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
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
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
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
Given a closed cone C in the Euclidean n-space, the completely positive cone of C is the convex cone K generated by matrices of the form uu^T as u varies over C. Examples of completely positive cones include the positive semidefinite cone (when C is the entire Euclidean n-space) and the cone of completely positive … Read more
We study a new geometric graph parameter $\egd(G)$, defined as the smallest integer $r\ge 1$ for which any partial symmetric matrix which is completable to a correlation matrix and whose entries are specified at the positions of the edges of $G$, can be completed to a matrix in the convex hull of correlation matrices of … Read more
We will derive two main results related to the primal simplex method for an LP on a 0-1 polytope. One of the results is that, for any 0-1 polytope and any two vertices of it, there exists an LP instance for which the simplex method finds a path between them, whose length is at most … Read more
We study the computational question whether a given polytope or spectrahedron $S_A$ (as given by the positive semidefiniteness region of a linear matrix pencil $A(x)$) is contained in another one $S_B$. First we classify the computational complexity, extending results on the polytope/poly\-tope-case by Gritzmann and Klee to the polytope/spectrahedron-case. For various restricted containment problems, NP-hardness … Read more
In this paper, we consider the solution of the standard linear programming (LP). A remarkable result in LP claims that all optimal solutions form an optimal face of the underlying polyhedron. In practice, many real-world problems have infinitely many optimal solutions and pursuing the optimal face, not just an optimal vertex, is quite desirable. The … Read more