ROL
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Implements the computation of optimization steps using Moreau-Yosida regularized bound constraints. More...
#include <ROL_MoreauYosidaPenaltyStep.hpp>
Public Member Functions | |
~MoreauYosidaPenaltyStep () | |
MoreauYosidaPenaltyStep (Teuchos::ParameterList &parlist) | |
void | initialize (Vector< Real > &x, const Vector< Real > &g, Vector< Real > &l, const Vector< Real > &c, Objective< Real > &obj, EqualityConstraint< Real > &con, BoundConstraint< Real > &bnd, AlgorithmState< Real > &algo_state) |
Initialize step with equality constraint. More... | |
void | compute (Vector< Real > &s, const Vector< Real > &x, const Vector< Real > &l, Objective< Real > &obj, EqualityConstraint< Real > &con, BoundConstraint< Real > &bnd, AlgorithmState< Real > &algo_state) |
Compute step (equality and bound constraints). More... | |
void | update (Vector< Real > &x, Vector< Real > &l, const Vector< Real > &s, Objective< Real > &obj, EqualityConstraint< Real > &con, BoundConstraint< Real > &bnd, AlgorithmState< Real > &algo_state) |
Update step, if successful (equality and 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... | |
void | compute (Vector< Real > &s, const Vector< Real > &x, Objective< Real > &obj, BoundConstraint< Real > &con, AlgorithmState< Real > &algo_state) |
Compute step for bound constraints; here only to satisfy the interface requirements, does nothing, needs refactoring. More... | |
void | update (Vector< Real > &x, const Vector< Real > &s, Objective< Real > &obj, BoundConstraint< Real > &con, AlgorithmState< Real > &algo_state) |
Update step, for bound constraints; here only to satisfy the interface requirements, does nothing, needs refactoring. More... | |
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virtual | ~Step () |
Step (void) | |
virtual void | initialize (Vector< Real > &x, 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 > &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, EqualityConstraint< 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, EqualityConstraint< 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, EqualityConstraint< Real > &con, AlgorithmState< Real > &algo_state) |
Update step, if successful (equality constraints). More... | |
virtual Teuchos::RCP< const StepState< Real > > | getStepState (void) const |
Get state for step object. More... | |
Private Member Functions | |
void | updateState (const Vector< Real > &x, const Vector< Real > &l, Objective< Real > &obj, EqualityConstraint< Real > &con, BoundConstraint< Real > &bnd, AlgorithmState< Real > &algo_state) |
Private Attributes | |
Teuchos::RCP< MoreauYosidaPenalty< Real > > | myPen_ |
Teuchos::RCP< Algorithm< Real > > | algo_ |
Teuchos::RCP< Vector< Real > > | x_ |
Teuchos::RCP< Vector< Real > > | g_ |
Teuchos::RCP< Vector< Real > > | l_ |
Real | tau_ |
bool | print_ |
Teuchos::ParameterList | parlist_ |
int | subproblemIter_ |
Additional Inherited Members | |
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Teuchos::RCP< StepState< Real > > | getState (void) |
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:
Definition at line 121 of file ROL_MoreauYosidaPenaltyStep.hpp.
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Definition at line 170 of file ROL_MoreauYosidaPenaltyStep.hpp.
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Definition at line 172 of file ROL_MoreauYosidaPenaltyStep.hpp.
References ROL::Step< Real >::getState().
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Definition at line 135 of file ROL_MoreauYosidaPenaltyStep.hpp.
References ROL::EqualityConstraint< Real >::applyAdjointJacobian(), ROL::AlgorithmState< Real >::cnorm, ROL::Step< Real >::getState(), ROL::AlgorithmState< Real >::gnorm, ROL::BoundConstraint< Real >::isActivated(), ROL::AlgorithmState< Real >::iter, ROL::AlgorithmState< Real >::ncval, ROL::AlgorithmState< Real >::nfval, ROL::AlgorithmState< Real >::ngrad, ROL::BoundConstraint< Real >::project(), ROL::ROL_EPSILON, ROL::Objective< Real >::update(), ROL::EqualityConstraint< Real >::update(), ROL::AlgorithmState< Real >::value, and ROL::EqualityConstraint< Real >::value().
Referenced by ROL::MoreauYosidaPenaltyStep< Real >::initialize(), and ROL::MoreauYosidaPenaltyStep< Real >::update().
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Initialize step with equality constraint.
