Ideal formulations for constrained convex optimization problems with indicator variables

TL;DR

This paper develops ideal convex formulations for complex convex optimization problems involving indicator variables and combinatorial constraints, enhancing relaxation quality in regression tasks.

Contribution

It provides the convex hull description of a broad class of problems with nonlinear objectives and combinatorial constraints, extending previous convexification methods.

Findings

Improved relaxation quality in regression with hierarchy constraints

Convex hull description for problems with combinatorial constraints

Efficient computational experiments demonstrating practical benefits

Abstract

Motivated by modern regression applications, in this paper, we study the convexification of a class of convex optimization problems with indicator variables and combinatorial constraints on the indicators. Unlike most of the previous work on convexification of sparse regression problems, we simultaneously consider the nonlinear non-separable objective, indicator variables, and combinatorial constraints. Specifically, we give the convex hull description of the epigraph of the composition of a one-dimensional convex function and an affine function under arbitrary combinatorial constraints. As special cases of this result, we derive ideal convexifications for problems with hierarchy, multi-collinearity, and sparsity constraints. Moreover, we also give a short proof that for a separable objective function, the perspective reformulation is ideal independent from the constraints of the problem. Our computational experiments with regression problems under hierarchy constraints on real datasets demonstrate the potential of the proposed approach in improving the relaxation quality without significant computational overhead.