Implements the computation of optimization steps using Moreau-Yosida regularized bound constraints. More...

#include <ROL_AugmentedLagrangianStep.hpp>

Inheritance diagram for ROL::MoreauYosidaPenaltyStep< Real >:

Public Member Functions
	~MoreauYosidaPenaltyStep ()

	MoreauYosidaPenaltyStep (ROL::ParameterList &parlist)

void	initialize (Vector< Real > &x, const Vector< Real > &g, Vector< Real > &l, const Vector< Real > &c, Objective< Real > &obj, Constraint< Real > &con, BoundConstraint< Real > &bnd, AlgorithmState< Real > &algo_state)
	Initialize step with equality constraint. More...

void	initialize (Vector< Real > &x, const Vector< Real > &g, Objective< Real > &obj, BoundConstraint< Real > &bnd, AlgorithmState< Real > &algo_state)
	Initialize step without equality constraint. More...

void	compute (Vector< Real > &s, const Vector< Real > &x, const Vector< Real > &l, Objective< Real > &obj, Constraint< Real > &con, BoundConstraint< Real > &bnd, AlgorithmState< Real > &algo_state)
	Compute step (equality and bound constraints). More...

void	compute (Vector< Real > &s, const Vector< Real > &x, Objective< Real > &obj, BoundConstraint< Real > &bnd, AlgorithmState< Real > &algo_state)
	Compute step for bound constraints. More...

void	update (Vector< Real > &x, Vector< Real > &l, const Vector< Real > &s, Objective< Real > &obj, Constraint< Real > &con, BoundConstraint< Real > &bnd, AlgorithmState< Real > &algo_state)
	Update step, if successful (equality and bound constraints). More...

void	update (Vector< Real > &x, const Vector< Real > &s, Objective< Real > &obj, BoundConstraint< Real > &bnd, AlgorithmState< Real > &algo_state)
	Update step, for bound constraints. More...

std::string	printHeader (void) const
	Print iterate header. More...

std::string	printName (void) const
	Print step name. More...

std::string	print (AlgorithmState< Real > &algo_state, bool pHeader=false) const
	Print iterate status. More...

Public Member Functions inherited from ROL::Step< Real >
virtual	~Step ()

	Step (void)

virtual void	initialize (Vector< Real > &x, const Vector< Real > &s, const Vector< Real > &g, Objective< Real > &obj, BoundConstraint< Real > &con, AlgorithmState< Real > &algo_state)
	Initialize step with bound constraint. More...

virtual void	initialize (Vector< Real > &x, const Vector< Real > &g, Vector< Real > &l, const Vector< Real > &c, Objective< Real > &obj, Constraint< Real > &con, AlgorithmState< Real > &algo_state)
	Initialize step with equality constraint. More...

virtual void	compute (Vector< Real > &s, const Vector< Real > &x, const Vector< Real > &l, Objective< Real > &obj, Constraint< Real > &con, AlgorithmState< Real > &algo_state)
	Compute step (equality constraints). More...

virtual void	update (Vector< Real > &x, Vector< Real > &l, const Vector< Real > &s, Objective< Real > &obj, Constraint< Real > &con, AlgorithmState< Real > &algo_state)
	Update step, if successful (equality constraints). More...

const ROL::Ptr< const StepState< Real > >	getStepState (void) const
	Get state for step object. More...

void	reset (const Real searchSize=1.0)
	Get state for step object. More...

Private Member Functions
void	updateState (const Vector< Real > &x, const Vector< Real > &l, Objective< Real > &obj, Constraint< Real > &con, BoundConstraint< Real > &bnd, AlgorithmState< Real > &algo_state)

void	updateState (const Vector< Real > &x, Objective< Real > &obj, BoundConstraint< Real > &bnd, AlgorithmState< Real > &algo_state)

