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.

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.