OptimizationResult¶

class OptimizationResult(*args)¶

Optimization result.

Returned by optimization solvers, see OptimizationAlgorithm.

Available constructors:

OptimizationResult(optimalPoint, optimalValue, evaluationNumber, absoluteError, relativeError, residualError, constraintError)

OptimizationResult(inputDimension, outputDimension)

Parameters

optimalPointsequence of float: Optimal point.
optimalValuesequence of float: Value at optimal point.
evaluationNumberint: Number of evaluations.
absoluteErrorfloat: Absolute error of the input $\vect{x}$ . See the getAbsoluteError method for details.
relativeErrorfloat: Relative error of the input $\vect{x}$ . See the getRelativeError method for details.
residualErrorfloat: Residual error of the output of the objective function $f(\vect{x})$ . See the getResidualError method for details.
constraintErrorfloat: Constraint error.
inputDimensionint: Dimension of the input of the objective function.
outputDimensionint: Dimension of the output of the objective function.

Methods

`computeLagrangeMultipliers`(*args)	Compute the Lagrange multipliers.
`drawErrorHistory`()	Draw the convergence criteria history.
`drawOptimalValueHistory`()	Draw the optimal value history.
`getAbsoluteError`()	Accessor to the absolute error.
`getAbsoluteErrorHistory`()	Accessor to the evolution of the absolute error.
`getClassName`()	Accessor to the object’s name.
`getConstraintError`()	Accessor to the constraint error.
`getConstraintErrorHistory`()	Accessor to the evolution of the constraint error.
`getEvaluationNumber`()	Accessor to the number of evaluations.
`getId`()	Accessor to the object’s id.
`getInputSample`()	Accessor to the input sample.
`getIterationNumber`()	Accessor to the number of iterations.
`getName`()	Accessor to the object’s name.
`getOptimalPoint`()	Accessor to the optimal point.
`getOptimalValue`()	Accessor to the optimal value.
`getOutputSample`()	Accessor to the output sample.
`getProblem`()	Accessor to the underlying optimization problem.
`getRelativeError`()	Accessor to the relative error.
`getRelativeErrorHistory`()	Accessor to the evolution of the relative error.
`getResidualError`()	Accessor to the residual error.
`getResidualErrorHistory`()	Accessor to the evolution of the residual error.
`getShadowedId`()	Accessor to the object’s shadowed id.
`getVisibility`()	Accessor to the object’s visibility state.
`hasName`()	Test if the object is named.
`hasVisibleName`()	Test if the object has a distinguishable name.
`setEvaluationNumber`(evaluationNumber)	Accessor to the number of evaluation.
`setIterationNumber`(iterationNumber)	Accessor to the number of iterations.
`setName`(name)	Accessor to the object’s name.
`setOptimalPoint`(optimalPoint)	Accessor to the optimal point.
`setOptimalValue`(optimalValue)	Accessor to the optimal value.
`setProblem`(problem)	Accessor to the underlying optimization problem.
`setShadowedId`(id)	Accessor to the object’s shadowed id.
`setVisibility`(visible)	Accessor to the object’s visibility state.

getLagrangeMultipliers
setLagrangeMultipliers
store

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

computeLagrangeMultipliers(*args)¶

Compute the Lagrange multipliers.

Parameters

xsequence of float, optional: Location where the multipliers are computed If not provided, the optimal point is used

Returns

lagrangeMultipliersequence of float: Lagrange multipliers of the problem at point x. It needs an extra call to the objective function gradient unless it can be computed during the optimization (AbdoRackwitz or SQP).

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.

drawErrorHistory()¶

Draw the convergence criteria history.

Returns

graphGraph: Convergence criteria history graph

drawOptimalValueHistory()¶

Draw the optimal value history.

Returns

graphGraph: Optimal value history graph

getAbsoluteError()¶

Accessor to the absolute error.

Returns

absoluteErrorfloat: Absolute error of the input point $\vect{x}$ , 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.

getAbsoluteErrorHistory()¶

Accessor to the evolution of the absolute error.

Returns

absoluteErrorHistorySample: Value of the absolute error at each function evaluation.

getClassName()¶

Accessor to the object’s name.

Returns

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

getConstraintError()¶

Accessor to the constraint error.

Returns

constraintErrorfloat: Constraint error, 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).

getConstraintErrorHistory()¶

Accessor to the evolution of the constraint error.

Returns

constraintErrorHistorySample: Value of the constraint error at each function evaluation.

getEvaluationNumber()¶

Accessor to the number of evaluations.

Returns

evaluationNumberint: Number of evaluations.

getId()¶

Accessor to the object’s id.

Returns

idint: Internal unique identifier.

getInputSample()¶

Accessor to the input sample.

Returns

inputSampleSample: Input points used by the solver

getIterationNumber()¶

Accessor to the number of iterations.

Returns

iterationNumberint: Number of iterations.

getName()¶

Accessor to the object’s name.

Returns

namestr: The name of the object.

getOptimalPoint()¶

Accessor to the optimal point.

Returns

optimalPointPoint: Optimal point

getOptimalValue()¶

Accessor to the optimal value.

Returns

optimalValuePoint: Value at the optimal point

getOutputSample()¶

Accessor to the output sample.

Returns

outputSampleSample: Output points used by the solver

getProblem()¶

Accessor to the underlying optimization problem.

Returns

problemOptimizationProblem: Problem corresponding to the result

getRelativeError()¶

Accessor to the relative error.

Returns

relativeErrorfloat: Relative error of the input point $\vect{x}$ . If $\|\vect{x}_{n+1}\|_{\infty}\neq 0$ , then the relative error is $\epsilon^r_n=\epsilon^a_n/\|\vect{x}_{n+1}\|_{\infty}$ where $\epsilon^a_n=\|\vect{x}_{n+1}-\vect{x}_n\|_{\infty}$ is the absolute error. Otherwise, the relative error is $\epsilon^r_n=-1$ .

getRelativeErrorHistory()¶

Accessor to the evolution of the relative error.

Returns

relativeErrorHistorySample: Value of the relative error at each function evaluation.

getResidualError()¶

Accessor to the residual error.

Returns

residualErrorfloat: Relative error, 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 $\epsilon^r_n=-1$ .

getResidualErrorHistory()¶

Accessor to the evolution of the residual error.

Returns

residualErrorHistorySample: Value of the residual error at each function evaluation.

getShadowedId()¶

Accessor to the object’s shadowed id.

Returns

idint: Internal unique identifier.

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.

setEvaluationNumber(evaluationNumber)¶

Accessor to the number of evaluation.

Parameters

evaluationNumberint: Number of evaluations.

setIterationNumber(iterationNumber)¶

Accessor to the number of iterations.

Parameters

iterationNumberint: Number of iterations.

setName(name)¶

Accessor to the object’s name.

Parameters

namestr: The name of the object.

setOptimalPoint(optimalPoint)¶

Accessor to the optimal point.

Parameters

optimalPointPoint: Optimal point

setOptimalValue(optimalValue)¶

Accessor to the optimal value.

Parameters

optimalValuePoint: Value at the optimal point

setProblem(problem)¶

Accessor to the underlying optimization problem.

Parameters

problemOptimizationProblem: Problem corresponding to the result

setShadowedId(id)¶

Accessor to the object’s shadowed id.

Parameters

idint: Internal unique identifier.

setVisibility(visible)¶

Accessor to the object’s visibility state.

Parameters

visiblebool: Visibility flag.

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