Private Attributes
ROL::Ptr< StatusTest< Real > >	status_

ROL::Ptr< Step< Real > >	step_

ROL::Ptr< Algorithm< Real > >	algo_

ROL::Ptr< Vector< Real > >	x_

ROL::Ptr< Vector< Real > >	g_

ROL::Ptr< Vector< Real > >	l_

ROL::Ptr< BoundConstraint< Real > >	bnd_

Real	compViolation_

Real	gLnorm_

Real	tau_

bool	print_

bool	updatePenalty_

ROL::ParameterList	parlist_

int	subproblemIter_

bool	hasEquality_

EStep	stepType_

std::string	stepname_

Additional Inherited Members
Protected Member Functions inherited from ROL::Step< Real >
ROL::Ptr< StepState< Real > >	getState (void)

Detailed Description

template<class Real>
class ROL::MoreauYosidaPenaltyStep< Real >

Implements the computation of optimization steps using Moreau-Yosida regularized bound constraints.

To describe the generalized Moreau-Yosida penalty method, we consider the following abstract setting. Suppose \(\mathcal{X}\) is a Hilbert space of functions mapping \(\Xi\) to \(\mathbb{R}\). For example, \(\Xi\subset\mathbb{R}^n\) and \(\mathcal{X}=L^2(\Xi)\) or \(\Xi = \{1,\ldots,n\}\) and \(\mathcal{X}=\mathbb{R}^n\). We assume \( f:\mathcal{X}\to\mathbb{R}\) is twice-continuously Fréchet differentiable and \(a,\,b\in\mathcal{X}\) with \(a\le b\) almost everywhere in \(\Xi\). Note that the generalized Moreau-Yosida penalty method will also work with secant approximations of the Hessian.

The generalized Moreau-Yosida penalty method is a proveably convergent algorithm for convex optimization problems and may not converge for general nonlinear, nonconvex problems. The algorithm solves

\[ \min_x \quad f(x) \quad \text{s.t.} \quad c(x) = 0, \quad a \le x \le b. \]

We can respresent the bound constraints using the indicator function \(\iota_{[a,b]}(x) = 0\) if \(a \le x \le b\) and equals \(\infty\) otherwise. Using this indicator function, we can write our optimization problem as the (nonsmooth) equality constrained program

\[ \min_x \quad f(x) + \iota_{[a,b]}(x) \quad \text{s.t.}\quad c(x) = 0. \]

Since the indicator function is not continuously Fréchet differentiable, we cannot apply our existing algorithms (such as, Composite Step SQP) to the above equality constrained problem. To circumvent this issue, we smooth the indicator function using generalized Moreau-Yosida regularization, i.e., we replace \(\iota_{[a,b]}\) in the objective function with

\[ \varphi(x,\mu,c) = \inf_y\; \{\; \iota_{[a,b]}(x-y) + \langle \mu, y\rangle_{\mathcal{X}} + \frac{c}{2}\|y\|_{\mathcal{X}}^2 \;\}. \]

One can show that \(\varphi(\cdot,\mu,c)\) for any \(\mu\in\mathcal{X}\) and \(c > 0\) is continuously Fréchet differentiable with respect to \(x\). Thus, using this penalty, Step::compute solves the following subproblem: given \(c_k>0\) and \(\mu_k\in\mathcal{X}\), determine \(x_k\in\mathcal{X}\) that solves

\[ \min_{x} \quad f(x) + \varphi(x,\mu_k,c_k)\quad\text{s.t.} c(x) = 0. \]

The multipliers \(\mu_k\) are then updated in Step::update as \(\mu_{k+1} = \nabla_x\varphi(x_k,\mu_k,c_k)\) and \(c_k\) is potentially increased (although this is not always necessary).

For more information on this method see:

D. P. Bertsekas. "Approximations Procedures Based on the Method of Multipliers." Journal of Optimization Theory and Applications, Vol. 23(4), 1977.
K. Ito, K. Kunisch. "Augmented Lagrangian Methods for Nonsmooth, Convex, Optimization in Hilbert Space." Nonlinear Analysis, 2000.

Definition at line 154 of file ROL_AugmentedLagrangianStep.hpp.

