On the convex hull of convex quadratic optimization problems with indicators

We consider the convex quadratic optimization problem in R n with indicator variables and arbitrary constraints on the indicators. We show that a convex hull description of the associated mixed-integer set in an extended space with a quadratic number of additional variables consists of an ( n + 1 )...

Full description

Saved in:
Bibliographic Details
Published in:Mathematical programming Vol. 204; no. 1-2; pp. 703 - 737
Main Authors: Wei, Linchuan, Atamtürk, Alper, Gómez, Andrés, Küçükyavuz, Simge
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2024
Springer
Subjects:
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Full Length Paper
Investment analysis
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Numerical Analysis
Theoretical
90C25
90C11
ISSN:0025-5610, 1436-4646
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:We consider the convex quadratic optimization problem in R n with indicator variables and arbitrary constraints on the indicators. We show that a convex hull description of the associated mixed-integer set in an extended space with a quadratic number of additional variables consists of an ( n + 1 ) × ( n + 1 ) positive semidefinite constraint (explicitly stated) and linear constraints. In particular, convexification of this class of problems reduces to describing a polyhedral set in an extended formulation. While the vertex representation of this polyhedral set is exponential and an explicit linear inequality description may not be readily available in general, we derive a compact mixed-integer linear formulation whose solutions coincide with the vertices of the polyhedral set. We also give descriptions in the original space of variables: we provide a description based on an infinite number of conic-quadratic inequalities, which are “finitely generated.” In particular, it is possible to characterize whether a given inequality is necessary to describe the convex hull. The new theory presented here unifies several previously established results, and paves the way toward utilizing polyhedral methods to analyze the convex hull of mixed-integer nonlinear sets.
ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-023-01982-0