Title: Function optimisation and Brouwer Fixed-Points on acute convex sets

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

Addresses: Department of Management and Information Systems, Kent State University, PO Box 5190, Room 426A, Business Administration Building, Kent, OH 44240, USA. ' Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, S.A.R., People's Republic of China. ' Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, S.A.R., People's Republic of China. ' Faculty of Policy Studies, Sakushin Gakuin University, Utsunomiya, Tochigi 171-0022, Japan

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.

DOI: 10.1504/IJOR.2008.019728

International Journal of Operational Research, 2008 Vol.3 No.6, pp.605 - 613

Published online: 27 Jul 2008 *

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