A Variational Discretization Concept in Control Constrained Optimization: The Linear-Quadratic Case

By Michael Hinze. Computational optimization and applications, 2005

608 citations recorded by the source corpus.

A new discretization concept for optimal control problems with control constraints is introduced which utilizes for the discretization of the control variable the relation between adjoint state and control. Its key feature is not to discretize the space of admissible controls but to implicitly utilize the first order optimality conditions and the discretization of the state and adjoint equations for the discretization of the control. For discrete controls obtained in this way an optimal error estimate is proved. The application to control of elliptic equations is discussed. Finally it is shown that the new concept is numerically implementable with only slight increase in program management. A numerical test confirms the theoretical investigations.

Subfields: Computational Mechanics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/772fddce51a32515abb94587dc334bb21b6d3495

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In Computational Mechanics

1. Dominik Meidner
2. Martin Gubisch
3. Yan
4. Michael
5. Zhou
6. Daniel Wachsmuth
7. Stefan Volkwein
8. Hinze

In Numerical Analysis

1. Yan
2. Michael
3. Zhou
4. Martin Gubisch
5. Daniel Wachsmuth
6. Dominik Meidner
7. Stefan Volkwein
8. Hinze

Citations in the corpus

• Proper Orthogonal Decomposition for Linear-Quadratic Optimal Control (2013, methodology): “Proper orthogonal decomposition, model-order reduction, apriori and a-posteriori error analysis, linear-quadratic optimal control, primal-dual active set strategy.”
• VARIATIONAL DISCRETIZATION FOR OPTIMAL CONTROL GOVERNED BY CONVECTION DOMINATED DIFFUSION EQUATIONS (2009, methodology): “A projection gradient method (see [14] and [15]) is used to compute the solution of the infinite dimensional optimization problem (2.”
• Error bounds for Euler approximation of linear-quadratic control problems with bang-bang solutions (2012, unknown): “[27, 22, 5]) to derive an upper bound for JN .”
• Adaptive Space-Time Finite Element Methods for Optimization Problems Governed by Nonlinear Parabolic Systems (2007, background): “Details of this approach for linear-quadratic elliptic optimization problems can be found in Hinze [44].”
• A priori error estimates for elliptic optimal control problems with a bilinear state equation (2009, methodology): “In [14] a variational approach is proposed without explicitly discretizing the control variable and in [24] a post-processing step is used to obtain the desired order of convergence.”
• Convergence and regularization results for optimal control problems with sparsity functional (2011, methodology): “The recently introduced variational discretization concept by Hinze [9] gives error estimate ‖u−uh‖L2 = O(h(2)).”
• Finite Element Approximation of Elliptic Dirichlet Optimal Control Problems (2007, background): “The analysis of finite element approximations of optimization problems governed by partial differential equations is an area of active research, see, e.g., [1, 12 , 17, 18].”