Strong formulations for quadratic optimization with M-matrices and indicator variables

• 成果类型: Article; Proceedings Paper
• 署名作者: Atamturk, Alper; Gomez, Andres
• 署名单位: University of California System; University of California Berkeley; Pennsylvania Commonwealth System of Higher Education (PCSHE); University of Pittsburgh
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-018-1301-5
• 发表日期: 2018
• 页码: 141-176
• 关键词: convex relaxations; graph cuts; PROGRAMS; reformulations; minimization; constraints; algorithm; DESIGN; cone

摘要

We study quadratic optimization with indicator variables and an M-matrix, i.e., a PSD matrix with non-positive off-diagonal entries, which arises directly in image segmentation and portfolio optimization with transaction costs, as well as a substructure of general quadratic optimization problems. We prove, under mild assumptions, that the minimization problem is solvable in polynomial time by showing its equivalence to a submodular minimization problem. To strengthen the formulation, we decompose the quadratic function into a sum of simple quadratic functions with at most two indicator variables each, and provide the convex-hull descriptions of these sets. We also describe strong conic quadratic valid inequalities. Preliminary computational experiments indicate that the proposed inequalities can substantially improve the strength of the continuous relaxations with respect to the standard perspective reformulation.