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

Yuzhou Qiu () and E. Alper Yıldırım ()
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Yuzhou Qiu: The University of Edinburgh
E. Alper Yıldırım: The University of Edinburgh

Journal of Global Optimization, 2024, vol. 90, issue 2, No 1, 293-322

Abstract: Abstract 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 $$\{0,\frac{1}{2},1\}$$ { 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.

Keywords: Quadratic programming with box constraints; Reformulation–linearization technique; Semidefinite relaxation; Convex underestimator (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10898-024-01407-y

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