EfficientGlobalOptimization¶

class
EfficientGlobalOptimization
(*args)¶ Efficient Global Optimization algorithm.
The EGO algorithm [jones1998] is an adaptative optimization method based on kriging. An initial design of experiment is used to build a first metamodel. At each iteration a new point that maximizes a criterion is chosen as optimizer candidate. The criterion uses a tradeoff between the metamodel value and the conditional variance. Then the new point is evaluated using the original model and the metamodel is relearnt on the extended design of experiment.
 Available constructors:
EfficientGlobalOptimization(problem, krigingResult)
 Parameters
 problem
OptimizationProblem
The optimization problem to solve optionally, a 2nd objective marginal can be used as noise
 krigingResult
KrigingResult
The result of the metamodel on the first design of experiment
 problem
Notes
Each point added to the metamodel design seeks to improve the current minimum. We chose the point so as to maximize an improvement criterion based on the metamodel.
The default criteria is called EI (Expected Improvement) and aims at maximizing the mean improvement:
This criterion is explicited using the kriging mean and variance:
An observation noise variance can be provided thanks to a 2nd objective marginal.
In that case the AEI (Augmented Expected Improvement) formulation is used. As we don’t have access to the real minimum of the function anymore a quantile of the kriging prediction is used, with the constant :
This criterion is minimized over the design points:
The AEI criterion reads:
with
A less computationally expensive noise function can be provided through
setNoiseModel()
to evaluate for the improvement criterion optimization, the objective being only used to compute values and associated noise at design points.By default the criteria is minimized using
MultiStart
with starting points uniformly sampled in the optimization problem bounds, seesetMultiStartExperimentSize()
andsetMultiStartNumber()
. This behavior can be overridden by using another solver withsetOptimizationAlgorithm()
.Examples
>>> import openturns as ot >>> ot.RandomGenerator.SetSeed(0) >>> dim = 4 >>> model = ot.SymbolicFunction(['x1', 'x2', 'x3', 'x4'], ... ['x1*x1+x2^3*x1+x3+x4']) >>> model = ot.MemoizeFunction(model) >>> bounds = ot.Interval([5.0] * dim, [5.0] * dim) >>> problem = ot.OptimizationProblem() >>> problem.setObjective(model) >>> problem.setBounds(bounds) >>> experiment = ot.Composite([0.0] * dim, [1.0, 2.0, 4.0]) >>> inputSample = experiment.generate() >>> outputSample = model(inputSample) >>> covarianceModel = ot.SquaredExponential([2.0] * dim, [0.1]) >>> basis = ot.ConstantBasisFactory(dim).build() >>> kriging = ot.KrigingAlgorithm(inputSample, outputSample, covarianceModel, basis) >>> kriging.run() >>> algo = ot.EfficientGlobalOptimization(problem, kriging.getResult()) >>> algo.setMaximumEvaluationNumber(2) >>> algo.run() >>> result = algo.getResult()
 Attributes
thisown
The membership flag
Methods
Compute the Lagrange multipliers of a problem at a given point.
AEI tradeoff constant accessor.
Accessor to the object’s name.
Correlation length stopping criterion factor accessor.
Expected improvement values.
getId
()Accessor to the object’s id.
Improvement criterion factor accessor.
Accessor to maximum allowed absolute error.
Accessor to maximum allowed constraint error.
Accessor to maximum allowed number of evaluations.
Accessor to maximum allowed number of iterations.
Accessor to maximum allowed relative error.
Accessor to maximum allowed residual error.
Size of the design to draw starting points.
Number of starting points for the criterion optimization.
getName
()Accessor to the object’s name.
Improvement noise model accessor.
Expected improvement solver accessor.
Parameter estimation period accessor.
Accessor to optimization problem.
Accessor to optimization result.
Accessor to the object’s shadowed id.
Accessor to starting point.
Accessor to the verbosity flag.
Accessor to the object’s visibility state.
hasName
()Test if the object is named.
Test if the object has a distinguishable name.
run
()Launch the optimization.
AEI tradeoff constant accessor.
Correlation length stopping criterion factor accessor.
setImprovementFactor
(improvementFactor)Improvement criterion factor accessor.
setMaximumAbsoluteError
(maximumAbsoluteError)Accessor to maximum allowed absolute error.
setMaximumConstraintError
(maximumConstraintError)Accessor to maximum allowed constraint error.
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.
Size of the design to draw starting points.
setMultiStartNumber
(multiStartNumberSize)Number of starting points for the criterion optimization.
setName
(name)Accessor to the object’s name.
setNoiseModel
(noiseModel)Improvement noise model accessor.
setOptimizationAlgorithm
(solver)Expected improvement solver accessor.
Parameter estimation period accessor.
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)¶ Initialize self. See help(type(self)) for accurate signature.

