## Abstract

A reformulation of quadratically constrained binary programs as duals of set-copositive linear optimization problems is derived using either \(\{0,1\}\)-formulations or \(\{-1,1\}\)-formulations. The latter representation allows an extension of the randomization technique by Goemans and Williamson. An application to the max-clique problem shows that the max-clique problem is equivalent to a linear program over the max-cut polytope with one additional linear constraint. This transformation allows the solution of a semidefinite relaxation of the max-clique problem with about the same computational effort as the semidefinite relaxation of the max-cut problem—independent of the number of edges in the underlying graph. A numerical comparison of this approach to the standard Lovasz number concludes the paper.

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## Notes

In particular, nonnegative continuous variables are not considered here, since the nonnegativity is lost for \(\{-1,1\}\)-formulations. It is possible to extend the approach of the present paper to problems with nonnegative variables and with \(\{-1,1\}\)-variables using set-completely-positive programs with a cone \(\mathcal{{C}}_{n}^*(M)\) where \(M\) not only depends on the dimension but also on the partition of continuous and binary variables. In such generalization, also for a \(\{0,1\}\)-formulation the cone \(\mathcal{{C}}_{n}^*(M)\) will depend on the partition of continuous and binary variables, and the transformations \(T,\ \mathcal{{T}}\) of the previous section will have a block structure. For simplicity, the presentation in this paper restricts itself to plain binary problems.

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## Acknowledgments

The authors would like to thank two anonymous referees for their criticism which helped to improve the presentation of the paper.

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Lieder, F., Rad, F.B.A. & Jarre, F. Unifying semidefinite and set-copositive relaxations of binary problems and randomization techniques.
*Comput Optim Appl* **61**, 669–688 (2015). https://doi.org/10.1007/s10589-015-9731-y

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DOI: https://doi.org/10.1007/s10589-015-9731-y

### Keywords

- Max-cut relaxation
- Max-clique problem
- Semidefinite program
- Completely positive program