Invariants of SDP exactness in quadratic programming

In this paper we study the Shor relaxation of quadratic programs by fixing a feasible set and considering the space of objective functions for which the Shor relaxation is exact. We first give conditions under which this region is invariant under the choice of generators defining the feasible set. We then describe this region when the feasible set is invariant under the action of a subgroup of the general linear group. We conclude by applying these results to quadratic binary programs. We give an explicit description of objective functions where the Shor relaxation is exact and use this knowledge to design an algorithm that produces candidate solutions for binary quadratic programs.(c) 2023 Elsevier Ltd. All rights reserved.

Automated summary

In this paper, we investigate the Shor relaxation of quadratic programs by fixing a feasible set and examining the space of objective functions for which the Shor relaxation is exact. We establish conditions for the invariance of this region under the choice of generators defining the feasible set and describe its characteristics when the feasible set is invariant under the action of a subgroup of the general linear group. Furthermore, we apply these findings to quadratic binary programs and present an algorithm that generates candidate solutions based on an explicit description of objective functions where the Shor relaxation is exact.