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
We consider a Newton-CG augmented Lagrangian method for solving semidefinite programming (SDP) problems from the perspective of approximate semismooth Newton methods. In order to analyze the rate of convergence of our proposed method, we characterize the Lipschitz continuity of the corresponding solution mapping at the origin. For the inner problems, we show that the positive … Read more
Expected utility models in portfolio optimization is based on the assumption of complete knowledge of the distribution of random returns. In this paper, we relax this assumption to the knowledge of only the mean, covariance and support information. No additional assumption on the type of distribution such as normality is made. The investor’s utility is … Read more
In this paper we present the algorithmic framework and practical aspects of implementing a parallel version of a primal-dual semidefinite programming solver on a distributed memory computer cluster. Our implementation is based on the CSDP solver and uses a message passing interface (MPI), and the ScaLAPACK library. A new feature is implemented to deal with … Read more
In \cite{aYOSHISE06}, the author proposed a homogeneous model for standard monotone nonlinear complementarity problems over symmetric cones and show that the following properties hold: (a) There is a path that is bounded and has a trivial starting point without any regularity assumption concerning the existence of feasible or strictly feasible solutions. (b) Any accumulation point … Read more
Mehrotra and Ozevin computationally found that a weighted primal barrier decomposition algorithm significantly outperforms the barrier decomposition proposed and analyzed in Zhao, and Mehrotra and Ozevin. This paper provides a theoretical foundation for the weighted barrier decomposition algorithm (WBDA). Although the worst case analysis of the WBDA achieves a first-stage iteration complexity bound that is … Read more
Given a 0-1 integer programming problem, several authors have introduced sequential relaxation techniques — based on linear and/or semidefinite programming — that generate the convex hull of integer points in at most $n$ steps. In this paper, we introduce a sequential relaxation technique, which is based on $p$-order cone programming ($1 \le p \le \infty$). … Read more