Predictor-corrector algorithm for solvingP _*(κ)-matrix LCP from arbitrary positive starting points

Abstract

A new predictor-corrector algorithm is proposed for solvingP _*(κ)-matrix linear complementarity problems. If the problem is solvable, then the algorithm converges from an arbitrary positive starting point (x ⁰,s ⁰). The computational complexity of the algorithm depends on the quality of the starting point. If the starting point is feasible or close to being feasible, it has$O((1 + \kappa )\sqrt {n/\rho _0 } L)$-iteration complexity, whereρ ₀ is the ratio of the smallest and average coordinate ofX ⁰ s ⁰. With appropriate initialization, a modified version of the algorithm terminates in O((1+κ)²(n/ρ ₀)L) steps either by finding a solution or by determining that the problem has no solution in a predetermined, arbitrarily large, region. The algorithm is quadratically convergent for problems having a strictly complementary solution. We also propose an extension of a recent algorithm of Mizuno toP _*(κ)-matrix linear complementarity problems such that it can start from arbitrary positive points and has superlinear convergence without a strictly complementary condition.