- A robust Lagrangian-DNN method for a class of quadratic optimization problems Naohiko Arima(nao_arimame.com) Sunyoung Kim(skimewha.ac.kr) Masakazu Kojima(masakazukojimamac.com) Kim-Chuan Toh(mattohkcnus.edu.sg) 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 (QOPs) using the bisection method combined with first-order methods by Kim, Kojima and Toh in 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 $\epsilon > 0$. 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 $\epsilon > 0$. 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 the binary QOPs, the multiple knapsack problems, the maximal stable set problems, and the quadratic assignment problems (QAPs) illustrate the robustness of the proposed method. In particular, a tight bound for QAPs with size $n=50$ could be obtained. Keywords: Nonconvex quadratic optimization problems with nonnegative variables, the Lagrangian-DNN relaxation, improved bisection method, the validity of lower bounds. Category 1: Linear, Cone and Semidefinite Programming (Other ) Category 2: Nonlinear Optimization (Quadratic Programming ) Citation: Research report B-475, Tokyo Institute of Technology, Department of Mathematical and Computing Sciences, Oh-Okayama, Meguro-ku, Tokyo 152-8552 Download: [PDF]Entry Submitted: 02/29/2016Entry Accepted: 02/29/2016Entry Last Modified: 02/29/2016Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Optmization Society.