Solving Constrained Affine Heaviside Composite Optimization Problems by a Progressive IP Approach

TL;DR

This paper introduces a progressive integer programming method to solve complex affine Heaviside composite optimization problems, addressing discontinuity and non-closed feasible sets, with proven convergence and demonstrated effectiveness in classification tasks.

Contribution

The paper proposes a novel resolution approach combining approximation and progressive IP to handle discontinuities and non-closed sets in Heaviside composite optimization problems.

Findings

Method converges to local optima of Heaviside problems.

Effective in score-based and tree-based multiclass classification with constraints.

Numerical results validate the approach's efficiency.

Abstract

This paper discusses the computational resolution and presents numerical results for solving affine combinations of Heaviside composite optimization problems (abbreviated as A-HSCOPs) by a progressive integer programming (abbreviated as PIP) method. The characteristics of these problems are that the Heaviside functions, which appear in the objective and define the constraints, are discontinuous, and their mixed-signed combinations result in the overall objective lacking the matching semicontinuity needed for the optimization and in the feasible set being not necessarily closed. Added to these challenging properties is the nondifferentiability of the inner functions in the composition. In this paper, we propose resolutions to all these challenges by first an approximation to remedy the lack of semicontinuity in the objective and closedness in the constraints, followed by a progressive integer programming approach with successive decomposition to handle the intrinsically discrete nature of the Heaviside function. Convergence to the local optimizers of the given Heaviside optimization problem is established. The effectiveness of the overall solution strategy is supported by extensive computational experiments on the score-based and tree-based multiclass classification problems with precision constraints.