Abstract
Interior point methods provide an attractive class of approaches for solving linear, quadratic and nonlinear programming problems, due to their excellent efficiency and wide applicability. In this paper, we consider PDE-constrained optimization problems with bound constraints on the state and control variables, and their representation on the discrete level as quadratic programming problems. To tackle complex problems and achieve high accuracy in the solution, one is required to solve matrix systems of huge scale resulting from Newton iteration, and hence fast and robust methods for these systems are required. We present preconditioned iterative techniques for solving a number of these problems using Krylov subspace methods, considering in what circumstances one may predict rapid convergence of the solvers in theory, as well as the solutions observed from practical computations.
Highlights
We are concerned with optimization problems which involve partial differential equations
The alternative optimize-thendiscretize method will guarantee an accurate solution of the continuous first order optimality conditions, when applied in conjunction with interior point methods the resulting matrix systems are not necessarily symmetric, nor can they be reduced to such low dimensions for these problems as the matrix systems illustrated later
We find it is advantageous to apply the discretize-thenoptimize approach for the interior point solution of PDE-constrained optimization problems—we highlight that this follows the approach used in important literature on the field such as [5,28]
Summary
We are concerned with optimization problems which involve partial differential equations. Interior point methods (IPMs) are very well-suited to solving quadratic optimization problems and they excel when sizes of problems grow large [17,52], which makes them perfect candidates for discretized PDE-constrained optimal control problems. Because of the unavoidable ill-conditioning of these equations the success of any iterative scheme for their solution depends on the ability to design efficient preconditioners which can improve spectral properties of linear systems. Our approach is derived from the matching strategy originally developed for a particular Poisson control problem [37] We adapt it to much more challenging circumstances of saddle point systems arising in IPMs applied to solve the PDE-constrained optimal control problems.
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