We present an extension of the proximal point method with Bregman distances to solve Variational Inequality Problems (VIP) on Hadamard manifolds (simply connected finite dimensional Riemannian manifold with nonpositive sectional curvature). Under some natural assumption, as for example, the existence of solutions of the (VIP) and the monotonicity of the multivalued vector field, we prove … Read more
Our first contribution in this paper is to prove that three natural sum of squares (sos) based sufficient conditions for convexity of polynomials via the definition of convexity, its first order characterization, and its second order characterization are equivalent. These three equivalent algebraic conditions, henceforth referred to as sos-convexity, can be checked by semidefinite programming … Read more
We present a fully polynomial time approximation scheme (FPTAS) for optimizing a very general class of nonlinear functions of low rank over a polytope. Our approximation scheme relies on constructing an approximate Pareto-optimal front of the linear functions which constitute the given low-rank function. In contrast to existing results in the literature, our approximation scheme … Read more
We show that any nonconvex quadratically constrained quadratic program(QCQP) can be represented as a generalized copositive program. In fact,we provide two representations. The first is based on the concept of completely positive (CP) matrices over second order cones, while the second is based on CP matrices over the positive semidefinte cone. Our analysis assumes that … Read more
Let $A \in \Omega_n$ be doubly-stochastic $n \times n$ matrix. Alexander Schrijver proved in 1998 the following remarkable inequality \begin{equation} \label{le} per(\widetilde{A}) \geq \prod_{1 \leq i,j \leq n} (1- A(i,j)); \widetilde{A}(i,j) =: A(i,j)(1-A(i,j)), 1 \leq i,j \leq n \end{equation} We prove in this paper the following generalization (or just clever reformulation) of (\ref{le}):\\ For all … Read more
Copositive optimization is a quickly expanding scientific research domain with wide-spread applications ranging from global nonconvex problems in engineering to NP-hard combinatorial optimization. It falls into the category of conic programming (optimizing a linear functional over a convex cone subject to linear constraints), namely the cone of all completely positive symmetric nxn matrices, and its … Read more
We propose a new subgradient method for the minimization of convex functions over a convex set. Common subgradient algorithms require an exact projection onto the feasible region in every iteration, which can be efficient only for problems that admit a fast projection. In our method we use inexact adaptive projections requiring to move within a … Read more
We first show that a continuous function “f” is nonnegative on a closed set K if and only if (countably many) moment matrices of some signed measure dnu = fdmu are all positive semidefinite (if K is compact mu is an arbitrary finite Borel measure with support exactly K). In particular, we obtain a convergent … Read more
For a given linear matrix function $A_1 – B_1XB^*_1$, where $X$ is a variable Hermitian matrix, this paper derives a group of closed-form formulas for calculating the global maximum and minimum ranks and inertias of the matrix function subject to a pair of consistent matrix equations $B_2XB^*_2 = A_2$ and $B_3XB_3^* = A_3$. As applications, … Read more
We consider the inverse optimization problem associated with the polynomial program $f^*=\min \{f(x):x\inK\}$ and a given current feasible solution $y\in K$. We provide a numerical scheme to compute an inverse optimal solution. That is, we compute a polynomial $\tilde{f}$ (which may be of same degree as $f$ if desired) with the following properties: (a) $y$ … Read more