Discretization of interior point methods for state constrained elliptic optimal control problems: optimal error estimates and parameter adjustment

Hinze, Michael; Schiela, Anton

doi:10.1007/s10589-009-9278-x

Cited by 30 publications

(15 citation statements)

References 24 publications

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“…Common are so called regularization methods which relax the state constraints in (11) by either substituting it by a mixed control-state constraint (Lavrentiev relaxation [41]), or by adding suitable penalty terms to the cost functional instead of requiring the state constraints (barrier methods [29,44], penalty methods [23,25].…”

Section: If In Addition Bumentioning

confidence: 99%

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Discrete concepts versus error analysis in PDE‐constrained optimization

Hinze

Tröltzsch

2010

GAMM-Mitteilungen

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Key words elliptic optimal control problem, state & control constraints, error analysis. MSC (2000) 49J20, 49K20, 35B37Solutions to optimization problems with pde constraints inherit special properties; the associated state solves the pde which in the optimization problem takes the role of a equality constraint, and this state together with the associated control solves an optimization problem, i.e. together with multipliers satisfies first and second order necessary optimality conditions. In this note we review the state of the art in designing discrete concepts for optimization problems with pde constraints with emphasis on structure conservation of solutions on the discrete level, and on error analysis for the discrete variables involved. As model problem for the state we consider an elliptic pde which is well understood from the analytical point of view. This allows to focus on structural aspects in discretization. We discuss the approaches First discretize, then optimize and First optimize, then discretize, and consider in detail two variants of the First discretize, then optimize approach, namely variational discretization, a discrete concept which avoids explicit discretization of the controls, and piecewise constant control approximations. We consider general constraints on the control, and also consider pointwise bounds on the state. We outline the basic ideas for providing optimal error analysis and complement our analytical findings with numerical examples which confirm our analytical results.

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Section: If In Addition Bumentioning

confidence: 99%

“…Let us allow also a semilinear equation of the form (29). Now, the mappingsĴ and g j : u → y(x j ) = y(G(Bu))(x j ) are real-valued and smooth functions depending on u ∈ R n so that this optimal control problem is equivalent to a finite-dimensional nonlinear programming problem.…”

Section: Finite-dimensional Control and State Constraints In Finitelymentioning

confidence: 99%

Discrete concepts versus error analysis in PDE‐constrained optimization

Hinze

Tröltzsch

2010

GAMM-Mitteilungen

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“…The paper [72] also contains strategies based on a posteriori error estimates for the automatic adjustment of the regularization parameters in treating state constraints. The proper adjustment of parameters is also considered in Hinze & Schiela [43] and Hintermueller & Hinze [39]. The extension of the DWR method for nonstationary problems with control and/or state constraints is currently under development.…”

Section: Control and State Constraintsmentioning

confidence: 99%

Adaptive finite element discretization in PDE‐based optimization

Rannacher

Vexler

2010

GAMM-Mitteilungen

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Key words PDE-based optimization, finite element method, goal-oriented adaptivity MSC (2000) 49K20, 49M15, 49M15, 65M50, 65N50This article surveys recent developments in the adaptive numerical solution of optimal control problems governed by partial differential equations (PDE). By the Euler-Lagrange formalism the optimization problem is reformulated as a saddle-point problem (KKT system) that is discretized by a Galerkin finite element method (FEM). Following the Dual Weighted Residual (DWR) approach the accuracy of the approximation is controlled by residual-based a posteriori error estimates. This opens the way toward systematic complexity reduction in the solution of PDE-based optimal control problems occurring in science and engineering.

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“…Finite element discretizations are only briefly mentioned. The interaction between parameter selection rules for the embedding and the discretization step size in case of certain barrier methods has been recently analyzed in [16].…”

Section: Problem and Optimality Characterizationmentioning

confidence: 99%

Elliptic Control by Penalty Techniques with Control Reduction

Großmann¹,

Kunz²,

Meischner³

2009

IFIP Advances in Information and Communication Technology

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Abstract. The paper deals with the numerical treatment of optimal control problems with bounded distributed controls and elliptic state equations by a wider class of barrier-penalty methods. If the constraints are treated by barrier-penalty techniques then the necessary and sufficient optimality condition forms a coupled system of nonlinear equations which contain not only the usual adjoint and the state equation, but also an approximate projection by means of barrier-penalty terms. Under the made assumptions from the last one the control can be eliminated. This reduced optimality system which does not contain explicitly the controls, but the more regular states and adjoints only, is studied in detail.

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Discretization of interior point methods for state constrained elliptic optimal control problems: optimal error estimates and parameter adjustment

Abstract: Elliptic optimal control problem, Error estimates, Interior point method, Pointwise state constraints,

Cited by 30 publications

References 24 publications

Discrete concepts versus error analysis in PDE‐constrained optimization

Discrete concepts versus error analysis in PDE‐constrained optimization

Adaptive finite element discretization in PDE‐based optimization

Elliptic Control by Penalty Techniques with Control Reduction

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