Reimplemented from ROL::Step< Real >.
Definition at line 194 of file ROL_MoreauYosidaPenaltyStep.hpp.
References ROL::Vector< Real >::clone(), ROL::Step< Real >::getState(), ROL::BoundConstraint< Real >::isActivated(), ROL::AlgorithmState< Real >::ncval, ROL::AlgorithmState< Real >::nfval, ROL::AlgorithmState< Real >::ngrad, ROL::BoundConstraint< Real >::project(), and ROL::MoreauYosidaPenaltyStep< Real >::updateState().
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Compute step (equality and bound constraints).
Reimplemented from ROL::Step< Real >.
Definition at line 222 of file ROL_MoreauYosidaPenaltyStep.hpp.
References ROL::Vector< Real >::axpy(), and ROL::Vector< Real >::set().
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Update step, if successful (equality and bound constraints).
Reimplemented from ROL::Step< Real >.
Definition at line 235 of file ROL_MoreauYosidaPenaltyStep.hpp.
References ROL::Step< Real >::getState(), ROL::AlgorithmState< Real >::iter, ROL::AlgorithmState< Real >::iterateVec, ROL::AlgorithmState< Real >::lagmultVec, ROL::AlgorithmState< Real >::ncval, ROL::AlgorithmState< Real >::nfval, ROL::AlgorithmState< Real >::ngrad, ROL::Vector< Real >::norm(), ROL::Vector< Real >::plus(), ROL::Vector< Real >::set(), ROL::AlgorithmState< Real >::snorm, ROL::MoreauYosidaPenaltyStep< Real >::tau_, ROL::EqualityConstraint< Real >::update(), and ROL::MoreauYosidaPenaltyStep< Real >::updateState().
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Print iterate header.
Implements ROL::Step< Real >.
Definition at line 263 of file ROL_MoreauYosidaPenaltyStep.hpp.
Referenced by ROL::MoreauYosidaPenaltyStep< Real >::print().
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Print step name.
Implements ROL::Step< Real >.
Definition at line 282 of file ROL_MoreauYosidaPenaltyStep.hpp.
Referenced by ROL::MoreauYosidaPenaltyStep< Real >::print().
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Print iterate status.
Implements ROL::Step< Real >.
Definition at line 291 of file ROL_MoreauYosidaPenaltyStep.hpp.
References ROL::AlgorithmState< Real >::cnorm, ROL::AlgorithmState< Real >::gnorm, ROL::AlgorithmState< Real >::iter, ROL::AlgorithmState< Real >::ncval, ROL::AlgorithmState< Real >::nfval, ROL::AlgorithmState< Real >::ngrad, ROL::MoreauYosidaPenaltyStep< Real >::printHeader(), ROL::MoreauYosidaPenaltyStep< Real >::printName(), ROL::AlgorithmState< Real >::snorm, ROL::MoreauYosidaPenaltyStep< Real >::subproblemIter_, and ROL::AlgorithmState< Real >::value.
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Compute step for bound constraints; here only to satisfy the interface requirements, does nothing, needs refactoring.
Implements ROL::Step< Real >.
Definition at line 333 of file ROL_MoreauYosidaPenaltyStep.hpp.
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Update step, for bound constraints; here only to satisfy the interface requirements, does nothing, needs refactoring.
Implements ROL::Step< Real >.
Definition at line 340 of file ROL_MoreauYosidaPenaltyStep.hpp.
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Definition at line 123 of file ROL_MoreauYosidaPenaltyStep.hpp.
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Definition at line 124 of file ROL_MoreauYosidaPenaltyStep.hpp.
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Definition at line 125 of file ROL_MoreauYosidaPenaltyStep.hpp.
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Definition at line 126 of file ROL_MoreauYosidaPenaltyStep.hpp.
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Definition at line 127 of file ROL_MoreauYosidaPenaltyStep.hpp.
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Definition at line 129 of file ROL_MoreauYosidaPenaltyStep.hpp.
Referenced by ROL::MoreauYosidaPenaltyStep< Real >::update().
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Definition at line 130 of file ROL_MoreauYosidaPenaltyStep.hpp.
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Definition at line 132 of file ROL_MoreauYosidaPenaltyStep.hpp.
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Definition at line 133 of file ROL_MoreauYosidaPenaltyStep.hpp.
Referenced by ROL::MoreauYosidaPenaltyStep< Real >::print().