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.