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Binary quadratic optimization problems that are difficult to solve by conic relaxations

Title
Binary quadratic optimization problems that are difficult to solve by conic relaxations
Authors
Kim S.Kojima M.
Ewha Authors
김선영
SCOPUS Author ID
김선영scopus
Issue Date
2017
Journal Title
Discrete Optimization
ISSN
1572-5286JCR Link
Citation
vol. 24, pp. 170 - 183
Keywords
A hierarchy of semidefinite relaxationsBinary integer quadratic programConic relaxationsInexact optimal valuesThe max-cut problem with equal weight
Publisher
Elsevier B.V.
Indexed
SCIE; SCOPUS scopus
Abstract
We study conic relaxations including semidefinite programming (SDP) relaxations and doubly nonnegative programming (DNN) relaxations to find the optimal values of binary QOPs. The main focus of the study is on how the relaxations perform with respect to the rank of the coefficient matrix in the objective of a binary QOP. More precisely, for a class of binary QOP instances, which include the max-cut problem of a graph with an odd number of nodes and equal weight, we show numerically that (1) neither the standard DNN relaxation nor the DNN relaxation derived from the completely positive formulation by Burer performs better than the standard SDP relaxation, and (2) Lasserre's hierarchy of SDP relaxations requires solving the SDP with the relaxation order at least ⌈n/2⌉ to attain the optimal value. The bound ⌈n/2⌉ for the max-cut problem of a graph with equal weight is consistent with Laurent's conjecture in 2003, which was proved recently by Fawzi, Saunderson and Parrilo in 2015. © 2016 Elsevier B.V.
DOI
10.1016/j.disopt.2016.08.001
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자연과학대학 > 수학전공 > Journal papers
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