SQP

class SQP(*args)

Sequential Quadratic Programming solver.

Available constructors:

SQP(problem)

SQP(problem, tau, omega, smooth)

Parameters:

problem : OptimizationProblem

Optimization problem to solve.

tau : float

Multiplicative decrease of linear step.

omega : float

Armijo factor.

smooth : float

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.OptimizationProblem(model, -0.5)
>>> algo = ot.SQP(problem)
>>> algo.setStartingPoint([0.0] * 4)
>>> algo.run()
>>> result = algo.getResult()

Methods

computeLagrangeMultipliers(x) Compute the Lagrange multipliers of a problem at a given point.
getClassName() Accessor to the object’s name.
getId() Accessor to the object’s id.
getMaximumAbsoluteError() Accessor to maximum allowed absolute error.
getMaximumConstraintError() Accessor to maximum allowed constraint error.
getMaximumEvaluationNumber() Accessor to maximum allowed number of evaluations.
getMaximumIterationNumber() Accessor to maximum allowed number of iterations.
getMaximumRelativeError() Accessor to maximum allowed relative error.
getMaximumResidualError() Accessor to maximum allowed residual error.
getName() Accessor to the object’s name.
getOmega() Accessor to omega parameter.
getProblem() Accessor to optimization problem.
getResult() Accessor to optimization result.
getShadowedId() Accessor to the object’s shadowed id.
getSmooth() Accessor to smooth parameter.
getStartingPoint() Accessor to starting point.
getTau() Accessor to tau parameter.
getVerbose() Accessor to the verbosity flag.
getVisibility() Accessor to the object’s visibility state.
hasName() Test if the object is named.
hasVisibleName() Test if the object has a distinguishable name.
run() Launch the optimization.
setMaximumAbsoluteError(maximumAbsoluteError) Accessor to maximum allowed absolute error.
setMaximumConstraintError(maximumConstraintError) Accessor to maximum allowed constraint error.
setMaximumEvaluationNumber(…) 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.
setName(name) Accessor to the object’s name.
setOmega(tau) Accessor to omega parameter.
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.
setSmooth(tau) Accessor to smooth parameter.
setStartingPoint(startingPoint) Accessor to starting point.
setStopCallback(*args) Set up a stop callback.
setTau(tau) Accessor to tau parameter.
setVerbose(verbose) Accessor to the verbosity flag.
setVisibility(visible) Accessor to the object’s visibility state.
__init__(*args)

x.__init__(…) initializes x; see help(type(x)) for signature

computeLagrangeMultipliers(x)

Compute the Lagrange multipliers of a problem at a given point.

Parameters:

x : sequence of float

Point at which the Lagrange multipliers are computed.

Returns:

lagrangeMultiplier : sequence 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()

Accessor to the object’s name.

Returns:

class_name : str

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

getId()

Accessor to the object’s id.

Returns:

id : int

Internal unique identifier.

getMaximumAbsoluteError()

Accessor to maximum allowed absolute error.

Returns:

maximumAbsoluteError : float

Maximum allowed absolute error.

getMaximumConstraintError()

Accessor to maximum allowed constraint error.

Returns:

maximumConstraintError : float

Maximum allowed constraint error.

getMaximumEvaluationNumber()

Accessor to maximum allowed number of evaluations.

Returns:

N : int

Maximum allowed number of evaluations.

getMaximumIterationNumber()

Accessor to maximum allowed number of iterations.

Returns:

N : int

Maximum allowed number of iterations.

getMaximumRelativeError()

Accessor to maximum allowed relative error.

Returns:

maximumRelativeError : float

Maximum allowed relative error.

getMaximumResidualError()

Accessor to maximum allowed residual error.

Returns:

maximumResidualError : float

Maximum allowed residual error.

getName()

Accessor to the object’s name.

Returns:

name : str

The name of the object.

getOmega()

Accessor to omega parameter.

Returns:

omega : float

Armijo factor.

getProblem()

Accessor to optimization problem.

Returns:

problem : OptimizationProblem

Optimization problem.

getResult()

Accessor to optimization result.

Returns:

result : OptimizationResult

Result class.

getShadowedId()

Accessor to the object’s shadowed id.

Returns:

id : int

Internal unique identifier.

getSmooth()

Accessor to smooth parameter.

Returns:

smooth : float

Growing factor in penalization term.

getStartingPoint()

Accessor to starting point.

Returns:

startingPoint : Point

Starting point.

getTau()

Accessor to tau parameter.

Returns:

tau : float

Multiplicative decrease of linear step.

getVerbose()

Accessor to the verbosity flag.

Returns:

verbose : bool

Verbosity flag state.

getVisibility()

Accessor to the object’s visibility state.

Returns:

visible : bool

Visibility flag.

hasName()

Test if the object is named.

Returns:

hasName : bool

True if the name is not empty.

hasVisibleName()

Test if the object has a distinguishable name.

Returns:

hasVisibleName : bool

True if the name is not empty and not the default one.

run()

Launch the optimization.

setMaximumAbsoluteError(maximumAbsoluteError)

Accessor to maximum allowed absolute error.

Parameters:

maximumAbsoluteError : float

Maximum allowed absolute error.

setMaximumConstraintError(maximumConstraintError)

Accessor to maximum allowed constraint error.

Parameters:

maximumConstraintError : float

Maximum allowed constraint error.

setMaximumEvaluationNumber(maximumEvaluationNumber)

Accessor to maximum allowed number of evaluations.

Parameters:

N : int

Maximum allowed number of evaluations.

setMaximumIterationNumber(maximumIterationNumber)

Accessor to maximum allowed number of iterations.

Parameters:

N : int

Maximum allowed number of iterations.

setMaximumRelativeError(maximumRelativeError)

Accessor to maximum allowed relative error.

Parameters:

maximumRelativeError : float

Maximum allowed relative error.

setMaximumResidualError(maximumResidualError)

Accessor to maximum allowed residual error.

Parameters:

maximumResidualError : float

Maximum allowed residual error.

setName(name)

Accessor to the object’s name.

Parameters:

name : str

The name of the object.

setOmega(tau)

Accessor to omega parameter.

Parameters:

omega : float

Armijo factor.

setProblem(problem)

Accessor to optimization problem.

Parameters:

problem : OptimizationProblem

Optimization problem.

setProgressCallback(*args)

Set up a progress callback.

Parameters:

callback : callable

Takes a float as argument as percentage of progress.

Notes

May not be implemented by all solvers, refer to the solver documentation.

setResult(result)

Accessor to optimization result.

Parameters:

result : OptimizationResult

Result class.

setShadowedId(id)

Accessor to the object’s shadowed id.

Parameters:

id : int

Internal unique identifier.

setSmooth(tau)

Accessor to smooth parameter.

Parameters:

smooth : float

Growing factor in penalization term.

setStartingPoint(startingPoint)

Accessor to starting point.

Parameters:

startingPoint : Point

Starting point.

setStopCallback(*args)

Set up a stop callback.

Parameters:

callback : callable

Returns an int deciding whether to stop or continue.

Notes

May not be implemented by all solvers, refer to the solver documentation.

setTau(tau)

Accessor to tau parameter.

Parameters:

tau : float

Multiplicative decrease of linear step.

setVerbose(verbose)

Accessor to the verbosity flag.

Parameters:

verbose : bool

Verbosity flag state.

setVisibility(visible)

Accessor to the object’s visibility state.

Parameters:

visible : bool

Visibility flag.