Constructor & Destructor Documentation

template<class Real >

ROL::MoreauYosidaPenaltyStep< Real >::~MoreauYosidaPenaltyStep ( )

inline

Definition at line 208 of file ROL_MoreauYosidaPenaltyStep.hpp.

template<class Real >

ROL::MoreauYosidaPenaltyStep< Real >::MoreauYosidaPenaltyStep ( ROL::ParameterList & parlist )

inline

Definition at line 210 of file ROL_MoreauYosidaPenaltyStep.hpp.

References ROL::Step< Real >::getState(), ROL::MoreauYosidaPenaltyStep< Real >::parlist_, ROL::MoreauYosidaPenaltyStep< Real >::print_, ROL::MoreauYosidaPenaltyStep< Real >::stepname_, ROL::MoreauYosidaPenaltyStep< Real >::stepType_, ROL::StringToEStep(), ROL::MoreauYosidaPenaltyStep< Real >::tau_, and ROL::MoreauYosidaPenaltyStep< Real >::updatePenalty_.

Member Function Documentation

template<class Real >

void ROL::MoreauYosidaPenaltyStep< Real >::updateState	(	const Vector< Real > &	x,
		const Vector< Real > &	l,
		Objective< Real > &	obj,
		Constraint< Real > &	con,
		BoundConstraint< Real > &	bnd,
		AlgorithmState< Real > &	algo_state
	)

inlineprivate

Definition at line 151 of file ROL_MoreauYosidaPenaltyStep.hpp.

References ROL::Constraint< Real >::applyAdjointJacobian(), ROL::AlgorithmState< Real >::cnorm, ROL::MoreauYosidaPenaltyStep< Real >::compViolation_, ROL::MoreauYosidaPenaltyStep< Real >::g_, ROL::Step< Real >::getState(), ROL::MoreauYosidaPenaltyStep< Real >::gLnorm_, ROL::AlgorithmState< Real >::gnorm, ROL::MoreauYosidaPenalty< Real >::gradient(), ROL::AlgorithmState< Real >::iter, ROL::AlgorithmState< Real >::ncval, ROL::AlgorithmState< Real >::nfval, ROL::AlgorithmState< Real >::ngrad, ROL::MoreauYosidaPenalty< Real >::testComplementarity(), ROL::Constraint< Real >::update(), ROL::MoreauYosidaPenalty< Real >::update(), ROL::Constraint< Real >::value(), ROL::AlgorithmState< Real >::value, and ROL::MoreauYosidaPenalty< Real >::value().

Referenced by ROL::MoreauYosidaPenaltyStep< Real >::initialize(), and ROL::MoreauYosidaPenaltyStep< Real >::update().

template<class Real >

void ROL::MoreauYosidaPenaltyStep< Real >::updateState	(	const Vector< Real > &	x,
		Objective< Real > &	obj,
		BoundConstraint< Real > &	bnd,
		AlgorithmState< Real > &	algo_state
	)

inlineprivate

Definition at line 179 of file ROL_MoreauYosidaPenaltyStep.hpp.

References ROL::AlgorithmState< Real >::cnorm, ROL::MoreauYosidaPenaltyStep< Real >::compViolation_, ROL::Step< Real >::getState(), ROL::MoreauYosidaPenaltyStep< Real >::gLnorm_, ROL::AlgorithmState< Real >::gnorm, ROL::MoreauYosidaPenalty< Real >::gradient(), ROL::AlgorithmState< Real >::iter, ROL::AlgorithmState< Real >::nfval, ROL::AlgorithmState< Real >::ngrad, ROL::MoreauYosidaPenalty< Real >::testComplementarity(), ROL::MoreauYosidaPenalty< Real >::update(), ROL::AlgorithmState< Real >::value, and ROL::MoreauYosidaPenalty< Real >::value().

template<class Real >

void ROL::MoreauYosidaPenaltyStep< Real >::initialize	(	Vector< Real > &	x,
		const Vector< Real > &	g,
		Vector< Real > &	l,
		const Vector< Real > &	c,
		Objective< Real > &	obj,
		Constraint< Real > &	con,
		BoundConstraint< Real > &	bnd,
		AlgorithmState< Real > &	algo_state
	)

inlinevirtual

Initialize step with equality constraint.

