A new merit function and an SQP method for non-strictly monotone variational inequalities

Abstract

Merit functions utilized to monitor the convergence of sequential quadratic programming (SQP) methods for nonlinear programs and variational inequality problems have in common that they include a penalty function for the explicit constraints, the value of the penalty parameter for which is subject to the requirement of being large enough compared to estimates of the optimal Lagrange multipliers. In existing applications of the SQP algorithm to variational inequality problems, the possible choices of merit functions used in descent algorithms are also limited by conditions which require the knowledge of problem parameters.This paper introduces a merit function for use in SQP methods for possibly non-strictly monotone variational inequality problems which does not include an explicit penalty function, and whose application in descent algorithms does not require the knowledge of any problem parameters. This function is an extension of the merit function of Marcotte and Zhu (1995) to primal—dual variational inequality problems. Among its other properties, it is directionally differentiable, and exact for strongly monotone problems. Based on the new merit function, we establish the convergence of a general algorithm for primal—dual variational inequalities which includes an adaptive SQP algorithm as a special case.KeywordsVariational inequalitiesgap functiondescent algorithmsequential quadratic programming.