# Ipopt¶

class Ipopt(*args)

Ipopt optimization solver Ipopt.

Parameters
problemOptimizationProblem, optional

Optimization problem to solve. Default is an empty problem.

Bonmin

Notes

Algorithms parameters:

Ipopt algorithms can be adapted using numerous parameters, described here. These parameters can be modified using the ResourceMap. For every option optionName, one simply add a key named Ipopt-optionName with the value to use, as shown below:

>>> import openturns as ot

Convergence criteria:

To estimate the convergence of the algorithm during the optimization process, Ipopt uses specific tolerance parameters, different from the standard absolute/relative/residual errors used in OpenTURNS. The definition of Ipopt’s parameters can be found in this paper, page 3.

Thus the attributes maximumAbsoluteError, maximumRelativeError, maximumResidualError and maximumConstraintError defined in’ OptimizationAlgorithm are not used in this case. The tolerances used by Ipopt can be set using specific options (e.g. tol, dual_inf_tol …).

Examples

The code below ensures the optimization of the following problem:

subject to

>>> import openturns as ot
>>> # Definition of objective function
>>> objectiveFunction = ot.SymbolicFunction(['x0','x1','x2','x3'], ['-x0 -x1 -x2'])
>>> # Definition of variables bounds
>>> bounds = ot.Interval([0,0,0,0],[1,1e308,1e308,5],[True,True,True,True],[True,False,False,True])

Inequality constraints are defined by a function such that . The inequality expression above has to be modified to match this formulation.

>>> # Definition of constraints
>>> h = ot.SymbolicFunction(['x0','x1','x2','x3'], ['-(x1-1/2)^2 - (x2-1/2)^2 + 1/4', '-x0 + x1', '-x0 - x2 - x3 + 2'])
>>> # Setting up Ipopt problem
>>> problem = ot.OptimizationProblem(objectiveFunction)
>>> problem.setBounds(bounds)
>>> problem.setInequalityConstraint(h)
>>> algo = ot.Ipopt(problem)
>>> algo.setStartingPoint([0,0,0,0])
>>> algo.setMaximumEvaluationNumber(1000)
>>> algo.setMaximumIterationNumber(1000)
>>> # Running the solver
>>> algo.run()
>>> # Retrieving the results
>>> result = algo.getResult()
>>> optimalPoint = result.getOptimalPoint()
>>> optimalValue = result.getOptimalValue()
>>> evaluationNumber = result.getInputSample().getSize()

Methods

 Retrieves the names of the available optimization algorithms. 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. Accessor to maximum allowed constraint error. Accessor to maximum allowed number of evaluations. 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. getProblem(self) Accessor to optimization problem. getResult(self) Accessor to optimization result. getShadowedId(self) Accessor to the object’s shadowed id. getStartingPoint(self) Accessor to starting point. 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. 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. setStartingPoint(self, startingPoint) Accessor to starting point. 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.

static IsAvailable()

Retrieves the names of the available optimization algorithms.

Returns
algoNamebool

Returns true if Ipopt support is available, false if not.

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 are associated with the following Lagrangian formulation of the optimization problem:

where .

The Lagrange multipliers are stored as , where:
• is of dimension 0 if there is no equality constraint, else of dimension the dimension of ie the number of scalar equality constraints

• and are of dimension 0 if there is no bound constraint, else of dimension of

• is of dimension 0 if there is no inequality constraint, else of dimension the dimension of ie the number of scalar inequality constraints

The vector is solution of the following linear system:

If there is no constraint of any kind, 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 where and 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 where is the current approximation of the optimum and 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 if , else .

getMaximumResidualError(self)

Accessor to maximum allowed residual error.

Returns
maximumResidualErrorfloat

Maximum allowed residual error, where the residual error is defined by if , else .

getName(self)

Accessor to the object’s name.

Returns
namestr

The name of the object.

getProblem(self)

Accessor to optimization problem.

Returns
problemOptimizationProblem

Optimization problem.

getResult(self)

Accessor to optimization result.

Returns
resultOptimizationResult

Result class.

Accessor to the object’s shadowed id.

Returns
idint

Internal unique identifier.

getStartingPoint(self)

Accessor to starting point.

Returns
startingPointPoint

Starting point.

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 where and 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 where is the current approximation of the optimum and 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 if , else .

setMaximumResidualError(self, maximumResidualError)

Accessor to maximum allowed residual error.

Parameters
Maximum allowed residual error, where the residual error is defined by

if , else .

setName(self, name)

Accessor to the object’s name.

Parameters
namestr

The name of the object.

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.

Accessor to the object’s shadowed id.

Parameters
idint

Internal unique identifier.

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)
...     return True
>>> 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.