Tight SDP relaxations for cardinality-constrained problems

TL;DR

This paper demonstrates that semidefinite programming relaxations can provide tight bounds and often optimal solutions for cardinality-constrained portfolio problems, highlighting SDP's effectiveness in mixed-integer quadratic optimization.

Contribution

It introduces a semidefinite programming relaxation for cardinality-constrained problems and shows its tightness and optimality in many cases, advancing SDP applications in combinatorial optimization.

Findings

Relaxation yields tight lower bounds

Achieves optimal solutions on many instances

Highlights SDP's modeling power for mixed-integer quadratic problems

Abstract

We model the cardinality-constrained portfolio problem using semidefinite matrices and investigate a relaxation using semidefinite programming. Experimental results show that this relaxation generates tight lower bounds and even achieves optimality on many instances from the literature. This underlines the modeling power of semidefinite programming for mixed-integer quadratic problems.