Results 1  10
of
13
Barrier methods for optimal control problems with state constraints
, 2007
"... We study barrier methods for state constrained optimal control problems with PDEs. In the focus of our analysis is the path of minimizers of the barrier subproblems with the aim to provide a solid theoretical basis for function space oriented pathfollowing algorithms. We establish results on exist ..."
Abstract

Cited by 20 (6 self)
 Add to MetaCart
We study barrier methods for state constrained optimal control problems with PDEs. In the focus of our analysis is the path of minimizers of the barrier subproblems with the aim to provide a solid theoretical basis for function space oriented pathfollowing algorithms. We establish results on existence, continuity, and convergence of this path. Moreover, we consider the structure of barrier subdierentials, which play the role of dual variables.
A control reduced primal interior point method for a class of control constrained optimal control problems
 Computational Optimization and Applications
"... A primal interior point method for control constrained optimal control problems with PDE constraints is considered. Pointwise elimination of the control leads to a homotopy in the remaining state and dual variables, which is addressed by a short step pathfollowing method. The algorithm is applied ..."
Abstract

Cited by 13 (3 self)
 Add to MetaCart
(Show Context)
A primal interior point method for control constrained optimal control problems with PDE constraints is considered. Pointwise elimination of the control leads to a homotopy in the remaining state and dual variables, which is addressed by a short step pathfollowing method. The algorithm is applied to the continuous, infinite dimensional problem, where discretization is performed only in the innermost loop when solving linear equations. The a priori elimination of the least regular control permits to obtain the required accuracy with comparatively coarse meshes. Convergence of the method and discretization errors are studied, and the method is illustrated at two numerical examples.
Primaldual interiorpoint methods for pdeconstrained optimization
"... Abstract. This paper provides a detailed analysis of a primaldual interiorpoint method for PDEconstrained optimization. Considered are optimal control problems with control constraints in L p. It is shown that the developed primaldual interiorpoint method converges globally and locally superlin ..."
Abstract

Cited by 11 (0 self)
 Add to MetaCart
(Show Context)
Abstract. This paper provides a detailed analysis of a primaldual interiorpoint method for PDEconstrained optimization. Considered are optimal control problems with control constraints in L p. It is shown that the developed primaldual interiorpoint method converges globally and locally superlinearly. Not only the easier L ∞setting is analyzed, but also a more involved L qanalysis, q < ∞, is presented. In L ∞ , the set of feasible controls contains interior points and the Fréchet differentiability of the perturbed optimality system can be shown. In the L qsetting, which is highly relevant for PDEconstrained optimization, these nice properties are no longer available. Nevertheless, a convergence analysis is developed using refined techniques. In particular, twonorm techniques and a smoothing step are required.
Stationary optimal control problems with pointwise state constraints
 SIAM Journal on Optimization, to appear, 2007. ROLAND GRIESSE AND KARL KUNISCH
"... Abstract. First order optimality conditions for stationary pointwise state constrained optimal control problems are considered. It is shown that the Lagrange multiplier associated with the pointwise inequality state constraint is a regular Borel measure only, in general. In case of sufficiently smoo ..."
Abstract

