This paper presents various worst-case results on the positive semidefinite (psd) rank of a nonnegative matrix, primarily in the context of polytopes. We prove that the psd rank of a generic n-dimensional polytope with v vertices is at least (nv)^(1/4) improving on previous lower bounds. For polygons with v vertices, we show that psd rank … Read more
We consider a property of positive polynomials on a compact set with a small perturbation. When applied to a Polynomial Optimization Problem (POP), the property implies that the optimal value of the corresponding SemiDefinite Programming (SDP) relaxation with sufficiently large relaxation order is bounded from below by $(f^¥ast – ¥epsilon)$ and from above by $f^¥ast … Read more
We address the following generalization $P$ of the Lowner-John’s ellipsoid problem. Given a (non necessarily convex) compact set $K\subset R^n$ and an even integer $d, find an homogeneous polynomial $g$ of degree $d$ such that $K\subset G:=\{x:g(x)\leq1\}$ and $G$ has minimum volume among all such sets. We show that $P$ is a convex optimization problem … Read more
This paper addresses the following three topics: positive semidefinite (psd) matrix completions, universal rigidity of frameworks, and the Strong Arnold Property (SAP). We show some strong connections among these topics, using semidefinite programming as unifying theme. Our main contribution is a sufficient condition for constructing partial psd matrices which admit a unique completion to a … Read more
The facial reduction algorithm of Borwein and Wolkowicz and the extended dual of Ramana provide a strong dual for the conic linear program (P) \sup { | Ax \leq_K b} in the absence of any constraint qualification. The facial reduction algorithm solves a sequence of auxiliary optimization problems to obtain such a dual. Ramana’s dual … Read more
We extend and characterize the concept of $s$-semigoodness for a sensing matrix in sparse nonnegative recovery (proposed by Juditsky , Karzan and Nemirovski [Math Program, 2011]) to the linear transformations in low-rank semidefinite matrix recovery. We show that s-semigoodness is not only a necessary and sufficient condition for exact $s$-rank semidefinite matrix recovery by a … Read more
The one-to-one relation between the points fulfilling the KKT conditions of an optimization problem and the zeros of the corresponding Kojima function is well-known. In the present paper we study the interplay between metric regularity and strong regularity of this a priori nonsmooth function in the context of semidefinite programming. Having in mind the topological … Read more
Interior-point methods in semidefinite programming (SDP) require the solution of a sequence of linear systems which are used to derive the search directions. Safeguards are typically required in order to handle rank-deficient Jacobians and free variables. We generalize the primal-dual regularization of \cite{friedlander-orban-2012} to SDP and show that it is possible to recover an optimal … Read more
We derive a new lower bound for the bandwidth of a graph that is based on a new lower bound for the minimum cut problem. Our new semide finite programming relaxation of the minimum cut problem is obtained by strengthening the well-known semide nite programming relaxation for the quadratic assignment problem by fixing two vertices in the … Read more
A stable symmetrization of the linear systems arising in interior-point methods for solving linear programs is introduced. A comparison of the condition numbers of the resulting interior-point linear systems with other commonly used approaches indicates that the new approach may be best suitable for an iterative solution. It is shown that there is a natural … Read more