An augmented Lagrange method for elliptic state constrained optimal control problems

Abstract

In this paper we apply an augmented Lagrange method to a class of semilinear elliptic optimal control problems with pointwise state constraints. We show strong convergence of subsequences of the primal variables to a local solution of the original problem as well as weak convergence of the adjoint states and weak* convergence of the multipliers associated to the state constraint. Moreover, we show existence of stationary points in arbitrary small neighborhoods of local solutions of the original problem. Additionally, various numerical results are presented.

A Joint Tikhonov Regularization and Augmented Lagrange Approach for Ill-Posed State Constrained Control Problems with Sparse Controls

“…With a similar argument we can establish the next lemma. Again the proof can be found in [18,Lemma 3.7]. If we now combine these two results we can show that our algorithm produces infinitely many successful steps.…”

“…Proof. The proof just differs from the one of [18,Theorem 3.3] concerning the additional subdifferential in (7b). Hence, we give just the most important steps here.…”

“…Our aim is to modify and extend the method presented in [18] to obtain a numerical scheme to solve (P ). The main idea is the following.…”

Optimal bilinear control problem related to a chemo-repulsion system in 2D domains

“…There is a wide collection of publications dealing with optimal control of PDEs. See, for example, [1,7,11,13,15,17,20,24,28,29] and the references therein. In all previous publications, the control variable enters the state equation either on the right-hand side (distributed controls) or is part of the boundary conditions (boundary controls).…”

Quasi-Variational Inequalities in Banach Spaces: Theory and Augmented Lagrangian Methods

“…A comparison of the classical ALM and its safeguarded analogue can be found in [35]. Moreover, the safeguarded ALM has been extended to quasi-variational inequalities in finite dimensions [32,36,51] and to constrained optimization problems and variational inequalities in Banach spaces [33,37,38,40].…”