A recipe for semidefinite relaxation for \((0,1)\)-quadratic programming.

*(English)*Zbl 0843.90088Summary: We review various relaxations of \((0,1)\)-quadratic programming problems. These include semidefinite programs, parametric trust region problems and concave quadratic maximization. All relaxations that we consider lead to efficiency solvable problems. The main contributions of the paper are the following. Using Lagrangian duality, we prove equivalence of the relaxations in a unified and simple way. Some of these equivalences have been known previously, but our approach leads to short and transparent proofs. Moreover, we extend the approach to the case of equality constrained problems by taking the squared linear constraints into the objective function. We show how this technique can be applied to the Quadratic Assignment Problem, the Graph Partitioning Problem and the Max-Clique Problem. Finally, we show our relaxation to be best possible among all quadratic majorants with zero trace.

##### Keywords:

minmax eigenvalue problems; theta function; quadratic assignment; graph partitioning; max-clique problem; relaxations of \((0,1)\)-quadratic programming problems; semidefinite programs; parametric trust region problems; concave quadratic maximization; Lagrangian duality##### Software:

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\textit{S. Poljak} et al., J. Glob. Optim. 7, No. 1, 51--73 (1995; Zbl 0843.90088)

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