Ideal formulations for constrained convex optimization problems with indicator variables

• 成果类型: Article
• 署名作者: Wei, Linchuan; Gomez, Andres; Kucukyavuz, Simge
• 署名单位: Northwestern University; University of Southern California
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-021-01734-y
• 发表日期: 2022
• 页码: 57-88
• 关键词: scalable algorithms; perspective reformulations; quadratic programs; subset-selection; 2nd-order cone; integer; regression; cuts; REPRESENTATION; relaxations

摘要

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 sparse regression problems demonstrate the potential of the proposed approach in improving the relaxation quality without significant computational overhead.