ResourceMap.AddAsString('Bonmin-mu_oracle','loqo')
>>> ot.ResourceMap.SetAsScalar('Bonmin-bonmin.time_limit', 5.0)

>>> # Running the solver
>>> bonminAlgorithm.run() 

>>> # Retrieving the results
>>> result = bonminAlgorithm.getResult() 
>>> optimalPoint = result.getOptimalPoint() 
>>> optimalValue = result.getOptimalValue() 
>>> evaluationNumber = result.getInputSample().getSize() 

Methods

`GetAlgorithmNames`()	Retrieves the names of the available optimization algorithms.
`IsAvailable`()	Retrieves the names of the available optimization algorithms.
`computeLagrangeMultipliers`(self, x)	Compute the Lagrange multipliers of a problem at a given point.
`getAlgorithmName`(self)	Accessor to algoName parameter.
`getClassName`(self)	Accessor to the object’s name.
`getId`(self)	Accessor to the object’s id.
`getMaximumAbsoluteError`(self)	Accessor to maximum allowed absolute error.
`getMaximumConstraintError`(self)	Accessor to maximum allowed constraint error.
`getMaximumEvaluationNumber`(self)	Accessor to maximum allowed number of evaluations.
`getMaximumIterationNumber`(self)	Accessor to maximum allowed number of iterations.
`getMaximumRelativeError`(self)	Accessor to maximum allowed relative error.
`getMaximumResidualError`(self)	Accessor to maximum allowed residual error.
`getName`(self)	Accessor to the object’s name.
`getProblem`(self)	Accessor to optimization problem.
`getResult`(self)	Accessor to optimization result.
`getShadowedId`(self)	Accessor to the object’s shadowed id.
`getStartingPoint`(self)	Accessor to starting point.
`getVerbose`(self)	Accessor to the verbosity flag.
`getVisibility`(self)	Accessor to the object’s visibility state.
`hasName`(self)	Test if the object is named.
`hasVisibleName`(self)	Test if the object has a distinguishable name.
`run`(self)	Launch the optimization.
`setAlgorithmName`(self, algoName)	Accessor to algoName parameter.
`setMaximumAbsoluteError`(self, …)	Accessor to maximum allowed absolute error.
`setMaximumConstraintError`(self, …)	Accessor to maximum allowed constraint error.
`setMaximumEvaluationNumber`(self, …)	Accessor to maximum allowed number of evaluations.
`setMaximumIterationNumber`(self, …)	Accessor to maximum allowed number of iterations.
`setMaximumRelativeError`(self, …)	Accessor to maximum allowed relative error.
`setMaximumResidualError`(self, …)	Accessor to maximum allowed residual error.
`setName`(self, name)	Accessor to the object’s name.
`setProblem`(self, problem)	Accessor to optimization problem.
`setProgressCallback`(self, \*args)	Set up a progress callback.
`setResult`(self, result)	Accessor to optimization result.
`setShadowedId`(self, id)	Accessor to the object’s shadowed id.
`setStartingPoint`(self, startingPoint)	Accessor to starting point.
`setStopCallback`(self, \*args)	Set up a stop callback.
`setVerbose`(self, verbose)	Accessor to the verbosity flag.
`setVisibility`(self, visible)	Accessor to the object’s visibility state.

__init__(self, \*args)¶: Initialize self. See help(type(self)) for accurate signature.

static GetAlgorithmNames()¶

Retrieves the names of the available optimization algorithms.

Returns

algoNameDescription: The names of the available optimization algorithms.

static IsAvailable()¶

Retrieves the names of the available optimization algorithms.

Returns

algoNamebool: Returns true if Bonmin support is available, false if not.

computeLagrangeMultipliers(self, x)¶

Compute the Lagrange multipliers of a problem at a given point.

Parameters

xsequence of float: Point at which the Lagrange multipliers are computed.

Returns

lagrangeMultipliersequence of float: Lagrange multipliers of the problem at the given point.

Notes

The Lagrange multipliers $\vect{\lambda}$ are associated with the following Lagrangian formulation of the optimization problem:

$\cL(\vect{x}, \vect{\lambda}_{eq}, \vect{\lambda}_{\ell}, \vect{\lambda}_{u}, \vect{\lambda}_{ineq}) = J(\vect{x}) + \Tr{\vect{\lambda}}_{eq} g(\vect{x}) + \Tr{\vect{\lambda}}_{\ell} (\vect{x}-\vect{\ell})^{+} + \Tr{\vect{\lambda}}_{u} (\vect{u}-\vect{x})^{+} + \Tr{\vect{\lambda}}_{ineq} h^{+}(\vect{x})$

where $\vect{\alpha}^{+}=(\max(0,\alpha_1),\hdots,\max(0,\alpha_n))$ .

