We propose a new interior point based method to minimize a linear function of a matrix variable subject to linear equality and inequality constraints over the set of positive semidefinite matrices. We show that the approach is very efficient for graph bisection problems, such as max-cut. Other applications include max-min eigenvalue problems and relaxations for the stable set problem.

Optimization problems involving the eigenvalues of symmetric and nonsymmetric matrices present a fascinating mathematical challenge. Such problems arise often in theory and practice, particularly in engineering design, and are amenable to a rich blend of classical mathematical techniques and contemporary optimization theory. This essay presents a personal choice of some central mathematical ideas, outlined for the broad optimization community. I discuss the convex analysis of spectral functions and invariant matrix norms, touching briey on semide nite representability, and then outlining two broader algebraic viewpoints based on hyperbolic polynomials and Lie algebra. Analogous nonconvex notions lead into eigenvalue perturbation theory. The last third of the article concerns stability, for polynomials, matrices, and associated dynamical systems, ending with a section on robustness. The powerful and elegant language of nonsmooth analysis appears throughout, as a unifying narrative thread.

We survey results on edge isoperimetric problems on graphs, present some new results and show some applications of such problems in combinatorics and computer science. 1 Introduction Let G = (V G ; EG ) be a simple connected graph. For a subset A ` VG denote I G (A) = f(u; v) 2 EG j u; v 2 Ag; ` G (A) = f(u; v) 2 EG j u 2 A; v 62 Ag: We omit the subscript G if the graph is uniquely defined by the context. By edge isoperimetric problems we mean the problem of estimation of the maximum and minimum of the functions I and ` respectively, taken over all subsets of VG of the same cardinality. The subsets on which the extremal values of I (or `) are attained are called isoperimetric subsets. These problems are discrete analogies of some continuous problems, many of which can be found in the book of P'olya and Szego [99] devoted to continuous isoperimetric inequalities and their numerous applications. Although the continuous isoperimetric problems have a history of thousand years, the dis...

Lower bounds on the bandwidth, the size of a vertex separator of general undirected graphs, and the largest common subgraph of two undirected (weighted) graphs are obtained. The bounds are based on a projection technique developed for the quadratic assignment problem, and once more demonstrate the importance of the extreme eigenvalues of the Laplacian. They will be shown to be strict for certain classes of graphs and compare favourably to bounds already known in literature. Further improvement is gained by applying nonlinear optimization methods.

separator problems in graphs