Approximate Analytic Center Quadratic Cut Method for Strongly Monotone Variational Inequalities

Abstract

For strongly monotone variational inequality problems (VIP) convergence of an algorithm is investigated which, at each iteration k, adds a quadratic cut through an approximate analytic center x _k of the subsequently shrinking convex body. First it is shown that the sequence of x _k converges to the unique solution x* of the VIP at $O(1/\sqrt k )$ . As an interesting detail note that — for increasingly accurate analytic centers — the complexity constants converge to the quantities obtained for ACQCM with exact centers. Secondly we show that the arithmetic complexity to update from x _k to x _{k +1} after inserting a quadratic cut through x _k is bounded by a constant number of Newton iterations plus $O(n\cdot \ln \ln [\varsigma \cdot \frac{{{k^2}}}{{{\varepsilon ^4}}}])$ , where n is the space-dimension, ε is the final solution accuracy ‖x _k — x*‖, and ζ depends on some problem-specific constants only.