Exposed faces of semidefinitely representable sets

A linear matrix inequality (LMI) is a condition stating that a symmetric matrix whose entries are affine linear combinations of variables is positive semidefinite. Motivated by the fact that diagonal LMIs define polyhedra, the solution set of an LMI is called a spectrahedron. Linear images of spectrahedra are called semidefinite representable sets. Part of the interest in spectrahedra and semidefinite representable sets arises from the fact that one can efficiently optimize linear functions on them by semidefinite programming, like one can do on polyhedra by linear programming. It is known that every face of a spectrahedron is exposed. This is also true in the general context of rigidly convex sets. We study the same question for semidefinite representable sets. Lasserre proposed a moment matrix method to construct semidefinite representations for certain sets. Our main result is that this method can only work if all faces of the considered set are exposed. This necessary condition complements sufficient conditions recently proved by Lasserre, Helton and Nie

Conditions for Existence of Dual Certificates in Rank-One Semidefinite Problems

Several signal recovery tasks can be relaxed into semidefinite programs with rank-one minimizers. A common technique for proving these programs succeed is to construct a dual certificate. Unfortunately, dual certificates may not exist under some formulations of semidefinite programs. In order to put problems into a form where dual certificate arguments are possible, it is important to develop conditions under which the certificates exist. In this paper, we provide an example where dual certificates do not exist. We then present a completeness condition under which they are guaranteed to exist. For programs that do not satisfy the completeness condition, we present a completion process which produces an equivalent program that does satisfy the condition. The important message of this paper is that dual certificates may not exist for semidefinite programs that involve orthogonal measurements with respect to positive-semidefinite matrices. Such measurements can interact with the positive-semidefinite constraint in a way that implies additional linear measurements. If these additional measurements are not included in the problem formulation, then dual certificates may fail to exist. As an illustration, we present a semidefinite relaxation for the task of finding the sparsest element in a subspace. One formulation of this program does not admit dual certificates. The completion process produces an equivalent formulation which does admit dual certificates

A sequential semidefinite programming method and an application in passive reduced-order modeling

We consider the solution of nonlinear programs with nonlinear semidefiniteness constraints. The need for an efficient exploitation of the cone of positive semidefinite matrices makes the solution of such nonlinear semidefinite programs more complicated than the solution of standard nonlinear programs. In particular, a suitable symmetrization procedure needs to be chosen for the linearization of the complementarity condition. The choice of the symmetrization procedure can be shifted in a very natural way to certain linear semidefinite subproblems, and can thus be reduced to a well-studied problem. The resulting sequential semidefinite programming (SSP) method is a generalization of the well-known SQP method for standard nonlinear programs. We present a sensitivity result for nonlinear semidefinite programs, and then based on this result, we give a self-contained proof of local quadratic convergence of the SSP method. We also describe a class of nonlinear semidefinite programs that arise in passive reduced-order modeling, and we report results of some numerical experiments with the SSP method applied to problems in that class

Logarithmic barriers for sparse matrix cones

Algorithms are presented for evaluating gradients and Hessians of logarithmic barrier functions for two types of convex cones: the cone of positive semidefinite matrices with a given sparsity pattern, and its dual cone, the cone of sparse matrices with the same pattern that have a positive semidefinite completion. Efficient large-scale algorithms for evaluating these barriers and their derivatives are important in interior-point methods for nonsymmetric conic formulations of sparse semidefinite programs. The algorithms are based on the multifrontal method for sparse Cholesky factorization

Fractional Zero Forcing via Three-color Forcing Games

An $r$-fold analogue of the positive semidefinite zero forcing process that is carried out on the $r$-blowup of a graph is introduced and used to define the fractional positive semidefinite forcing number. Properties of the graph blowup when colored with a fractional positive semidefinite forcing set are examined and used to define a three-color forcing game that directly computes the fractional positive semidefinite forcing number of a graph. We develop a fractional parameter based on the standard zero forcing process and it is shown that this parameter is exactly the skew zero forcing number with a three-color approach. This approach and an algorithm are used to characterize graphs whose skew zero forcing number equals zero.Comment: 24 page

Exact Solution Methods for the kk-item Quadratic Knapsack Problem

The purpose of this paper is to solve the 0-1 $k$-item quadratic knapsack problem $(kQKP)$, a problem of maximizing a quadratic function subject to two linear constraints. We propose an exact method based on semidefinite optimization. The semidefinite relaxation used in our approach includes simple rank one constraints, which can be handled efficiently by interior point methods. Furthermore, we strengthen the relaxation by polyhedral constraints and obtain approximate solutions to this semidefinite problem by applying a bundle method. We review other exact solution methods and compare all these approaches by experimenting with instances of various sizes and densities.Comment: 12 page