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Title: Function optimisation and brouwer fixed-points on acute convex sets
Authors: Troutt, MD
Hou, SH
Pang, WK
Higuchi, T
Keywords: Acute convex sets
Brouwer Fixed-Point
Frank-Wolfe algorithm
Method of steepest descent
Issue Date: 2008
Source: International journal of operational research, 2008, v. 3, no. 6, p. 605-613 How to cite?
Journal: International Journal of Operational Research 
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.
ISSN: 1745-7645
DOI: 10.1504/IJOR.2008.019728
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