This software package is a Matlab implementation of infeasible path-following algorithms for solving standard semidefinite programming (SDP) problems. Mehrotratype predictor-corrector variants are included. Analogous algorithms for the homogeneous formulation of the standard SDP problem are also implemented. Four types of search directions are available, namely, the AHO, HKM, NT, and GT directions. A few classes of SDP problems are included as well. Numerical results for these classes show that our algorithms are fairly efficient and robust on problems with dimensions of the order of a few hundreds.

We consider the problem of finding a linear iteration that yields distributed averaging consensus over a network, i.e., that asymptotically computes the average of some initial values given at the nodes. When the iteration is assumed symmetric, the problem of finding the fastest converging linear iteration can be cast as a semidefinite program, and therefore efficiently and globally solved. These optimal linear iterations are often substantially faster than several common heuristics that are based on the Laplacian of the associated graph.

Abstract. Given a covariance matrix, we consider the problem of maximizing the variance explained by a particular linear combination of the input variables while constraining the number of nonzero coefficients in this combination. This problem arises in the decomposition of a covariance matrix into sparse factors or sparse principal component analysis (PCA), and has wide applications ranging from biology to finance. We use a modification of the classical variational representation of the largest eigenvalue of a symmetric matrix, where cardinality is constrained, and derive a semidefinite programming–based relaxation for our problem. We also discuss Nesterov’s smooth minimization technique applied to the semidefinite program arising in the semidefinite relaxation of the sparse PCA problem. The method has complexity O(n 4 √ log(n)/ɛ), where n is the size of the underlying covariance matrix and ɛ is the desired absolute accuracy on the optimal value of the problem.

We present a dual-scaling interior-point algorithm and show how it exploits the structure and sparsity of some large scale problems. We solve the positive semidefinite relaxation of combinatorial and quadratic optimization problems subject to boolean constraints. We report the first computational results of interior-point algorithms for approximating the maximum cut semidefinite programs with dimension up-to 3000.

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.

In this paper, we present a nonlinear programming algorithm for solving semidefinite programs (SDPs) in standard form. The algorithm's distinguishing feature is a change of variables that replaces the symmetric, positive semidefinite variable X of the SDP with a rectangular variable R according to the factorization X = RR T . The rank of the factorization, i.e., the number of columns of R, is chosen minimally so as to enhance computational speed while maintaining equivalence with the SDP. Fundamental results concerning the convergence of the algorithm are derived, and encouraging computational results on some large-scale test problems are also presented. Keywords: semidefinite programming, low-rank factorization, nonlinear programming, augmented Lagrangian, limited memory BFGS. 1 Introduction In the past few years, the topic of semidefinite programming, or SDP, has received considerable attention in the optimization community, where interest in SDP has included the investigation of...

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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...

The Goemans-Williamson randomized algorithm guarantees a high-quality approximation to the Max-Cut problem, but the cost associated with such an approximation can be excessively high for large-scale problems due to the need for solving an expensive semidefinite relaxation. In order to achieve better practical performance, we propose an alternative, rank-two relaxation and develop a specialized version of the Goemans-Williamson technique. The proposed approach leads to continuous optimization heuristics applicable to Max-Cut as well as other binary quadratic programs, for example the Max-Bisection problem. A computer code based on the rank-two relaxation heuristics is compared with two state-of-the-art semidefinite programming codes that implement the Goemans-Williamson randomized algorithm, as well as with a purely heuristic code for effectively solving a particular Max-Cut problem arising in physics. Computational results show that the proposed approach is fast and scalable and, more importantly, attains a higher approximation quality in practice than that of the Goemans-Williamson randomized algorithm. An extension to Max-Bisection is also discussed as well as an important difference between the proposed approach and the Goemans-Williamson algorithm, namely that the new approach does not guarantee an upper bound on the Max-Cut optimal value. Key words. Binary quadratic programs, Max-Cut and Max-Bisection, semidefinite relaxation, rank-two relaxation, continuous optimization heuristics. AMS subject classifications. 90C06, 90C27, 90C30 1.