OptimizationResult¶

class OptimizationResult(*args)¶

Optimization result.

Returned by optimization solvers, see OptimizationAlgorithm.

Parameters:

problemOptimizationProblem: Problem being solved.

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.
`getFinalPoints`()	Accessor to the final points.
`getFinalValues`()	Accessor to the final values.
`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.
`getParetoFrontsIndices`()	Accessor to the Pareto fronts indices in the final population.
`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.
`setFinalPoints`(finalPoints)	Accessor to the final points.
`setFinalValues`(finalValues)	Accessor to the final values.
`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.
`setParetoFrontsIndices`(indices)	Accessor to the Pareto fronts indices in the final population.
`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.

__init__(*args)¶

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.

getFinalPoints()¶

Accessor to the final points.

Returns:

finalPointsSample: Final population. For non-evolutionary algorithms this will return the optimal point.

getFinalValues()¶

Accessor to the final values.

Returns:

finalValuesSample: Values at the final points. For non-evolutionary algorithms this will return the optimal value.

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

getParetoFrontsIndices()¶

Accessor to the Pareto fronts indices in the final population.

In the multi-objective case, it consists of stratas of points in the final population. The first front contains the best candidates according to the objectives.

Returns:

indiceslist of Indices: Pareto fronts indices

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: