A Class Of SQP Algorithms For Non-Strictly Monotone Variational
Inequalities

Michael Patriksson
Department of Mathematics
Chalmers University of Technology
S-412 96 Göteborg
Sweden

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.

KEY WORDS: Variational Inequality, Sequential Quadratic Programming, Cost
Approximation, Modified Descent Algorithm, Penalty Method