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 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 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 could be obtained.
Bibliographical noteFunding Information:
The research of Sunyoung Kim was supported by NRF 2014-R1A2A1A11049618. The research of Masakazu Kojima was supported by Grant-in-Aid for Scientific Research (A) 26242027 and the Japan Science and Technology Agency (JST), the Core Research of Evolutionary Science and Technology (CREST) research project. Research of Kim-Chuan Toh was supported in part by the Ministry of Education, Singapore, Academic Research Fund under Grant R-146-000-194-112.
© 2016, Springer Science+Business Media New York.
- Improved bisection method
- Nonconvex quadratic optimization problems with nonnegative variables
- The Lagrangian-DNN relaxation
- The validity of lower bounds