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.
__init__(*args)

Initialize self. See help(type(self)) for accurate signature.

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.

thisown

The membership flag