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- Title
Supermodularity and valid inequalities for quadratic optimization with indicators.
- Authors
Atamtürk, Alper; Gómez, Andrés
- Abstract
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.
- Subjects
SET functions; QUADRATIC forms; NONCONVEX programming; NONLINEAR functions; QUADRATIC programming; MODULAR forms; QUADRATIC differentials
- Publication
Mathematical Programming, 2023, Vol 201, Issue 1/2, p295
- ISSN
0025-5610
- Publication type
Article
- DOI
10.1007/s10107-022-01908-2