SDP Relaxations in Combinatorial Optimization from a Lagrangian Viewpoint

Abstract

A successful technique for some problems in combinatorial optimization is the so-called SDP relaxation, essentially due to L. Lovasz, and much developed by M. Goemans and D.P. Williamson. As observed by S. Poljak, F. Rendl and H. Wolkowicz, this technique can be interpreted from the point of view of Lagrangian duality. A central tool for this is dualization of quadratic constraints, an operation pioneered by N.Z. Shor. We synthesize these various operations, in a language close to that of nonlinear programming. Then we show how the approach can be applied to general combinatorial problems.