Enhancements of Discretization Approaches for Non-Convex Mixed-Integer Quadratically Constraint Quadratic Programming: Part II

TL;DR

This paper introduces an improved discretization method called D-NMDT for non-convex MIQCQPs, demonstrating its theoretical advantages and superior performance in producing tight dual bounds compared to existing methods.

Contribution

The paper extends the NMDT approach to a doubly discretized version (D-NMDT), providing a theoretical analysis and computational evidence of its improved effectiveness.

Findings

D-NMDT produces tighter dual bounds than NMDT.

D-NMDT outperforms separable relaxations from Part I.

D-NMDT is competitive with state-of-the-art MIQCQP solvers.

Abstract

This is Part II of a study on mixed-integer programming (MIP) relaxation techniques for the solution of non-convex mixed-integer quadratically constrained quadratic programs (MIQCQPs). We set the focus on MIP relaxation methods for non-convex continuous variable products and extend the well-known MIP relaxation normalized multi-parametric disaggregation technique (NMDT), applying a sophisticated discretization to both variables. We refer to this approach as doubly discretized normalized multiparametric disaggregation technique (D-NMDT). In a comprehensive theoretical analysis, we underline the theoretical advantages of the enhanced method D-NMDT compared to NMDT. Furthermore, we perform a broad computational study to demonstrate its effectiveness in terms of producing tight dual bounds for MIQCQPs. Finally, we compare D-NMDT to the separable MIP relaxations from Part I and a state-of-the-art MIQCQP solver.