computeLagrangeMultipliers
(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 are associated with the following Lagrangian formulation of the optimization problem:
where .
 The Lagrange multipliers are stored as , where:
is of dimension 0 if there is no equality constraint, else of dimension the dimension of ie the number of scalar equality constraints
and are of dimension 0 if there is no bound constraint, else of dimension of
is of dimension 0 if there is no inequality constraint, else of dimension the dimension of ie the number of scalar inequality constraints
The vector is solution of the following linear system:
If there is no constraint of any kind, is of dimension 0, as well as if no constraint is active.

getAEITradeoff
()¶ AEI tradeoff constant accessor.
 Returns
 cfloat
Used to define a quantile of the kriging prediction at the design points.

getClassName
()¶ Accessor to the object’s name.
 Returns
 class_namestr
The object class name (object.__class__.__name__).

getCorrelationLengthFactor
()¶ Correlation length stopping criterion factor accessor.
When a correlation length becomes smaller than the minimal distance between design point for a single component that means the model tends to be noisy, and the EGO formulation is not adapted anymore.
 Returns
 bfloat
Used to define a stopping criterion on the minimum correlation length: with the minimum distance between design points.

getExpectedImprovement
()¶ Expected improvement values.
 Returns
 ei
Sample
The expected improvement optimal values.
 ei

getId
()¶ Accessor to the object’s id.
 Returns
 idint
Internal unique identifier.

getImprovementFactor
()¶ Improvement criterion factor accessor.
 Returns
 afloat
Used to define a stopping criterion on the improvement criterion: with the current maximum of the improvement and the current optimum.

getMaximumAbsoluteError
()¶ Accessor to maximum allowed absolute error.
 Returns
 maximumAbsoluteErrorfloat
Maximum allowed absolute error, where the absolute error is defined by where and 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 where is the current approximation of the optimum and 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 if , else .

getMaximumResidualError
()¶ Accessor to maximum allowed residual error.
 Returns
 maximumResidualErrorfloat
Maximum allowed residual error, where the residual error is defined by if , else .

getMultiStartExperimentSize
()¶ Size of the design to draw starting points.
 Returns
 multiStartExperimentSizeint
The size of the Monte Carlo design from which to select the best starting points.

getMultiStartNumber
()¶ Number of starting points for the criterion optimization.
 Returns
 multiStartNumberint
The number of starting points for the criterion optimization.

getName
()¶ Accessor to the object’s name.
 Returns
 namestr
The name of the object.

getNoiseModel
()¶ Improvement noise model accessor.
 Returns
 noiseVariance
Function
The noise variance used for the AEI criterion optimization only. Of same input dimension than the objective and 1d output.
 noiseVariance

getOptimizationAlgorithm
()¶ Expected improvement solver accessor.
 Returns
 solver
OptimizationSolver
The solver used to optimize the expected improvement
 solver

getParameterEstimationPeriod
()¶ Parameter estimation period accessor.
 Returns
 periodint
The number of iterations between covariance parameters relearn. Default is 1 (each iteration). Can be set to 0 (never).

getProblem
()¶ Accessor to optimization problem.
 Returns
 problem
OptimizationProblem
Optimization problem.
 problem

getResult
()¶ Accessor to optimization result.
 Returns
 result
OptimizationResult
Result class.
 result

getShadowedId
()¶ Accessor to the object’s shadowed id.
 Returns
 idint
Internal unique identifier.

