Lagrangian-Penalization Algorithm for Constrained Optimization and Variational Inequalities

Abstract

Let X, Y be real Hilbert spaces. Consider a bounded linear operator A : X → Y and a nonempty closed convex set \(\mathcal{C}\subset Y\). In this paper we propose an inexact proximal-type algorithm to solve constrained optimization problems $$(\mathcal{P})\qquad\qquad\qquad \inf \{f(x)\ :\ Ax\in \mathcal{C}\}, $$ where f is a proper lower-semicontinuous convex function on X; and variational inequalities $$(\mathcal{VI})\qquad\qquad\qquad 0\in\mathcal{M} x+A^*N_{\mathcal{C}}(Ax), $$ where \(\mathcal{M}:X\rightrightarrows X\) is a maximal monotone operator and \(N_{\mathcal{C}}\) denotes the normal cone to the set \(\mathcal{C}\). Our method combines a penalization procedure involving a bounded sequence of parameters, with the predictor corrector proximal multiplier method of Chen and Teboulle (Math Program 64(1, Ser A):81–101, 1994). Under suitable assumptions the sequences generated by our algorithm are proved to converge weakly to solutions of \((\mathcal{P})\) and \((\mathcal{VI})\). As applications, we describe how the algorithm can be used to find sparse solutions of linear inequality systems and solve partial differential equations by domain decomposition.