Function optimisation and Brouwer Fixed-Points on acute convex sets. (28th July 2008)

• Record Type: Journal Article
• Authors: Troutt, Marvin D.; Hou, Shui-Hung; Pang, Wan-Kai; Higuchi, Toru
• Is Part Of: International journal of operational research. Volume 3:Number 6(2008)
• Page Start: 605
• Page End: 613
• Publication Date: 2008-07-28
• Subjects: acute convex sets -- Brouwer Fixed-Point theorem -- chaos -- Frank?Wolfe algorithm -- method of steepest descent -- function optimisation; Operations research -- Periodicals; 003.05
• Languages: English
• ISSNs: 1745-7645

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.