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

Sunyoung Kim, Masakazu Kojima

Research output: Contribution to journalArticlepeer-review

5 Scopus citations


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.

Original languageEnglish
Pages (from-to)170-183
Number of pages14
JournalDiscrete Optimization
StatePublished - 1 May 2017

Bibliographical note

Publisher Copyright:
© 2016 Elsevier B.V.


  • A hierarchy of semidefinite relaxations
  • Binary integer quadratic program
  • Conic relaxations
  • Inexact optimal values
  • The max-cut problem with equal weight


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