On exact and inexact RLT and SDP-RLT relaxations of quadratic programs with box constraints

• Published in: Journal of global optimization Vol. 90; no. 2; pp. 293 - 322
• Main Authors: Qiu, Yuzhou, Yıldırım, E. Alper
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.10.2024; Springer; Springer Nature B.V
• Subjects: Algorithms; Computer Science; Constraints; Descriptions; Lower bounds; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Optimization; Quadratic equations; Quadratic programming; Real Functions; Upper bounds; Quadratic programming with box constraints; Reformulation–linearization technique; Semidefinite relaxation; Convex underestimator
• ISSN: 0925-5001; 1573-2916; 1573-2916
• DOI: 10.1007/s10898-024-01407-y

Quadratic programs with box constraints involve minimizing a possibly nonconvex quadratic function subject to lower and upper bounds on each variable. This is a well-known NP-hard problem that frequently arises in various applications. We focus on two convex relaxations, namely the reformulation–linearization technique (RLT) relaxation and the SDP-RLT relaxation obtained by combining the Shor relaxation with the RLT relaxation. Both relaxations yield lower bounds on the optimal value of a quadratic program with box constraints. We show that each component of each vertex of the RLT relaxation lies in the set { 0 , 1 2 , 1 } . We present complete algebraic descriptions of the set of instances that admit exact RLT relaxations as well as those that admit exact SDP-RLT relaxations. We show that our descriptions can be converted into algorithms for efficiently constructing instances with (1) exact RLT relaxations, (2) inexact RLT relaxations, (3) exact SDP-RLT relaxations, and (4) exact SDP-RLT but inexact RLT relaxations. Our preliminary computational experiments illustrate that our algorithms are capable of generating computationally challenging instances for state-of-the-art solvers.