Supermodularity and valid inequalities for quadratic optimization with indicators

• 成果类型: Article
• 署名作者: Atamturk, Alper; Gomez, Andres
• 署名单位: University of California System; University of California Berkeley; University of Southern California
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-022-01908-2
• 发表日期: 2023
• 页码: 295-338
• 关键词: perspective cuts; PROGRAMS; reformulations; formulations; cardinality

摘要

We study the minimization of a rank-one quadratic with indicators and show that the underlying set function obtained by projecting out the continuous variables is supermodular. Although supermodular minimization is, in general, difficult, the specific set function for the rank-one quadratic can be minimized in linear time. We show that the convex hull of the epigraph of the quadratic can be obtained from inequalities for the underlying supermodular set function by lifting them into nonlinear inequalities in the original space of variables. Explicit forms of the convex-hull description are given, both in the original space of variables and in an extended formulation via conic quadratic-representable inequalities, along with a polynomial separation algorithm. Computational experiments indicate that the lifted supermodular inequalities in conic quadratic form are quite effective in reducing the integrality gap for quadratic optimization with indicators.