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

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...

Full description

Saved in:
Bibliographic Details
Published inMathematical programming Vol. 170; no. 1; pp. 141 - 176
Main Authors Atamtürk, Alper, Gómez, Andrés
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.07.2018
Springer Nature B.V
Subjects
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Formulations
Full Length Paper
Image segmentation
Inequalities
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Numerical Analysis
Optimization
Quadratic equations
Theoretical
90C20
Perspective formulation
Quadratic optimization
90C57
90C11
Convex piecewise nonlinear inequalities
Conic quadratic cuts
Submodularity
Online AccessGet full text

Cover

More Information
Summary: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.
ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-018-1301-5