Chance constrained programming is an effective and convenient approach to control risk in decision making under uncertainty. However, due to unknown probability distributions of random parameters, the solution obtained from a chance constrained optimization problem can be biased. In practice, instead of knowing the true distribution of a random parameter, only a series of historical … 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
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 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
We consider the decision problem asking whether a partial rational symmetric matrix with an all-ones diagonal can be completed to a full positive semidefinite matrix of rank at most $k$. We show that this problem is $\NP$-hard for any fixed integer $k\ge 2$. Equivalently, for $k\ge 2$, it is $\NP$-hard to test membership in the … Read more
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 … Read more
In this paper, we propose a primal IP method for solving the optimal experimental design problem with a large class of smooth convex optimality criteria, including A-, D- and p th mean criterion, and establish its global convergence. We also show that the Newton direction can be computed efficiently when the size of the moment … Read more
The contributions of the first half of this thesis are on the computational and algebraic aspects of convexity in polynomial optimization. We show that unless P=NP, there exists no polynomial time (or even pseudo-polynomial time) algorithm that can decide whether a multivariate polynomial of degree four (or higher even degree) is globally convex. This solves … Read more
This work is devoted to the study of existence and stability results of semidefinite linear complementarity problems (SDLCP). Our approach consists of approximating the variational inequality formulation of the SDLCP by a sequence of suitable chosen variational inequalities. This provides particular estimates for the asymptotic cone of the solution set of the SDLCP. We thus … Read more