NLopt

class NLopt(*args)

Interface to NLopt.

This class exposes the solvers from the non-linear optimization library [nlopt2009].

More details about available algorithms are available here.

Parameters
problemOptimizationProblem

Optimization problem to solve.

algoNamestr

The NLopt identifier of the algorithm. Use GetAlgorithmNames() to list available names.

See also

AbdoRackwitz, Cobyla, SQP, TNC

Notes

Here are some properties of the different algorithms:

Algorithm

Derivative info

Constraint support

AUGLAG

no derivative

all

AUGLAG_EQ

no derivative

all

GD_MLSL

first derivative

bounds required

GD_MLSL_LDS

first derivative

bounds required

GN_CRS2_LM

no derivative

bounds required

GN_DIRECT

no derivative

bounds required

GN_DIRECT_L

no derivative

bounds required

GN_DIRECT_L_NOSCAL

no derivative

bounds required

GN_DIRECT_L_RAND

no derivative

bounds required

GN_DIRECT_L_RAND_NOSCAL

no derivative

bounds required

GN_ESCH

no derivative

bounds

GN_ISRES

no derivative

all, bounds required

GN_MLSL

no derivative

bounds required

GN_MLSL_LDS

no derivative

bounds

GN_ORIG_DIRECT

no derivative

inequality, bounds required

GN_ORIG_DIRECT_L

no derivative

inequality, bounds required

G_MLSL

no derivative

bounds required

G_MLSL_LDS

no derivative

bounds required

LD_AUGLAG

first derivative

all

LD_AUGLAG_EQ

first derivative

all

LD_CCSAQ

first derivative

bounds, inequality

LD_LBFGS

first derivative

bounds

LD_LBFGS_NOCEDAL

first derivative

bounds

LD_MMA

first derivative

bounds, inequality

LD_SLSQP

first derivative

all

LD_TNEWTON

first derivative

bounds

LD_TNEWTON_PRECOND

first derivative

bounds

LD_TNEWTON_PRECOND_RESTART

first derivative

bounds

LD_TNEWTON_RESTART

first derivative

bounds

LD_VAR1

first derivative

bounds

LD_VAR2

first derivative

bounds

LN_AUGLAG

no derivative

all

LN_AUGLAG_EQ

no derivative

all

LN_BOBYQA

no derivative

bounds

LN_COBYLA

no derivative

all

LN_NELDERMEAD

no derivative

bounds

LN_NEWUOA

no derivative

bounds

LN_NEWUOA_BOUND

no derivative

bounds

LN_PRAXIS

no derivative

bounds

LN_SBPLX

no derivative

bounds

Examples

>>> import openturns as ot
>>> dim = 4
>>> bounds = ot.Interval([-3.0] * dim, [5.0] * dim)
>>> linear = ot.SymbolicFunction(['x1', 'x2', 'x3', 'x4'], ['x1+2*x2-3*x3+4*x4'])
>>> problem = ot.OptimizationProblem(linear, ot.Function(), ot.Function(), bounds)
>>> print(ot.NLopt.GetAlgorithmNames())
[AUGLAG,AUGLAG_EQ,GD_MLSL,GD_MLSL_LDS,...
>>> algo = ot.NLopt(problem, 'LD_MMA')
>>> algo.setStartingPoint([0.0] * 4)
>>> algo.run()
>>> result = algo.getResult()
>>> x_star = result.getOptimalPoint()
>>> y_star = result.getOptimalValue()
Attributes
thisown

The membership flag

Methods

GetAlgorithmNames()

Accessor to the list of algorithms provided by NLopt, by names.

IsAvailable()

Ask whether NLopt support is available.

SetSeed(seed)

Initialize the random generator seed.

computeLagrangeMultipliers(x)

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

getAlgorithmName()

Accessor to the algorithm name.

getClassName()

Accessor to the object’s name.

getId()

Accessor to the object’s id.

getInitialStep()

Initial local derivative-free algorithms step accessor.

getLocalSolver()

Local solver accessor.

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.

getProblem()

Accessor to optimization problem.

getResult()

Accessor to optimization result.

getShadowedId()

Accessor to the object’s shadowed id.

getStartingPoint()

Accessor to starting point.

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.

setAlgorithmName(algoName)

Accessor to the algorithm name.

setInitialStep(initialStep)

Initial local derivative-free algorithms step accessor.

setLocalSolver(localSolver)

Local solver accessor.

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.

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.

setStartingPoint(startingPoint)

Accessor to starting point.

setStopCallback(*args)

Set up a stop callback.

setVerbose(verbose)

Accessor to the verbosity flag.

setVisibility(visible)

Accessor to the object’s visibility state.

static GetAlgorithmNames()

Accessor to the list of algorithms provided by NLopt, by names.