Reimplemented from ROL::Step< Real >.

Definition at line 238 of file ROL_MoreauYosidaPenaltyStep.hpp.

References ROL::Vector< Real >::clone(), ROL::MoreauYosidaPenaltyStep< Real >::g_, ROL::Step< Real >::getState(), ROL::MoreauYosidaPenaltyStep< Real >::hasEquality_, ROL::BoundConstraint< Real >::isActivated(), ROL::MoreauYosidaPenaltyStep< Real >::l_, ROL::AlgorithmState< Real >::ncval, ROL::AlgorithmState< Real >::nfval, ROL::AlgorithmState< Real >::ngrad, ROL::BoundConstraint< Real >::project(), ROL::MoreauYosidaPenaltyStep< Real >::updateState(), and ROL::MoreauYosidaPenaltyStep< Real >::x_.

template<class Real >

void ROL::MoreauYosidaPenaltyStep< Real >::initialize	(	Vector< Real > &	x,
		const Vector< Real > &	g,
		Objective< Real > &	obj,
		BoundConstraint< Real > &	bnd,
		AlgorithmState< Real > &	algo_state
	)

inlinevirtual

Initialize step without equality constraint.

Reimplemented from ROL::Step< Real >.

Definition at line 264 of file ROL_MoreauYosidaPenaltyStep.hpp.

References ROL::MoreauYosidaPenaltyStep< Real >::bnd_, ROL::Vector< Real >::clone(), ROL::MoreauYosidaPenaltyStep< Real >::g_, ROL::Step< Real >::getState(), ROL::BoundConstraint< Real >::isActivated(), ROL::AlgorithmState< Real >::ncval, ROL::AlgorithmState< Real >::nfval, ROL::AlgorithmState< Real >::ngrad, ROL::BoundConstraint< Real >::project(), ROL::MoreauYosidaPenaltyStep< Real >::updateState(), and ROL::MoreauYosidaPenaltyStep< Real >::x_.

template<class Real >

void ROL::MoreauYosidaPenaltyStep< Real >::compute	(	Vector< Real > &	s,
		const Vector< Real > &	x,
		const Vector< Real > &	l,
		Objective< Real > &	obj,
		Constraint< Real > &	con,
		BoundConstraint< Real > &	bnd,
		AlgorithmState< Real > &	algo_state
	)

inlinevirtual

Compute step (equality and bound constraints).

Reimplemented from ROL::Step< Real >.

Definition at line 290 of file ROL_MoreauYosidaPenaltyStep.hpp.

template<class Real >

void ROL::MoreauYosidaPenaltyStep< Real >::compute	(	Vector< Real > &	s,
		const Vector< Real > &	x,
		Objective< Real > &	obj,
		BoundConstraint< Real > &	bnd,
		AlgorithmState< Real > &	algo_state
	)

inlinevirtual

Compute step for bound constraints.

Reimplemented from ROL::Step< Real >.

Definition at line 328 of file ROL_MoreauYosidaPenaltyStep.hpp.

template<class Real >

void ROL::MoreauYosidaPenaltyStep< Real >::update	(	Vector< Real > &	x,
		Vector< Real > &	l,
		const Vector< Real > &	s,
		Objective< Real > &	obj,
		Constraint< Real > &	con,
		BoundConstraint< Real > &	bnd,
		AlgorithmState< Real > &	algo_state
	)

inlinevirtual

Update step, if successful (equality and bound constraints).

Reimplemented from ROL::Step< Real >.

Definition at line 356 of file ROL_MoreauYosidaPenaltyStep.hpp.

template<class Real >

void ROL::MoreauYosidaPenaltyStep< Real >::update	(	Vector< Real > &	x,
		const Vector< Real > &	s,
		Objective< Real > &	obj,
		BoundConstraint< Real > &	bnd,
		AlgorithmState< Real > &	algo_state
	)