均衡点問題に対するパス追跡型算法

Abstract

We propose a path following algorithm for the stationary point problem: given a polytope n ~ Rn and an affine function f: Rn -+ Rn find a point x E n such that x·f(x) ~ x·f(x) for any point x E n. The linear system to be handled in the algorithm has only n+l equations while the linear complementarity problem to which the problem is reduced has n+m equations, where m is the number of constraints defining n. The algorithm is a variable dimension fIXed point algorithm having as many rays as the vertices of n. It first leaves the starting point wEn toward a vertex of n chosen by solving the linear programming problem: minimizef(w)·x subjects to x En, and then moves on convex hulls of wand higher dimensional faces of n. Generally speaking, it terminates as soon as it hits the boundary of n or it fmds a zero of f.