Linear programming is arguably one of the most basic forms of optimization. Its theory and algorithms can not only be applied to linear optimization problems but also to relaxations of nonlinear problems and branch-and-bound methods for mixed-integer and global optimization problems. Recent research shows that against intuition bad condition numbers frequently occur in linear programming. … Read more
The problem of approximately factoring a real or complex multivariate polynomial $f$ seeks minimal perturbations $\Delta f$ to the coefficients of the input polynomial $f$ so that the deformed polynomial $f + \Delta f$ has the desired factorization properties. Efficient algorithms exist that compute the nearest real or complex polynomials that has non-trivial factors. (see … Read more
In this paper, we analyze the limiting behavior of the weighted least squares problem $\min_{x\in\Re^n}\sum_{i=1}^p\|D_i(A_ix-b_i)\|^2$, where each $D_i$ is a positive definite diagonal matrix. We consider the situation where the magnitude of the weights are drastically different block-wisely so that $\max(D_1)\geq\min(D_1) \gg \max(D_2) \geq \min(D_2) \gg \max(D_3) \geq \ldots \gg \max(D_{p-1}) \geq \min(D_{p-1}) \gg \max(D_p)$. … Read more
The convex cone of $n \times n$ completely positive (CPP) matrices and its dual cone of copositive matrices arise in several areas of applied mathematics, including optimization. Every CPP matrix is doubly nonnegative (DNN), i.e., positive semidefinite and component-wise nonnegative, and it is known that, for $n \le 4$ only, every DNN matrix is CPP. … Read more
In the context of simulating the frictional contact dynamics of large systems of rigid bodies, this paper reviews a novel method for solving large cone complementarity problems by means of a fixed-point iteration algorithm. The method is an extension of the Gauss-Seidel and Gauss-Jacobimethods with overrelaxation for symmetric convex linear complementarity problems. Convergent under fairly … Read more
Quadratic assignment problems (QAPs) with a Hamming distance matrix of a hypercube or a Manhattan distance matrix of rectangular grids arise frequently from communications and facility locations and are known to be among the hardest discrete optimization problems. In this paper we consider the issue of how to obtain lower bounds for those two classes … Read more
In this paper, we propose a semidefinite optimization (SDP) based model for the class of minimax two-stage stochastic linear optimization problems with risk aversion. The distribution of the second-stage random variables is assumed to be chosen from a set of multivariate distributions with known mean and second moment matrix. For the minimax stochastic problem with … Read more
The cone of completely positive matrices $C^*$ is the convex hull of all symmetric rank-1-matrices $xx^T$ with nonnegative entries. Determining whether a given matrix $B$ is completely positive is an $\cal NP$-complete problem. We examine a simple algorithm which — for a given input $B$ — either determines a certificate proving that $B\in C^*$ or … Read more
The Lovász theta number of a graph G can be viewed as a semidefinite programming relaxation of the stability number of G. It has recently been shown that a copositive strengthening of this semidefinite program in fact equals the stability number of G. We introduce a related strengthening of the Lovász theta number toward the … Read more
We demonstrate the use of linear programming techniques in the analysis of mechanism design problems. We use these techniques to analyze the extent to which a first-price auction is robust to collusion when, contrary to some prior literature on collusion at first-price auctions, the cartel cannot prevent its members from bidding at the auction. In … Read more