Proximal Point Method on Finslerian Manifolds and the “Effort–Accuracy” Trade-off

Abstract

In this paper we consider minimization problems with constraints. We will show that if the set of constraints is a Finslerian manifold of non positive flag curvature, and the objective function is differentiable and satisfies the property Kurdyka-Lojasiewicz, then the proximal point method is naturally extended to solve that class of problems. We will prove that the sequence generated by our method is well defined and converges to a minimizer point. We show how tools of Finslerian geometry, more specifically non symmetrical metrics, can be used to solve nonconvex constrained problems in Euclidean spaces. As an application, we give one result on the speed of decision and making and costs to change.