TNC¶

class TNC(*args)¶

Truncated Newton Constrained solver.

Tunrcated-Newton method Non-linear optimizer. This solver uses no derivative information and only supports bound constraints.

Available constructors:

TNC(problem)

TNC(problem, scale, offset, maxCGit, eta, stepmx, accuracy, fmin, rescale)

Parameters

problemOptimizationProblem: Optimization problem to solve.
specificParametersTNCSpecificParameters: Parameters for this solver.
scalesequence of float: Scaling factors to apply to each variables
offsetsequence of float: Constant to subtract to each variable
maxCGitint: Maximum number of hessian*vector evaluation per main iteration
etafloat: Severity of the line search.
stepmxfloat: Maximum step for the line search. may be increased during call
accuracyfloat: Relative precision for finite difference calculations
fminfloat: Minimum function value estimate.
rescalefloat: f scaling factor (in log10) used to trigger f value rescaling

See also

AbdoRackwitz, SQP, Cobyla, NLopt

Examples

>>> import openturns as ot
>>> model = ot.SymbolicFunction(['E', 'F', 'L', 'I'], ['-F*L^3/(3*E*I)'])
>>> bounds = ot.Interval([1.0]*4, [2.0]*4)
>>> problem = ot.OptimizationProblem(model, ot.Function(), ot.Function(), bounds)
>>> algo = ot.TNC(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.
`getAccuracy`(self)	Accessor to accuracy parameter.
`getClassName`(self)	Accessor to the object’s name.
`getEta`(self)	Accessor to eta parameter.
`getFmin`(self)	Accessor to fmin parameter.
`getId`(self)	Accessor to the object’s id.
`getMaxCGit`(self)	Accessor to maxCGit parameter.
`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.
`getOffset`(self)	Accessor to offset parameter.
`getProblem`(self)	Accessor to optimization problem.
`getRescale`(self)	Accessor to rescale parameter.
`getResult`(self)	Accessor to optimization result.
`getScale`(self)	Accessor to scale parameter.
`getShadowedId`(self)	Accessor to the object’s shadowed id.
`getStartingPoint`(self)	Accessor to starting point.
`getStepmx`(self)	Accessor to stepmx 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.
`setAccuracy`(self, accuracy)	Accessor to accuracy parameter.
`setEta`(self, eta)	Accessor to eta parameter.
`setFmin`(self, fmin)	Accessor to fmin parameter.
`setMaxCGit`(self, maxCGit)	Accessor to maxCGit parameter.
`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.
`setOffset`(self, offset)	Accessor to offset parameter.
`setProblem`(self, problem)	Accessor to optimization problem.
`setProgressCallback`(self, \*args)	Set up a progress callback.
`setRescale`(self, rescale)	Accessor to rescale parameter.
`setResult`(self, result)	Accessor to optimization result.
`setScale`(self, scale)	Accessor to scale parameter.
`setShadowedId`(self, id)	Accessor to the object’s shadowed id.
`setStartingPoint`(self, startingPoint)	Accessor to starting point.
`setStepmx`(self, stepmx)	Accessor to stepmx parameter.
`setStopCallback`(self, \*args)	Set up a stop callback.
`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.

getAccuracy(self)¶

Accessor to accuracy parameter.

Returns

accuracyfloat

Relative precision for finite difference calculations

if <= machine_precision, set to sqrt(machine_precision).

getClassName(self)¶

Accessor to the object’s name.

Returns

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

getEta(self)¶

Accessor to eta parameter.

Returns

etafloat

Severity of the line search.

if < 0 or > 1, set to 0.25.

getFmin(self)¶

Accessor to fmin parameter.

Returns

fminfloat: Minimum function value estimate.

getId(self)¶

Accessor to the object’s id.

Returns

idint: Internal unique identifier.

getMaxCGit(self)¶

Accessor to maxCGit parameter.

Returns

maxCGitint

Maximum number of hessian*vector evaluation per main iteration

if maxCGit = 0, the direction chosen is -gradient

if maxCGit < 0, maxCGit is set to max(1,min(50,n/2)).

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.

getOffset(self)¶

Accessor to offset parameter.

Returns

offsetPoint

Constant to subtract to each variable

if empty, the constant are (min-max)/2 for interval bounded

variables and x for the others.

getProblem(self)¶

Accessor to optimization problem.

Returns

problemOptimizationProblem: Optimization problem.

getRescale(self)¶

Accessor to rescale parameter.

Returns

rescalefloat

f scaling factor (in log10) used to trigger f value rescaling

if 0, rescale at each iteration

if a big value, never rescale

if < 0, rescale is set to 1.3.

getResult(self)¶

Accessor to optimization result.

Returns

resultOptimizationResult: Result class.

getScale(self)¶

Accessor to scale parameter.

Returns

scalePoint

Scaling factors to apply to each variable

if empty, the factors are min-max for interval bounded variables

and 1+|x] for the others.

getShadowedId(self)¶

Accessor to the object’s shadowed id.

Returns

idint: Internal unique identifier.

getStartingPoint(self)¶

Accessor to starting point.

Returns

startingPointPoint: Starting point.

getStepmx(self)¶

Accessor to stepmx parameter.

Returns

stepmxfloat

Maximum step for the line search. may be increased during call

if too small, will be set to 10.0.

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.

setAccuracy(self, accuracy)¶

Accessor to accuracy parameter.

Parameters

accuracyfloat

Relative precision for finite difference calculations

if <= machine_precision, set to sqrt(machine_precision).

setEta(self, eta)¶

Accessor to eta parameter.

Parameters

etafloat

Severity of the line search.

if < 0 or > 1, set to 0.25.

setFmin(self, fmin)¶

Accessor to fmin parameter.

Parameters

fminfloat: Minimum function value estimate.

setMaxCGit(self, maxCGit)¶

Accessor to maxCGit parameter.

Parameters

maxCGitint

Maximum number of hessian*vector evaluation per main iteration

if maxCGit = 0, the direction chosen is -gradient

if maxCGit < 0, maxCGit is set to max(1,min(50,n/2)).

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.

setOffset(self, offset)¶

Accessor to offset parameter.

Parameters

offsetsequence of float

Constant to subtract to each variable

if empty, the constant are (min-max)/2 for interval bounded

variables and x for the others.

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()

setRescale(self, rescale)¶

Accessor to rescale parameter.

Parameters

rescalefloat

f scaling factor (in log10) used to trigger f value rescaling

if 0, rescale at each iteration

if a big value, never rescale

if < 0, rescale is set to 1.3.

setResult(self, result)¶

Accessor to optimization result.

Parameters

resultOptimizationResult: Result class.

setScale(self, scale)¶

Accessor to scale parameter.

Parameters

scalesequence of float

Scaling factors to apply to each variable

if empty, the factors are min-max for interval bounded variables

and 1+|x] for the others.

setShadowedId(self, id)¶

Accessor to the object’s shadowed id.

Parameters

idint: Internal unique identifier.

setStartingPoint(self, startingPoint)¶

Accessor to starting point.

Parameters

startingPointPoint: Starting point.

setStepmx(self, stepmx)¶

Accessor to stepmx parameter.

Parameters

stepmxfloat

Maximum step for the line search. may be increased during call

if too small, will be set to 10.0.

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()

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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