A robust Lagrangian-DNN method for a class of quadratic optimization problems

Naohiko Arima (), Sunyoung Kim (), Masakazu Kojima () and Kim-Chuan Toh ()
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Naohiko Arima: Tokyo Institute of Technology
Sunyoung Kim: Ewha W. University
Masakazu Kojima: Chuo University
Kim-Chuan Toh: National University of Singapore

Computational Optimization and Applications, 2017, vol. 66, issue 3, No 3, 453-479

Abstract: Abstract The Lagrangian-doubly nonnegative (DNN) relaxation has recently been shown to provide effective lower bounds for a large class of nonconvex quadratic optimization problems (QAPs) using the bisection method combined with first-order methods by Kim et al. (Math Program 156:161–187, 2016). While the bisection method has demonstrated the computational efficiency, determining the validity of a computed lower bound for the QOP depends on a prescribed parameter $$\epsilon > 0$$ ϵ > 0 . To improve the performance of the bisection method for the Lagrangian-DNN relaxation, we propose a new technique that guarantees the validity of the computed lower bound at each iteration of the bisection method for any choice of $$\epsilon > 0$$ ϵ > 0 . It also accelerates the bisection method. Moreover, we present a method to retrieve a primal-dual pair of optimal solutions of the Lagrangian-DNN relaxation using the primal-dual interior-point method. As a result, the method provides a better lower bound and substantially increases the robustness as well as the effectiveness of the bisection method. Computational results on binary QOPs, multiple knapsack problems, maximal stable set problems, and quadratic assignment problems illustrate the robustness of the proposed method. In particular, a tight bound for QAPs with size $$n=50$$ n = 50 could be obtained.

Keywords: Nonconvex quadratic optimization problems with nonnegative variables; The Lagrangian-DNN relaxation; Improved bisection method; The validity of lower bounds; 90C20; 90C25; 90C26 (search for similar items in EconPapers)
Date: 2017
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Citations: View citations in EconPapers (3)

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DOI: 10.1007/s10589-016-9879-0

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