Operator-Splitting Methods for Monotone Affine Variational Inequalities, with a Parallel Application to Optimal Control

This paper applies splitting techniques developed for set-valued maximal monotone operators to monotone aane variational inequalities, including as a special case the classical linear comple-mentarity problem. We give a uniied presentation of several splitting algorithms for monotone operators, and then apply these results to obtain two classes of algorithms for aane variational inequalities. The second class resembles classical matrix splitting, but has a novel \under-relaxation" step, and converges under more general conditions. In particular, the convergence proofs do not require the aane operator to be symmetric. We specialize our matrix-splitting-like method to discrete-time optimal control problems formulated as extended linear-quadratic programs in the manner advocated by Rockafellar and Wets. The result is a highly parallel algorithm, which we implement and test on the Connection Machine CM{5 computer family. The aane variational inequality problem is to nd a vector x lying in a closed convex set B such that hMx + q; w ? xi 0 8 w 2 B ; (1) where M is a given n n matrix and q is a vector from < n. We denote this problem avi(M; q; B). A common and important special case occurs when B is a box, that is, It is well known that when`0 and u 1, the problem reduces to the linear complementarity problem (LCP) 5, 6, 24] of nding some x 2 < n satisfying x 0 Mx + q 0 hx; Mx + qi = 0 : (3) This paper is restricted to the monotone case of avi(M; q; B), where M is positive semideenite, although not necessarily symmetric.