Function optimisation and Brouwer fixed-points on acute convex sets (Q947335)

Summary: The Brouwer Fixed-Point (FP) theorem is as follows. Given a continuous function \(\phi (x)\) defined on a convex compact set S such that \(\phi (x)\) lies in \(S\) then, there exists a point \(x^*\) in \(S\) such that \(\phi (x^*) = x^*\). 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.