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.

setAEITradeoff
(c)¶ AEI tradeoff constant accessor.
 Parameters
 cfloat
Used to define a quantile of the kriging prediction at the design points.

setCorrelationLengthFactor
(b)¶ Correlation length stopping criterion factor accessor.
When a correlation length becomes smaller than the minimal distance between design point for a single component that means the model tends to be noisy, and the EGO formulation is not adapted anymore.
 Parameters
 bfloat
Used to define a stopping criterion on the minimum correlation length: with the minimum distance between design points.

setImprovementFactor
(improvementFactor)¶ Improvement criterion factor accessor.
 Parameters
 afloat
Used to define a stopping criterion on the improvement criterion: with the current maximum of the improvement and the current optimum.

setMaximumAbsoluteError
(maximumAbsoluteError)¶ Accessor to maximum allowed absolute error.
 Parameters
 maximumAbsoluteErrorfloat
Maximum allowed absolute error, where the absolute error is defined by where and 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 where is the current approximation of the optimum and 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 if , else .

setMaximumResidualError
(maximumResidualError)¶ Accessor to maximum allowed residual error.
 Parameters
 Maximum allowed residual error, where the residual error is defined by
if , else .

setMultiStartExperimentSize
(multiStartExperimentSize)¶ Size of the design to draw starting points.
 Parameters
 multiStartExperimentSizeint
The size of the Monte Carlo design from which to select the best starting points. The default number can be tweaked with the EfficientGlobalOptimizationDefaultMultiStartExperimentSize key from
ResourceMap
.

setMultiStartNumber
(multiStartNumberSize)¶ Number of starting points for the criterion optimization.
 Parameters
 multiStartNumberint
The number of starting points for the criterion optimization. The default number can be tweaked with the EfficientGlobalOptimizationDefaultMultiStartNumber key from
ResourceMap
.

setName
(name)¶ Accessor to the object’s name.
 Parameters
 namestr
The name of the object.

setNoiseModel
(noiseModel)¶ Improvement noise model accessor.
 Parameters
 noiseVariance
Function
The noise variance used for the AEI criterion optimization only. Of same input dimension than the objective and 1d output.
 noiseVariance

setOptimizationAlgorithm
(solver)¶ Expected improvement solver accessor.
 Parameters
 solver
OptimizationSolver
The solver used to optimize the expected improvement
 solver

setParameterEstimationPeriod
(parameterEstimationPeriod)¶ Parameter estimation period accessor.
 Parameters
 periodint
The number of iterations between covariance parameters relearn. Default is 1 (each iteration). Can be set to 0 (never). The default number can be tweaked with the EfficientGlobalOptimizationDefaultParameterEstimationPeriod key from
ResourceMap
.

setProblem
(problem)¶ Accessor to optimization problem.
 Parameters
 problem
OptimizationProblem
Optimization problem.
 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'], ['(1x1)^2+100*(x2x1^2)^2']) >>> problem = ot.OptimizationProblem(rosenbrock) >>> solver = ot.OptimizationAlgorithm(problem) >>> solver.setStartingPoint([0, 0]) >>> solver.setMaximumResidualError(1.e3) >>> solver.setMaximumIterationNumber(100) >>> def report_progress(progress): ... sys.stderr.write(' progress=' + str(progress) + '%\n') >>> solver.setProgressCallback(report_progress) >>> solver.run()

setResult
(result)¶ Accessor to optimization result.
 Parameters
 result
OptimizationResult
Result class.
 result

setShadowedId
(id)¶ Accessor to the object’s shadowed id.
 Parameters
 idint
Internal unique identifier.

setStartingPoint
(startingPoint)¶ Accessor to starting point.
 Parameters
 startingPoint
Point
Starting point.
 startingPoint

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'], ['(1x1)^2+100*(x2x1^2)^2']) >>> problem = ot.OptimizationProblem(rosenbrock) >>> solver = ot.OptimizationAlgorithm(problem) >>> solver.setStartingPoint([0, 0]) >>> solver.setMaximumResidualError(1.e3) >>> solver.setMaximumIterationNumber(100) >>> 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.

thisown
¶ The membership flag