Let P be a pointed, polyhedral cone in R_n. In this paper, we study the cone C = cone{xx^T: x \in P} of quadratic forms. Understanding the structure of C is important for globally solving NP-hard quadratic programs over P. We establish key characteristics of C and construct a separation algorithm for C provided one … Read more
The main topic addressed in this paper is trace-optimization of polynomials in noncommuting (nc) variables: given an nc polynomial f, what is the smallest trace f(A) can attain for a tuple of matrices A? A relaxation using semidefinite programming (SDP) based on sums of hermitian squares and commutators is proposed. While this relaxation is not … Read more
We consider the problem of minimizing a polynomial on the hypercube [0,1]^n and derive new error bounds for the hierarchy of semidefinite programming approximations to this problem corresponding to the Positivstellensatz of Schmuedgen (1991). The main tool we employ is Bernstein approximations of polynomials, which also gives constructive proofs and degree bounds for positivity certificates … Read more
Run time distributions or time-to-target plots are very useful tools to characterize the running times of stochastic algorithms for combinatorial optimization. We further explore run time distributions and describe a new tool to compare two algorithms based on stochastic local search. For the case where the running times of both algorithms fit exponential distributions, we … Read more
A new continuous location model is presented and embedded in the literature on robustness in facility location. The multimodality of the model is investigated, and a branch and bound method based on dc optimization is described. Numerical experience is reported, showing that the developed method allows one to solve in a few seconds problems with … Read more
The cone of Completely Positive (CP) matrices can be used to exactly formulate a variety of NP-Hard optimization problems. A tractable relaxation for CP matrices is provided by the cone of Doubly Nonnegative (DNN) matrices; that is, matrices that are both positive semidefinite and componentwise nonnegative. A natural problem in the optimization setting is then … Read more
This paper considers a portfolio selection problem in which portfolios with minimum number of active assets are sought. This problem is motivated by the need of inducing sparsity on the selected portfolio to reduce transaction costs, complexity of portfolio management, and instability of the solution. The resulting problem is a difficult combinatorial problem. We propose … Read more
Given a compact parameter set $Y\subset R^p$, we consider polynomial optimization problems $(P_\y$) on $R^n$ whose description depends on the parameter $y\in Y$. We assume that one can compute all moments of some probability measure $\varphi$ on $Y$, absolutely continuous with respect to the Lebesgue measure (e.g. $Y$ is a box or a simplex and … Read more
This paper is concerned with a class of ellipsoidal sets (ellipsoids and elliptic cylinders) in the m-dimensional Euclidean space which are determined by a freely chosen positive semidefinite matrix. All ellipsoidal sets in this class are similar to each other through a parallel transformation and a scaling around their centers by a constant factor. Based … Read more
Mathematica is an advanced software system that enables symbolic computing, numerics, program code development, model visualization and professional documentation in a unified framework. Our MathOptimizer software package serves to solve global and local optimization models developed using Mathematica. We introduce MathOptimizer’s key features and discuss its usage options that support a range of operational modes. … Read more