The convergence of an interior point method for an elliptic control problem with mixed control-state constraints

Abstract

Abstract. The paper addresses primal interior point method for state-constrained PDE optimal control problems. By a Lavrentiev regularization, the state constraint is transformed to a mixed control-state constraint with bounded Lagrange multiplier. Existence and convergence of the central path are established, and linear convergence of a short-step pathfollowing method is shown. The behaviour of the method is are demonstrated by numerical examples.Key words. interior point method, function space, optimal control, mixed control-state constraints, Lavrentiev regularization AMS subject classifications. 90C51, 49J20, 65M151. Introduction. The application of interior point methods to optimal control problems has received a good deal of interest in the past years. This parallels the fast development of numerical methods in large scale optimization where interior point methods play an important role. In the context of PDE control, their performance was carefully tested by Haddoux et al. [8] for discretized versions of elliptic control problems. Similarly, Grund and Rösch [7] considered different codes of interior point methods for elliptic control problems with pointwise state-constraints. Trust-region interior point techniques have been considered by M. Ulbrich, S. Ulbrich and M. Heinkenschloss in [15] for the optimal control of semilinear parabolic equations in a function space setting. Moreover, affine-scaling interior-point methods were presented for semilinear parabolic boundary control in [14].In [17,16] primal-dual interior point methods have been analysed for ODE problems in the infinite dimensional function space setting and their computational realization by inexact pathfollowing methods has been suggested. In [18] this method has been enhanced on the control of elliptic PDE problems with control constrains.A satisfactory convergence theory, however, had only been obtained for control constraints, whereas results for state constraints are scarce. The difficulty arises from the fact that Lagrange multipliers for state constraints are usually only measures, which hampers theoretical convergence analysis and affects the numerical solution.Concerning the regularity of Lagrange multipliers, the situation changes for mixed control-state constraints such as constraints of bottleneck type. Under natural assumptions, their multipliers can shown to be functions in certain L p -spaces, we only mention [12,4,3]. In [9], the idea came up to add a tiny fraction of the control to the state constraint such that a mixed control-state constraint results. The Lagrange multiplier to this mixed constraint is a bounded and measurable function. This Lavrentiev-regularization for state constraints has been analyzed in the context of primal-dual active set methods for elliptic control problems. Some results concerning the convergence of the solutions of the regularized problem to that of the origine state constrained can be found in [9].

On regularization methods for the numerical solution of parabolic control problems with pointwise state constraints

“…This might be of interest for the convergence analysis of numerical methods in function space. For instance, the convergence of a primal-dual interior point method with regularized pointwise state constraints was shown by Prüfert et al in [23]. It was recently pointed out by Schiela [29] that the interior point property cannot be expected for purely pointwise state constraints if the standard logarithmic barrier function is applied.…”

Adaptive finite element methods for mixed control-state constrained optimal control problems for elliptic boundary value problems

“…Therefore, it is a natural idea to regularize state constrained problems by means of mixed control-state constrained ones, since with regard to numerical solution techniques the regularized problems can be formally treated as in the case of pure control constraints (cf. e.g., [2,11,33,36,37,38,39,40]). However, so far an a posteriori error analysis of adaptive finite element approximations has not been provided for mixed control-state constrained control problems.…”

A control reduced primal interior point method for a class of control constrained optimal control problems

“…This result is a direct consequence of [14]. Alternatively, existence of central path solutions for µ > 0 can be shown directly by applying Schauder's fixed point theorem to u = u(S(Su − y d )).…”

“…Existence and convergence of the central path defined by primal interior point methods for control constrained problems has been established in [14], along with convergence of a function space oriented pathfollowing method. Again, we can use (8) and (9) in order to eliminate u = u(λ).…”

“…Both primal and primal-dual adaptive function space oriented linearly convergent interior-point methods for optimal control problems have been developed, analyzed and applied in [21,20,22,14]. Superlinear convergence has been obtained recently in [19] by means of an intermediate smoothing step closely related to the current setting.…”