We present polynomial-time interior-point algorithms for solving the Fisher and Arrow-Debreu competitive market equilibrium problems with linear utilities and $n$ players. Both of them have the arithmetic operation complexity bound of $O(n^4\log(1/\epsilon))$ for computing an $\epsilon$-equilibrium solution. If the problem data are rational numbers and their bit-length is $L$, then the bound to generate an … Read more
In this paper we discuss computational complexity and risk averse approaches to two and multistage stochastic programming problems. We argue that two stage (say linear) stochastic programming problems can be solved with a reasonable accuracy by Monte Carlo sampling techniques while there are indications that complexity of multistage programs grows fast with increase of the … Read more
We study a problem faced by a major beverage producer. The company produces and distributes several brands to various customers from its regional distributors. For some of these brands, most customers do not have enough demand to justify full pallet shipments. Therefore, the company decided to design a number of mixed or “rainbow” pallets so … Read more
In this paper we derive estimates of the sample sizes required to solve a multistage stochastic programming problem with a given accuracy by the (conditional sampling) sample average approximation method. The presented analysis is self contained and is based on a, relatively elementary, one dimensional Cramer’s Large Deviations Theorem. CitationWorking paper, Georgia Institute of Technology, … Read more
Given a convex body S and a point x \in S, let sym(x,S) denote the symmetry value of x in S: sym(x,S):= max{t : x + t(x – y) \in S for every y \in S}, which essentially measures how symmetric S is about the point x, and define sym(S):=\max{sym(x,S) : x \in S }. … Read more
Interior-point methods (IPMs) for semidefinite optimization (SDO) have been studied intensively, due to their polynomial complexity and practical efficiency. Recently, J.Peng et al. introduced so-called self-regular kernel (and barrier) functions and designed primal-dual interior-point algorithms based on self-regular proximity for linear optimization (LO) problems. They have also extended the approach for LO to SDO. In … Read more
Traditional stochastic programming is risk neutral in the sense that it is concerned with the optimization of an expectation criteria. A common approach to addressing risk in decision making problems is to consider a weighted mean-risk criterion, where some dispersion statistic is used as a measure of risk. We investigate the computational suitability of various … Read more
In this paper we study the distribution tails and the moments of a condition number which arises in the study of homogeneous systems of linear inequalities. We consider the case where this system is defined by a Gaussian random matrix and characterise the exact decay rates of the distribution tails, improve the existing moment estimates, … Read more
We study the limiting properties of the affine-scaling directions for linear programming problems. The worst-case angle between the affine-scaling directions and the objective function vector provides an interesting measure that has been very helpful in convergence analyses and in understanding the behaviour of various interior-point algorithms. We establish new relations between this measure and some … Read more
Virtual private networks (VPN) are often used to distribute live content, such as video or audio streams, from a single source to a large number of destinations. Streaming caches or splitters are deployed in these multicast networks to allow content distribution without overloading the network. In this paper, we consider two combinatorial optimization problems that … Read more