An$O(\sqrt n L)$ iteration potential reduction algorithm for linear complementarity problems

Abstract

This paper proposes an interior point algorithm for a positive semi-definite linear complementarity problem: find an (x, y)∈ℝ²ⁿ such thaty=Mx+q, (x,y)⩾0 andx ^T y=0. The algorithm reduces the potential function

$$f(x,y) = (n + \sqrt n )\log x^T y - \sum\limits_{i = 1}^n {\log x_i y_i } $$

by at least 0.2 in each iteration requiring O(n ³) arithmetic operations. If it starts from an interior feasible solution with the potential function value bounded by$O(\sqrt n L)$, it generates, in at most$O(\sqrt n L)$ iterations, an approximate solution with the potential function value$- O(\sqrt n L)$, from which we can compute an exact solution in O(n ³) arithmetic operations. The algorithm is closely related with the central path following algorithm recently given by the authors. We also suggest a unified model for both potential reduction and path following algorithms for positive semi-definite linear complementarity problems.