We describe a few applications of semide nite programming in combinatorial optimization.

Optimization problems in which the variable is not a vector but a symmetric matrix which is required to be positive semidefinite have been intensely studied in the last ten years. Part of the reason for the interest stems from the applicability of such problems to such diverse areas as designing the strongest column, checking the stability of a differential inclusion, and obtaining tight bounds for hard combinatorial optimization problems. Part also derives from great advances in our ability to solve such problems efficiently in theory and in practice (perhaps “or ” would be more appropriate: the most effective computational methods are not always provably efficient in theory, and vice versa). Here we describe this class of optimization problems, give a number of examples demonstrating its significance, outline its duality theory, and discuss algorithms for solving such problems.

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.

In the last decade important relations between Laplace eigenvalues and eigenvectors of graphs and several other graph parameters were discovered. In these notes we present some of these results and discuss their consequences. Attention is given to the partition and the isoperimetric properties of graphs, the max-cut problem and its relation to semidefinite programming, rapid mixing of Markov chains, and to extensions of the results to infinite graphs.

The graph partitioning problem is to divide the vertices of a graph into disjoint clusters to minimize the total cost of the edges cut by the clusters. A spectral partitioning heuristic uses the graph's eigenvectors to construct a geometric representation of the graph (e.g., linear orderings) which are subsequently partitioned. Our main result shows that when all the eigenvectors are used, graph partitioning reduces to a new vector partitioning problem. This result implies that as many eigenvectors as are practically possible should be used to construct a solution. This philosophy isincontrast to that of the widely-used spectral bipartitioning (SB) heuristic (which uses a single eigenvector to construct a 2-way partitioning) and several previous multiway partitioning heuristics [7][10][16][26][37] (which usek eigenvectors to construct a k-way partitioning). Our result motivates a simple ordering heuristic that is a multiple-eigenvector extension of SB. This heuristic not only signi cantly outperforms SB, but can also yield excellent multi-way VLSI circuit partitionings as compared to [1] [10]. Our experiments suggest that the vector partitioning perspective opens the door to new and effective heuristics.

Abstract—A variety of clustering algorithms have recently been proposed to handle data that is not linearly separable; spectral clustering and kernel k-means are two of the main methods. In this paper, we discuss an equivalence between the objective functions used in these seemingly different methods—in particular, a general weighted kernel k-means objective is mathematically equivalent to a weighted graph clustering objective. We exploit this equivalence to develop a fast high-quality multilevel algorithm that directly optimizes various weighted graph clustering objectives, such as the popular ratio cut, normalized cut, and ratio association criteria. This eliminates the need for any eigenvector computation for graph clustering problems, which can be prohibitive for very large graphs. Previous multilevel graph partitioning methods such as Metis have suffered from the restriction of equal-sized clusters; our multilevel algorithm removes this restriction by using kernel k-means to optimize weighted graph cuts. Experimental results show that our multilevel algorithm outperforms a state-of-the-art spectral clustering algorithm in terms of speed, memory usage, and quality. We demonstrate that our algorithm is applicable to large-scale clustering tasks such as image segmentation, social network analysis, and gene network analysis. Index Terms—Clustering, data mining, segmentation, kernel k-means, spectral clustering, graph partitioning. 1

Recently, a variety of clustering algorithms have been proposed to handle data that is not linearly separable. Spectral clustering and kernel k-means are two such methods that are seemingly quite different. In this paper, we show that a general weighted kernel k-means objective is mathematically equivalent to a weighted graph partitioning objective. Special cases of this graph partitioning objective include ratio cut, normalized cut and ratio association. Our equivalence has important consequences: the weighted kernel k-means algorithm may be used to directly optimize the graph partitioning objectives, and conversely, spectral methods may be used to optimize the weighted kernel k-means objective. Hence, in cases where eigenvector computation is prohibitive, we eliminate the need for any eigenvector computation for graph partitioning. Moreover, we show that the Kernighan-Lin objective can also be incorporated into our framework, leading to an incremental weighted kernel k-means algorithm for local optimization of the objective. We further discuss the issue of convergence of weighted kernel k-means for an arbitrary graph affinity matrix and provide a number of experimental results. These results show that non-spectral methods for graph partitioning are as effective as spectral methods and can be used for problems such as image segmentation in addition to data clustering.

The complexity of circuit designs has necessitated a top-down approach to layout synthesis. A large body of work shows that a good layout hierarchy, or partitioning tree, as measured by the associated Rent parameter, will correspond to an area-efficient layout. We define the intrinsic Rent parameter of a netlist to be the minimum possible Rent parameter of any partitioning tree for the netlist. Experimental results show that spectra-based ratio cut partitioning algorithms yield partitioning trees with the lowest observed Rent parameter over all benchmarks and over all algorithms tested. For examples where the intrinsic Rent parameter is known, spectral ratio cut partitioning yields a partitioning tree with Rent parameter essentially identical to this theoretical optimum. These results have deep implications withrespect to both the choice of partitioning algorithms for top-down layout, as well as new approaches to layout area estimation. The paper concludes with directions for future research, including several promising techniques for fast estimation of the (intrinsic) Rent parameter.

We study the problem of finding the minimum bisection of a graph into two parts of prescribed sizes. We formulate two lower bounds on the problem by relaxing node- and edge-incidence vectors of cuts. We prove that both relaxations provide the same bound. The main fact we prove is that the duality between the relaxed edge- and node-vectors preserves very natural cardinality constraints on cuts. We present an analogous result also for the max-cut problem, and show a relation between the edge relaxation and some other optimality criteria studied before. Finally, we briefly mention possible applications for a practical computational approach.

Quadratically constrained quadratic programs (QQP) play an important modeling role for many diverse problems. These problems are in general NP hard, and numerically intractable. Lagrangian relaxations often provide good approximate solutions to these hard problems. Such relaxations are equivalent to semidefinite programming relaxations. For several special cases of QQP, e.g. convex programs and trust region subproblems, the Lagrangian relaxation provides the exact optimal value, i.e. there is a zero duality gap. However this is not true for the general QQP, or even the QQP with two convex constraints, but a nonconvex objective. In this paper we consider a certain QQP where the quadratic constraints correspond to the matrix orthogonality condition XX T = I. For this problem we show that the Lagrangian dual based on relaxing the constraints XX T = I, and the seemingly redundant constraints X T X = I, has a zero duality gap. This result has natural applications to quadratic assignm...