We give theorems that can be used to upper bound the densities of packings of different spherical caps in the unit sphere and of translates of different convex bodies in Euclidean space. These theorems extend the linear programming bounds for packings of spherical caps and of convex bodies through the use of semidefinite programming. We … Read more
This paper describes a new approach for computing nonnegative matrix factorizations (NMFs) with linear programming. The key idea is a data-driven model for the factorization, in which the most salient features in the data are used to express the remaining features. More precisely, given a data matrix X, the algorithm identifies a matrix C that … Read more
We study the convex hull of the intersection of a convex set E and a linear disjunction. This intersection is at the core of solution techniques for Mixed Integer Conic Optimization. We prove that if there exists a cone K (resp., a cylinder C) that has the same intersection with the boundary of the disjunction … Read more
The set of polynomials which are nonnegative over a subset of the nonnegative orthant (we call them set semidefinite) have many uses in optimization. A common example of this type of set is the set of copositive matrices, where effectively we are considering nonnegativity over the entire nonnegative orthant and we limit the polynomials to … Read more
The positive semidefinite (psd) rank of a polytope is the smallest $k$ for which the cone of $k \times k$ real symmetric psd matrices admits an affine slice that projects onto the polytope. In this paper we show that the psd rank of a polytope is at least the dimension of the polytope plus one, … Read more
group of analytical formulas formulas for calculating the global maximal and minimal ranks and inertias of the quadratic matrix-valued function $$ \phi(X) = \left(\, AXB + C\,\right)\!M\!\left(\, AXB + C \right)^{*} + D $$ are established and their consequences are presented, where $A$, $B$, $C$ and $D$ are given complex matrices with $A$ and $C$ … Read more
On the interior of a regular convex cone $K \subset \mathbb R^n$ there exist two canonical Hessian metrics, the one generated by the logarithm of the characteristic function, and the Cheng-Yau metric. The former is associated with a self-concordant logarithmically homogeneous barrier on $K$ with parameter of order $O(n)$, the universal barrier. This barrier is … Read more
Given a symmetric and positive (semi)definite $n$-by-$n$ matrix $M$ and a vector, in this paper, we consider the matrix splitting method for solving the second-order cone linear complementarity problem (SOCLCP). The matrix splitting method is among the most widely used approaches for large scale and sparse classical linear complementarity problems (LCP), and its linear convergence … Read more
This paper introduces the notion of a weighted Complementarity Problem (wCP), which consists in finding a pair of vectors $(x,s)$ belonging to the intersection of a manifold with a cone, such that their product in a certain algebra, $x\circ s$, equals a given weight vector $w$. When $w$ is the zero vector, then wCP reduces … Read more
A real square matrix Q is a bilinear complementarity relation on a proper cone K in R^n if x in K, s in K^* with x perpendicular to s implies x^{T}Qs=0, where K^* is the dual of K. The bilinearity rank of K is the dimension of the space of all bilinear complementarity relations on … Read more