Returns
namesDescription

List of algorithm names provided by NLopt, according to its naming convention.

Examples

>>> import openturns as ot
>>> print(ot.NLopt.GetAlgorithmNames())
[AUGLAG,AUGLAG_EQ,GD_MLSL,GD_MLSL_LDS,GN_CRS2_LM,GN_DIRECT,GN_DIRECT_L,GN_DIRECT_L_NOSCAL,GN_DIRECT_L_RAND,GN_DIRECT_L_RAND_NOSCAL,GN_DIRECT_NOSCAL,GN_ESCH,GN_ISRES,GN_MLSL,GN_MLSL_LDS,GN_ORIG_DIRECT,GN_ORIG_DIRECT_L,G_MLSL,G_MLSL_LDS,LD_AUGLAG,LD_AUGLAG_EQ,LD_CCSAQ,LD_LBFGS,LD_LBFGS_NOCEDAL,LD_MMA,LD_SLSQP,LD_TNEWTON,LD_TNEWTON_PRECOND,LD_TNEWTON_PRECOND_RESTART,LD_TNEWTON_RESTART,LD_VAR1,LD_VAR2,LN_AUGLAG,LN_AUGLAG_EQ,LN_BOBYQA,LN_COBYLA,LN_NELDERMEAD,LN_NEWUOA,LN_NEWUOA_BOUND,LN_PRAXIS,LN_SBPLX]#41
static IsAvailable()

Ask whether NLopt support is available.

Returns
availablebool

Whether NLopt support is available.

static SetSeed(seed)

Initialize the random generator seed.

Parameters
seedint

The RNG seed.

computeLagrangeMultipliers(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.

getAlgorithmName()

Accessor to the algorithm name.

Returns
algoNamestr

The NLopt identifier of the algorithm.

getClassName()

Accessor to the object’s name.

Returns
class_namestr

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

getId()

Accessor to the object’s id.

Returns
idint

Internal unique identifier.

getInitialStep()

Initial local derivative-free algorithms step accessor.

Returns
dxPoint

The initial step.

getLocalSolver()

Local solver accessor.

Returns
solverNLopt

The local solver.

getMaximumAbsoluteError()

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()

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()

Accessor to maximum allowed number of evaluations.

Returns
Nint

Maximum allowed number of evaluations.

getMaximumIterationNumber()

Accessor to maximum allowed number of iterations.

Returns
Nint

Maximum allowed number of iterations.

getMaximumRelativeError()

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()

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()

Accessor to the object’s name.

Returns
namestr

The name of the object.

getProblem()

Accessor to optimization problem.

Returns
problemOptimizationProblem

Optimization problem.

getResult()

Accessor to optimization result.

Returns
resultOptimizationResult

Result class.

getShadowedId()

Accessor to the object’s shadowed id.

Returns
idint

Internal unique identifier.

getStartingPoint()

Accessor to starting point.

Returns
startingPointPoint

Starting point.

getVerbose()

Accessor to the verbosity flag.

Returns
verbosebool

Verbosity flag state.

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.

run()

Launch the optimization.

setAlgorithmName(algoName)

Accessor to the algorithm name.

Parameters
algoNamestr

The NLopt identifier of the algorithm.

setInitialStep(initialStep)

Initial local derivative-free algorithms step accessor.

Parameters
dxsequence of float

The initial step.

setLocalSolver(localSolver)

Local solver accessor.

Parameters
solverNLopt

The local solver.

setMaximumAbsoluteError(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(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(maximumEvaluationNumber)

Accessor to maximum allowed number of evaluations.

Parameters
Nint

Maximum allowed number of evaluations.

setMaximumIterationNumber(maximumIterationNumber)

Accessor to maximum allowed number of iterations.

Parameters
Nint

Maximum allowed number of iterations.

setMaximumRelativeError(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(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(name)

Accessor to the object’s name.

Parameters
namestr

The name of the object.

setProblem(problem)

Accessor to optimization problem.

Parameters
problemOptimizationProblem

Optimization problem.

setProgressCallback(*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(result)

Accessor to optimization result.

Parameters
resultOptimizationResult

Result class.

setShadowedId(id)

Accessor to the object’s shadowed id.

Parameters
idint

Internal unique identifier.

setStartingPoint(startingPoint)

Accessor to starting point.

Parameters
startingPointPoint

Starting point.

setStopCallback(*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()
setVerbose(verbose)

Accessor to the verbosity flag.

Parameters
verbosebool

Verbosity flag state.

setVisibility(visible)

Accessor to the object’s visibility state.

Parameters
visiblebool

Visibility flag.