Function optimisation and Brouwer Fixed-Points on acute convex sets

Marvin D. Troutt, Shui-Hung Hou, Wan-Kai Pang and Toru Higuchi

International Journal of Operational Research, 2008, vol. 3, issue 6, 605-613

Abstract: The Brouwer Fixed-Point (FP) theorem is as follows. Given a continuous function φ(x) defined on a convex compact set S such that φ(x) lies in S then, there exists a point x* in S such that φ(x*) = x*. It is well-known that many optimisation problems can be cast as problems of finding a Brouwer FP. Instead, we propose an approach to the reverse problem of finding an FP by optimisation. First, we define acuteness for convex sets and propose an algorithm for computing a Brouwer FP based on a direction of ascent of what we call a hypothetical function. The algorithm uses 1D search as in the Frank–Wolfe algorithm. We report on numerical experiments comparing results with the Banach-iteration or successive-substitution method. The proposed algorithm is convergent for some challenging chaos-based examples for which the Banach-iteration approach fails.

Keywords: acute convex sets; Brouwer Fixed-Point theorem; chaos; Frank–Wolfe algorithm; method of steepest descent; function optimisation. (search for similar items in EconPapers)
Date: 2008
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