Towards the optimum by semidefinite and copositive programming: New approach to approximate hard optimization problems

Semidefinite and copositive programming have attained an important role in combinatorial optimization in the last two decades. There is a strong evidence that semidefinite and copositive approximation models are significantly stronger than the purely linear ones for many combinatorial problems. In some cases the copositive models give even the exact value of the problem. The first part of the book contains beside a survey of standard results from linear algebra and conic programming also a new method to solve semidefinite programs, based on the augmented Lagrangian method. This method named the Boundary point method goes far beyond the reach of interior point methods when the linear constraints are nearly orthogonal. The second part demonstrates the application of semidefinite and copositive programming to the following NP-hard problems from combinatorial optimization: the bandwidth problem, the quadratic assignment problem, the min-cut problem and the general graph partitioning problem. The book also provides the ideas how to extend the approach to some other 0-1 problems, like the stability number problem and the balanced vertex separator problem.

Janez Povh (1973) studied mathematics at the Faculty of mathematics and physics at University in Ljubljana. After B.Sc. in 1998 and M.Sc. in 2002 he defended the Ph.D. thesis at the same faculty in 2006 under supervision of dr. Franz Rendl. From 2008 he is assistant professor at the Faculty of information studies in Novo mesto, Slovenia.