Paper Info
Title
A Lagrangian-DNN relaxation: a fast method for computing tight lower bounds for a class of quadratic optimization problems.
Abstract
We propose an efficient computational method for linearly constrained quadratic optimization problems (QOPs) with complementarity constraints based on their Lagrangian and doubly nonnegative (DNN) relaxation and first-order algorithms. The simplified Lagrangian---completely positive programming (CPP) relaxation of such QOPs proposed by Arima, Kim, and Kojima in 2012 takes one of the simplest forms, an unconstrained conic linear optimization problem with a single Lagrangian parameter in a CPP matrix variable with its upper-left element fixed to 1. Replacing the CPP matrix variable by a DNN matrix variable, we derive the Lagrangian---DNN relaxation, and establish the equivalence between the optimal value of the DNN relaxation of the original QOP and that of the Lagrangian---DNN relaxation. We then propose an efficient numerical method for the Lagrangian---DNN relaxation using a bisection method combined with the proximal alternating direction multiplier and the accelerated proximal gradient methods. Numerical results on binary QOPs, quadratic multiple knapsack problems, maximum stable set problems, and quadratic assignment problems illustrate the superior performance of the proposed method for attaining tight lower bounds in shorter computational time.
Year
DOI
Venue
2016
10.1007/s10107-015-0874-5
Mathematical Programming
Keywords
Field
DocType
Linearly constrained quadratic optimization problems with complementarity constraints, The Lagrangian–conic relaxation, Bisection method, Iterative solver, 90C20, 90C25, 90C26
Discrete mathematics,Mathematical optimization,Bisection method,Matrix (mathematics),Quadratic equation,Proximal Gradient Methods,Knapsack problem,Quadratic programming,Lagrangian relaxation,Numerical analysis,Mathematics
Journal
Volume
Issue
ISSN
156
1-2
1436-4646
Citations 
PageRank 
References 
15
0.62
17
Authors
3
Name
Order
Citations
PageRank
S. Kim124814.25
Masakazu Kojima21603222.51
Kim-Chuan Toh3109780.39