Derivative-free methods for mixed-integer nonsmooth constrained optimization

TL;DR

This paper introduces a new derivative-free, linesearch-based algorithmic framework for solving mixed-integer nonsmooth constrained optimization problems using only black-box function evaluations, with proven convergence and extensive testing.

Contribution

It presents a novel derivative-free algorithmic framework that handles nonsmoothness, discrete variables, and nonlinear constraints in mixed-integer optimization problems.

Findings

Algorithms converge globally to stationary points.

Extensive numerical tests demonstrate effectiveness.

Framework successfully manages nonsmooth and constrained problems.

Abstract

In this paper, we consider mixed-integer nonsmooth constrained optimization problems whose objective/constraint functions are available only as the output of a black-box zeroth-order oracle (i.e., an oracle that does not provide derivative information) and we propose a new derivative-free linesearch-based algorithmic framework to suitably handle those problems. We first describe a scheme for bound constrained problems that combines a dense sequence of directions (to handle the nonsmoothness of the objective function) with primitive directions (to handle discrete variables). Then, we embed an exact penalty approach in the scheme to suitably manage nonlinear (possibly nonsmooth) constraints. We analyze the global convergence properties of the proposed algorithms toward stationary points and we report the results of an extensive numerical experience on a set of mixed-integer test problems.