SQP¶

class SQP(*args)¶

Sequential Quadratic Programming solver.

This solver uses second derivative information and can only be used to solve level function problems.

Available constructors:

SQP(problem)

SQP(problem, tau, omega, smooth)

Parameters

problemOptimizationProblem: Optimization problem to solve.
taufloat: Multiplicative decrease of linear step.
omegafloat: Armijo factor.
smoothfloat: Growing factor in penalization term.

See also

AbdoRackwitz, Cobyla, TNC, NLopt

Notes

SQP methods solve a sequence of optimization subproblems, each of which optimizes a quadratic model of the objective subject to a linearization of the constraints.

Examples

>>> import openturns as ot
>>> model = ot.SymbolicFunction(['x1', 'x2', 'x3', 'x4'], ['x1*cos(x1)+2*x2*x3-3*x3+4*x3*x4'])
>>> problem = ot.NearestPointProblem(model, -0.5)
>>> algo = ot.SQP(problem)
>>> algo.setStartingPoint([1.0] * 4)
>>> algo.run()
>>> result = algo.getResult()

Methods

`computeLagrangeMultipliers`(self, x)	Compute the Lagrange multipliers of a problem at a given point.
`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.
`getOmega`(self)	Accessor to omega parameter.
`getProblem`(self)	Accessor to optimization problem.
`getResult`(self)	Accessor to optimization result.
`getShadowedId`(self)	Accessor to the object’s shadowed id.
`getSmooth`(self)	Accessor to smooth parameter.
`getStartingPoint`(self)	Accessor to starting point.
`getTau`(self)	Accessor to tau parameter.
`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.
`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.
`setOmega`(self, tau)	Accessor to omega parameter.
`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.
`setSmooth`(self, tau)	Accessor to smooth parameter.
`setStartingPoint`(self, startingPoint)	Accessor to starting point.
`setStopCallback`(self, \*args)	Set up a stop callback.
`setTau`(self, tau)	Accessor to tau parameter.
`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.

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.

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.

getOmega(self)¶

Accessor to omega parameter.

Returns

omegafloat: Armijo factor.

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.

getSmooth(self)¶

Accessor to smooth parameter.

Returns

smoothfloat: Growing factor in penalization term.

getStartingPoint(self)¶

Accessor to starting point.

Returns

startingPointPoint: Starting point.

getTau(self)¶

Accessor to tau parameter.

Returns

taufloat: Multiplicative decrease of linear step.

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.

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.

setOmega(self, tau)¶

Accessor to omega parameter.

Parameters

omegafloat: Armijo factor.

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.

setSmooth(self, tau)¶

Accessor to smooth parameter.

Parameters

smoothfloat: Growing factor in penalization term.

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()

setTau(self, tau)¶

Accessor to tau parameter.

Parameters

taufloat: Multiplicative decrease of linear step.

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.

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

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