Abstract
It is well known that by adding integrality constraints to the semidefinite programming (SDP) relaxation of the max-cut problem, the resulting integer semidefinite program is an exact formulation of the problem. In this paper we show similar results for a wide variety of discrete optimization problems for which SDP relaxations have been derived. Based on a comprehensive study on discrete positive semidefinite matrices, we introduce a generic approach to derive mixed-integer SDP (MISDP) formulations of binary quadratically constrained quadratic programs and binary quadratic matrix programs. Applying a problem-specific approach, we derive more compact MISDP formulations of several problems, such as the quadratic assignment problem, the graph partition problem, and the integer matrix completion problem. We also show that several structured problems allow for novel compact MISDP formulations through the notion of association schemes. Complementary to the recent advances on algorithmic aspects related to MISDP, this work opens new perspectives on solution approaches for the here considered problems.
| Original language | English |
|---|---|
| Pages (from-to) | 1071-1096 |
| Number of pages | 26 |
| Journal | SIAM Journal on Optimization |
| Volume | 34 |
| Issue number | 1 |
| DOIs | |
| Publication status | Published - 2023 |
Bibliographical note
Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project https://www.openaccess.nl/en/you-share-we-take-careOtherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.
Keywords
- mixed-integer semidefinite programming
- discrete positive semidefinite matrices
- binary quadratic programming
- quadratic matrix programming
- association schemes