In this paper we consider optimization problems where the objective function is given in a form of the expectation. A basic difficulty of solving such stochastic optimization problems is that the involved multidimensional integrals (expectations) cannot be computed with high accuracy. The aim of this paper is to compare two computational approaches based on Monte … Read more
In this paper we analyze several new methods for solving optimization problems with the objective function formed as a sum of two convex terms: one is smooth and given by a black-box oracle, and another is general but simple and its structure is known. Despite to the bad properties of the sum, such problems, both … Read more
In this paper we analyze several new methods for solving optimization problems with the objective function formed as a sum of two convex terms: one is smooth and given by a black-box oracle, and another is general but simple and its structure is known. Despite to the bad properties of the sum, such problems, both … Read more
We study the set $S = \{(x, y) \in \Re_{+} \times Z^{n}: x + B_{j} y_{j} \geq b_{j}, j = 1, \ldots, n\}$, where $B_{j}, b_{j} \in \Re_{+} – \{0\}$, $j = 1, \ldots, n$, and $B_{1} | \cdots | B_{n}$. The set $S$ generalizes the mixed-integer rounding (MIR) set of Nemhauser and Wolsey and … Read more
We consider the computational complexity of optimizing various classes of continuous functions over a simplex, hypercube or sphere. These relatively simple optimization problems have many applications. We review known approximation results as well as negative (inapproximability) results from the recent literature. Citation CentER Discussion paper 2006-85 Tilburg University THe Netherlands Article Download View The complexity … Read more
We consider a class of two-stage linear stochastic programming problems, introduced by Shmoys and Swamy (2004), motivated by a relaxation of a stochastic set cover problem. We show that the sample size required to solve this problem by the sample average approximation (SAA) method with a relative accuracy $\kappa>0$ and confidence $1-\alpha$ is polynomial in … Read more
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. Citation Working paper, Georgia Institute of … Read more