The Lagrange multipliers are stored as $(\vect{\lambda}_{eq}, \vect{\lambda}_{\ell}, \vect{\lambda}_{u}, \vect{\lambda}_{ineq})$ , where:

$\vect{\lambda}_{eq}$ is of dimension 0 if there is no equality constraint, else of dimension the dimension of $g(\vect{x})$ ie the number of scalar equality constraints
$\vect{\lambda}_{\ell}$ and $\vect{\lambda}_{u}$ are of dimension 0 if there is no bound constraint, else of dimension of $\vect{x}$
$\vect{\lambda}_{eq}$ is of dimension 0 if there is no inequality constraint, else of dimension the dimension of $h(\vect{x})$ ie the number of scalar inequality constraints

The vector $\vect{\lambda}$ is solution of the following linear system:

$\Tr{\vect{\lambda}}_{eq}\left[\dfrac{\partial g}{\partial\vect{x}}(\vect{x})\right]+ \Tr{\vect{\lambda}}_{\ell}\left[\dfrac{\partial (\vect{x}-\vect{\ell})^{+}}{\partial\vect{x}}(\vect{x})\right]+ \Tr{\vect{\lambda}}_{u}\left[\dfrac{\partial (\vect{u}-\vect{x})^{+}}{\partial\vect{x}}(\vect{x})\right]+ \Tr{\vect{\lambda}}_{ineq}\left[\dfrac{\partial h}{\partial\vect{x}}(\vect{x})\right]=-\dfrac{\partial J}{\partial\vect{x}}(\vect{x})$

If there is no constraint of any kind, $\vect{\lambda}$ is of dimension 0, as well as if no constraint is active.

getAlgorithmName(self)¶

Accessor to algoName parameter. Retrieves the name of the optimization algorithm used.

Returns

algoNamestr: The name of the optimization algorithm used.

getClassName(self)¶

Accessor to the object’s name.

Returns

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

getId(self)¶

Accessor to the object’s id.

Returns

idint: Internal unique identifier.

getMaximumAbsoluteError(self)¶

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(self)¶

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(self)¶

Accessor to maximum allowed number of evaluations.

Returns

Nint: Maximum allowed number of evaluations.

getMaximumIterationNumber(self)¶

Accessor to maximum allowed number of iterations.

Returns

Nint: Maximum allowed number of iterations.

getMaximumRelativeError(self)¶

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(self)¶

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(self)¶

Accessor to the object’s name.

Returns

namestr: The name of the object.

getProblem(self)¶

Accessor to optimization problem.

Returns

problemOptimizationProblem: Optimization problem.

getResult(self)¶

Accessor to optimization result.

Returns

resultOptimizationResult: Result class.

getShadowedId(self)¶

Accessor to the object’s shadowed id.

Returns

idint: Internal unique identifier.

getStartingPoint(self)¶

Accessor to starting point.

Returns

startingPointPoint: Starting point.

getVerbose(self)¶

Accessor to the verbosity flag.

Returns

verbosebool: Verbosity flag state.

getVisibility(self)¶

Accessor to the object’s visibility state.

Returns

visiblebool: Visibility flag.

hasName(self)¶

Test if the object is named.

Returns

hasNamebool: True if the name is not empty.

hasVisibleName(self)¶

Test if the object has a distinguishable name.

Returns

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

run(self)¶: Launch the optimization.

setAlgorithmName(self, algoName)¶

Accessor to algoName parameter. Sets the optimization algorithm to use. Possible values for algoName are B-BB, B-OA, B-QG, B-Hyb, B-Ecp, B-iFP. See Bonmin’s online documentation for more details. Defautl is B-BB.

Parameters

algoNamestr: The name of the optimization algorithm to use.

setMaximumAbsoluteError(self, 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(self, 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(self, maximumEvaluationNumber)¶

Accessor to maximum allowed number of evaluations.

Parameters

Nint: Maximum allowed number of evaluations.

setMaximumIterationNumber(self, maximumIterationNumber)¶

Accessor to maximum allowed number of iterations.

Parameters

Nint: Maximum allowed number of iterations.

setMaximumRelativeError(self, 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(self, maximumResidualError)¶

Accessor to maximum allowed residual error.

Parameters

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 $\epsilon^r_n=-1$ .

setName(self, name)¶

Accessor to the object’s name.

Parameters

namestr: The name of the object.

setProblem(self, problem)¶

Accessor to optimization problem.

Parameters

problemOptimizationProblem: Optimization problem.

setProgressCallback(self, \*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.setMaximumIterationNumber(100)
>>> def report_progress(progress):
...     sys.stderr.write('-- progress=' + str(progress) + '%\n')
>>> solver.setProgressCallback(report_progress)
>>> solver.run()

setResult(self, result)¶

Accessor to optimization result.

Parameters

resultOptimizationResult: Result class.

setShadowedId(self, id)¶

Accessor to the object’s shadowed id.

Parameters

idint: Internal unique identifier.

setStartingPoint(self, startingPoint)¶

Accessor to starting point.

Parameters

startingPointPoint: Starting point.

setStopCallback(self, \*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.setMaximumIterationNumber(100)
>>> def ask_stop():
...     return True
>>> solver.setStopCallback(ask_stop)
>>> solver.run()

setVerbose(self, verbose)¶

Accessor to the verbosity flag.

Parameters

verbosebool: Verbosity flag state.

setVisibility(self, visible)¶

Accessor to the object’s visibility state.

Parameters

visiblebool: Visibility flag.

OpenTURNS

An Open source initiative for the Treatment of Uncertainties, Risks'N Statistics

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Bonmin¶

Algorithm	Description
B-BB	NLP-based branch-and-bound
B-OA	Outer-approximation decomposition
B-QG	Quesada and Grossmann’s branch-and-cut
B-Hyb	Hybrid outer-approximation based branch-and-cut
B-iFP	Iterated feasibility pump algorithm