Function optimisation and Brouwer Fixed‐Points on acute convex sets

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.