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 Carlo sampling techniques, namely, the Stochastic Approximation (SA) and the Sample Average Approximation (SAA) methods. Both approaches, the SA and SAA methods, have a long history. Current opinion is that the SAA method can efficiently use a specific (say linear) structure of the considered problem, while the SA approach is a crude subgradient method which often performs poorly in practice. We intend to demonstrate that a properly modified SA approach can be competitive and even significantly outperform the SAA method for a certain class of convex stochastic problems. We extend the analysis to the case of convex-concave stochastic saddle point problems, and present (in our opinion highly encouraging) results of numerical experiments.

Abstract Stochastic optimization problems provide a means to model uncertainty in the input data where theuncertainty is modeled by a probability distribution over the possible realizations of the actual data. We

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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 Cramér’s Large Deviations Theorem.

A large class of stochastic optimization problems can be modeled as minimizing an objective function f that depends on a choice of a vector x ∈ X, as well as on a random external parameter ω ∈ Ω given by a probability distribution π. The value of the objective function is a random variable and often the goal is to find an x ∈ X to minimize the expected cost Eω[fω(x)]. Each ω is referred to as a scenario. We consider the case when Ω is large or infinite and we are allowed to sample from π in a black-box fashion. A common method, known as the SAA method (sample average approximation), is to pick sufficiently many independent samples from π and use them to approximate π and correspondingly Eω[fω(x)]. This is one of several scenario reduction methods used in practice. There has been substantial recent interest in two-stage stochastic versions of combinatorial optimization problems which can be modeled by the framework described above. In particular, we are interested in the

Stochastic optimization problems attempt to model uncertainty in the data by assuming that the input is specified by a probability distribution. We consider the well-studied paradigm of 2-stage models with recourse: first, given only distributional information about (some of) the data one commits on initial actions, and then once the actual data is realized (according to the distribution), further (recourse) actions can be taken. We show that for a broad class of 2-stage linear models with recourse, one can, for any ɛ> 0, in time polynomial in 1 ɛ and the size of the input, compute a solution of value within a factor (1 + ɛ) of the optimum, in spite of the fact that exponentially many second-stage scenarios may occur. In conjunction with a suitable rounding scheme, this yields the first approximation algorithms for 2-stage stochastic integer optimization problems where the underlying random data is given by a “black box ” and no restrictions are placed on the costs in the two stages. Our rounding approach for stochastic integer programs shows that an approximation algorithm for a deterministic analogue yields, with a small constant-factor loss, provably near-optimal solutions for the stochastic generalization. Among the range of applications we consider are stochastic versions of the multicommodity flow, set cover, vertex cover, and facility location problems.

We consider an optimization problem of minimization of a linear function subject to the chance constraint Prob{G(x, ξ) ∈ C} ≥ 1 − ε, where C is a convex set, G(x, ξ) is bi-affine mapping and ξ is a vector of random perturbations with known distribution. When C is multi-dimensional and ε is small, like 10 −6 or 10 −10, this problem is, generically, a problem of minimizing under a nonconvex and difficult to compute constraint and as such is computationally intractable. We investigate the potential of conceptually simple scenario approximation of the chance constraint. That is, approximation of the form G(x, η j) ∈ C, j = 1,..., N, where {η j} N j=1 is a sample drawn from a properly chosen trial distribution. The emphasis is on the situation where the solution to the approximation should, with probability at least 1 − δ, be feasible for the problem of interest, while the sample size N should be polynomial in the size of this problem and in ln(1/ε), ln(1/δ).

Clustering is a ubiquitous problem that arises in many applications in different fields such as data mining, image processing, machine learning, and bioinformatics. Clustering problems have been extensively studied as optimization problems with various objective functions in the Operations Research and Computer Science literature. We focus on a class of objective functions more commonly referred to as facility location problems. These problems arise in a wide range of applications such as, plant or warehouse location problems, cache placement problems, and network design problems where the costs obey economies of scale. In the simplest of these problems, the uncapacitated facility location (UFL) problem, we want to open facilities at some subset of a given set of locations and assign each client in a given set D to an open facility so as to minimize the sum of the facility opening costs and client assignment costs. This is a very well-studied problem; however it fails to address many of the requirements of real applications. In this thesis we consider various problems that build upon UFL and capture additional issues that arise in applications such as, uncertainties in the data, clients with different service needs, and facilities with interconnectivity requirements. By focusing initially on facility location problems in these new models, we develop new algorithmic techniques that will find application in a wide range of settings. We consider a widely used paradigm in stochastic programming to model settings where the underlying data, for example, the locations or demands of the clients, may be uncertain: the 2-stage with recourse model that involves making some initial decisions, observing additional information, and then augmenting the initial decisions, if necessary, by taking recourse actions. We present a randomized polynomial time

Abstract. Stochastic optimization is a leading approach to model optimization problems in which there is uncertainty in the input data, whether from measurement noise or an inability to know the future. In this survey, we outline some recent progress in the design of polynomialtime algorithms with performance guarantees on the quality of the solutions found for an important class of stochastic programming problems — 2-stage problems with recourse. In particular, we show that for a number of concrete problems, algorithmic approaches that have been applied for their deterministic analogues are also effective in this more challenging domain. More specifically, this work highlights the role of tools from linear programming, rounding techniques, primal-dual algorithms, and the role of randomization more generally. 1

More formally, given a probability distribution on scenarios A and a vector x describing the first stagedecisions, the cost incurred is given by