Christoph Helmberg

An Interior-Point Method for Semidefinite Programming

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....

A Spectral Bundle Method for Semidefinite Programming

A central drawback of primal-dual interior point methods for semidefinite programs is their lack of ability to exploit problem structure in cost and coefficient matrices. This restricts applicability to problems of small dimension. Typically, semidefinite relaxations arising in combinatorial applications have sparse and well-structured cost and coefficient matrices of huge order. We present a method that allows us to compute acceptable approximations to the optimal solution of large problems within reasonable time. Semidefinite programming problems with constant trace on the primal feasible set are equivalent to eigenvalue optimization problems. These are convex nonsmooth programming problems and can be solved by bundle methods. We propose replacing the traditional polyhedral cutting plane model constructed from subgradient information by a semidefinite model that is tailored to eigenvalue problems. Convergence follows from the traditional approach but a proof is included for completeness. We present numerical examples demonstrating the efficiency of the approach on combinatorial examples.

Solving quadratic (0,1)-problems by semidefinite programs and cutting planes

We present computational experiments for solving quadratic (0, 1) problems. Our approach combines a semidefinite relaxation with a cutting plane technique, and is applied in a Branch and Bound setting. Our experiments indicate that this type of approach is very robust, and allows to solve many moderately sized problems, having say, less than 100 binary variables, in a routine manner.

A spectral bundle method with bounds

Abstract.Semidefinite relaxations of quadratic 0-1 programming or graph partitioning problems are well known to be of high quality. However, solving them by primal-dual interior point methods can take much time even for problems of moderate size. The recent spectral bundle method of Helmberg and Rendl can solve quite efficiently large structured equality-constrained semidefinite programs if the trace of the primal matrix variable is fixed, as happens in many applications. We extend the method so that it can handle inequality constraints without seriously increasing computation time. In addition, we introduce inexact null steps. This abolishes the need of computing exact eigenvectors for subgradients, which brings along significant advantages in theory and in practice. Encouraging preliminary computational results are reported.

A Semidefinite Programming Approach to the Quadratic Knapsack Problem

In order to gain insight into the quality of semidefinite relaxations of constrained quadratic 0/1 programming problems we study the quadratic knapsack problem. We investigate several basic semidefinite relaxations of this problem and compare their strength in theory and in practice. Various possibilities to improve these basic relaxations by cutting planes are discussed. The cutting planes either arise from quadratic representations of linear inequalities or from linear inequalities in the quadratic model. In particular, a large family of combinatorial cuts is introduced for the linear formulation of the knapsack problem in quadratic space. Computational results on a small class of practical problems illustrate the quality of these relaxations and cutting planes.

Fixing Variables in Semidefinite Relaxations

The standard technique of reduced cost fixing from linear programming is not trivially extensible to semidefinite relaxations because the corresponding Lagrange multipliers are usually not available. We propose a general technique for computing reasonable Lagrange multipliers for constraints that are not part of the problem description. Its specialization to the semidefinite {-1,1} relaxation of quadratic 0-1 programming yields an efficient routine for fixing variables. The routine offers the possibility of exploiting problem structure. We extend the traditional bijective map between {0,1} and {-1,1} formulations to the constraints so that the dual variables remain the same and structural properties are preserved. Consequently, the fixing routine can be applied efficiently to optimal solutions of the semidefinite {0,1} relaxation of constrained quadratic 0-1 programming as well. We provide numerical results showing the efficacy of this approach.

Bundle Methods to Minimize the Maximum Eigenvalue Function

In the last ten years the study of interior point methods dominated algorithmic research in semidefinite programming. Only recently interest in nonsmooth optimization methods revived again, the impetus coming from two different directions. On the one hand alternative possibilities were sought to solve structured large scale semidefinite programs which were not amenable to current interior point codes [338], on the other hand new developments in the second order theory of nonsmooth convex optimization suggested the specialization of these theoretic techniques to semidefinite programming [597, 598]. We present these new methods under the common framework of bundle methods and survey the underlying theory as well as some implementational aspects. In order to illustrate the efficiency and potential of the algorithms we also present numerical results.

A spectral approach to bandwidth and separator problems in graphs

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.

Embedded in the Shadow of the Separator

Eigenvectors to the second smallest eigenvalue of the Laplace matrix of a graph, also known as Fiedler vectors, are the basic ingredient in spectral graph partitioning heuristics. Maximizing this second smallest eigenvalue over all nonnegative edge weightings with bounded total weight yields the absolute algebraic connectivity introduced by Fiedler, who proved tight connections of this value to the connectivity of the graph. Our objective is to gain a better understanding of the connections between separators and the eigenspace of this eigenvalue by studying the dual semidefinite optimization problem to the absolute algebraic connectivity. By exploiting optimality conditions we show that this problem is equivalent to finding an embedding of the

Quadratic knapsack relaxations using cutting planes and semidefinite programming

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