A Lagrangian–DNN relaxation: a fast method for computing tight lower bounds for a class of quadratic optimization problems

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)...

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Vydáno v:Mathematical programming Ročník 156; číslo 1-2; s. 161 - 187
Hlavní autoři: Kim, Sunyoung, Kojima, Masakazu, Toh, Kim-Chuan
Médium: Journal Article
Jazyk:angličtina
Vydáno: Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2016
Springer Nature B.V
Témata:
Algorithms
Approximation
Assignment problem
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Computation
Computer programming
Conics
Equivalence
Full Length Paper
Knapsack problem
Lower bounds
Mathematical analysis
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematical models
Mathematics
Mathematics and Statistics
Mathematics of Computing
Numerical Analysis
Optimization
Optimization techniques
Programming
Studies
Theoretical
Variables
United States > US
90C20
The Lagrangian–conic relaxation
Iterative solver
90C25
Bisection method
90C26
Linearly constrained quadratic optimization problems with complementarity constraints
ISSN:0025-5610, 1436-4646
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Popis
Shrnutí: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.
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-015-0874-5