Cited by 9 (3 self)
 Add to MetaCart
(Show Context)
Abstract. First order optimality conditions for stationary pointwise state constrained optimal control problems are considered. It is shown that the Lagrange multiplier associated with the pointwise inequality state constraint is a regular Borel measure only, in general. In case of sufficiently smooth data and under a regularity assumption on the active set, the Lagrange multiplier can be decomposed into a regular L2part concentrated on the active set and a singular part, which is concentrated on the interface between the active and the inactive set. In a second part of the paper numerical solution strategies are reviewed. These methods fall into two classes: the first class which includes interiorpoint methods as well as active set strategies is purely finite dimensional. The second class, however, admits an analysis in function space. The latter methods typically rely on regularization. In this respect, MoreauYosidabased and Lavrentievbased techniques are discussed. The paper ends by a numerical comparison of the presented solution algorithms.
Asymptotic expansions for interior penalty solutions of control constrained linearquadratic problems
, 2008
"... ..."
(Show Context)
Interior point methods in function space for state constraints  Inexact . . .
, 2009
"... We consider an interior point method in function space for PDE constrained optimal control problems with state constraints. Our emphasis is on the construction and analysis of an algorithm that integrates a Newton pathfollowing method with adaptive grid refinement. This is done in the framework of ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
We consider an interior point method in function space for PDE constrained optimal control problems with state constraints. Our emphasis is on the construction and analysis of an algorithm that integrates a Newton pathfollowing method with adaptive grid refinement. This is done in the framework of inexact Newton methods in function space, where the discretization error of each Newton step is controlled by adaptive grid refinement in the innermost loop. This allows to perform most of the required Newton steps on coarse grids, such that the overall computational time is dominated by the last few steps. For this purpose we propose an aposteriori error estimator for a problem suited norm.
Discretization of interior point methods for state constrained elliptic optimal control problems: optimal error estimates and parameter adjustment
, 2007
"... An adjustment scheme for the relaxation parameter of interior point approaches to the numerical solution of pointwise state constrained elliptic optimal control problems is introduced. The method is based on error estimates of an associated finite element discretization of the relaxed problems and ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
(Show Context)
An adjustment scheme for the relaxation parameter of interior point approaches to the numerical solution of pointwise state constrained elliptic optimal control problems is introduced. The method is based on error estimates of an associated finite element discretization of the relaxed problems and optimally selects the relaxation parameter in dependence on the mesh size of discretization. The finite element analysis for the relaxed problems is carried out and a numerical example is presented which confirms our analytical findings.
Nonsmooth Newton methods for setvalued saddle point problems
 SIAM J. Numer. Anal
"... Abstract. We present a new class of iterative schemes for large scale set– valued saddle point problems as arising, e.g., from optimization problems in the presence of linear and inequality constraints. Our algorithms can be either regarded as nonsmooth Newton–type methods for the nonlinear Schur c ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
Abstract. We present a new class of iterative schemes for large scale set– valued saddle point problems as arising, e.g., from optimization problems in the presence of linear and inequality constraints. Our algorithms can be either regarded as nonsmooth Newton–type methods for the nonlinear Schur complement or as Uzawa–type iterations with active set preconditioners. Numerical experiments with a control constrained optimal control problem and a discretized Cahn–Hilliard equation with obstacle potential illustrate the reliability and efficiency of the new approach. 1.
ÉquipesProjets Commands
"... apport de recherche ISSN 02496399 ISRN INRIA/RR7126FR+ENGinria00436768, version 1 27 Nov 2009Asymptotic expansion for the solution of a penalized control constrained semilinear elliptic problems ..."
Abstract
 Add to MetaCart
apport de recherche ISSN 02496399 ISRN INRIA/RR7126FR+ENGinria00436768, version 1 27 Nov 2009Asymptotic expansion for the solution of a penalized control constrained semilinear elliptic problems
Mathematisches Forschungsinstitut Oberwolfach Report No. 11/2006 Numerical Techniques for Optimization Problems with PDE Constraints
, 2006
"... Abstract. Optimization problems with partial differential equation (PDE) constraints arise in many science and engineering applications. Their robust and efficient solution present many mathematical challenges and requires a tight integration of properties and structures of the underlying problem, o ..."
Abstract
 Add to MetaCart
Abstract. Optimization problems with partial differential equation (PDE) constraints arise in many science and engineering applications. Their robust and efficient solution present many mathematical challenges and requires a tight integration of properties and structures of the underlying problem, of fast numerical PDE solvers, and of numerical nonlinear optimization. This workshop brought together experts in the above subdisciplines to exchange the latest research developments, discuss open questions, and to foster interactions between these subdisciplines.