Relaxations of KKT Conditions do not Strengthen Finite RLT and SDP-RLT Bounds for Nonconvex Quadratic Programs

TL;DR

This paper investigates whether adding optimality conditions to relaxations of nonconvex quadratic programs improves their bounds, concluding that it does not strengthen the RLT and SDP-RLT relaxations and can sometimes be misleading.

Contribution

It proves that incorporating first-order optimality conditions does not improve the bounds of RLT and SDP-RLT relaxations for nonconvex quadratic programs.

Findings

Relaxations with optimality conditions do not improve bounds.

Examples show unbounded relaxations and misleading information.

Relaxations of optimality conditions can be uninformative or deceptive.

Abstract

We consider linear and semidefinite programming relaxations of nonconvex quadratic programs given by the reformulation-linearization technique (RLT relaxation), and the Shor relaxation combined with the RLT relaxation (SDP-RLT relaxation). By incorporating the first-order optimality conditions, a quadratic program can be formulated as an optimization problem with complementarity constraints. We investigate the effect of incorporating optimality conditions on the strength of the RLT and SDP-RLT relaxations. Under the assumption that the original relaxations have a finite optimal value, we establish that the RLT and SDP-RLT bounds arising from the complementarity formulation agree with their counterparts. We present several classes of instances of quadratic programs with unbounded RLT and SDP-RLT relaxations to illustrate the different behavior of the relaxations of the complementarity formulation. In particular, our examples reveal that relaxations of optimality conditions may even yield misleading information on certain families of instances.