Eigenvalue Optimization for Solving the MAX-CUT Problem

The purpose of this semester project is to investigate the Spectral Bundle Method, which is a specialized subgradient method particularly suited for solving large scale semidefinite programs that can be cast as eigenvalue optimization problems of the form min y∈R aλmax(C − m ∑ i=1 Aiyi) + b T y, where C and Ai are given real symmetric matrices, b ∈ R allows to specify a linear cost term, and a > 0 is a constant multiplier for the maximum eigenvalue function λmax(·). In particular, a semidefinite relaxation of the well-known maxcut problem belongs to this class of problems. After a general description of the Spectral Bundle Method, a matlab implementation of the method designed for solving the eigenvalue relaxation of the max-cut problem is presented. Finally, it is explained how to extract an optimal solution of the original max-cut problem from the optimal solution of the eigenvalue relaxation.