A lower bound on the iterative complexity of the Harker and Pang globalization technique of the Newton-min algorithm for solving the linear complementarity problem

Abstract

The plain Newton-min algorithm for solving the linear complementarity problem (LCP) “$0\leqslant x\perp (Mx+q)\geqslant 0$” can be viewed as an instance of the plain semismooth Newton method on the equational version “$\min (x,Mx+q)=0$” of the problem. This algorithm converges for any q when M is an $\mathbf{M }$-matrix, but not when it is a $\mathbf{P }$-matrix. When convergence occurs, it is often very fast (in at most n iterations for an $\mathbf{M }$-matrix, where n is the number of variables, but often much faster in practice). In 1990, Harker and Pang proposed to improve the convergence ability of this algorithm by introducing a stepsize along the Newton-min direction that results in a jump over at least one of the encountered kinks of the min-function, in order to avoid its points of nondifferentiability. This paper shows that, for the Fathi problem (an LCP with a positive definite symmetric matrix M, hence a $\mathbf{P }$-matrix), an algorithmic scheme, including the algorithm of Harker and Pang, may require n iterations to converge, depending on the starting point.