Numerical infinitesimals in a variable metric method for convex nonsmooth optimization

The objective of the paper is to evaluate the impact of the infinity computing paradigm on practical solution of nonsmooth unconstrained optimization problems, where the objective function is assumed to be convex and not necessarily differentiable. For such family of problems, the occurrence of discontinuities in the derivatives may result in failures of the algorithms suited for smooth problems. We focus on a family of nonsmooth optimization methods based on a variable metric approach, and we use the infinity computing techniques for numerically dealing with some quantities which can assume values arbitrarily small or large, as a consequence of nonsmoothness. In particular we consider the case, treated in the literature, where the metric is defined via a diagonal matrix with positive entries. We provide the computational results of our implementation on a set of benchmark test-problems from scientific literature.