Abstract
We propose a path following algorithm for the stationary point problem: given a polytope Ω⫋︀R^n and an affine function f: R^n → R^n find a point x^^^⋴Ω such that x^^^・f(x^^^)<__=x・f(x^^^) for any point x⋴Ω. The linear system to be handled in the algorithm has only n+1 equations while the linear complementarity problem to which the problem is reduced has n+m equations, where m is the number of constraints defining Ω. The algorithm is a variable dimension fixed point algorithm having as many rays as the vertices of Ω. It first leaves the starting point w⋴Ω toward a vertex of Ω chosen by solving the linear programming problem: minimize f(w)・x subjects to x⋴Ω, and then moves on convex hulls of w and higher dimensional faces of Ω. Generally speaking, it terminates as soon as it hits the boundary of Ω or it finds a zero of f.
