Results 1  10
of
65
Stochastic Approximation Approach to Stochastic Programming
"... 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 th ..."
Abstract

Cited by 266 (18 self)
 Add to MetaCart
(Show Context)
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 convexconcave stochastic saddle point problems, and present (in our opinion highly encouraging) results of numerical experiments.
The sample average approximation method for stochastic discrete optimization
 SIAM Journal on Optimization
, 2001
"... Abstract. In this paper we study a Monte Carlo simulation based approach to stochastic discrete optimization problems. The basic idea of such methods is that a random sample is generated and consequently the expected value function is approximated by the corresponding sample average function. The ob ..."
Abstract

Cited by 207 (20 self)
 Add to MetaCart
Abstract. In this paper we study a Monte Carlo simulation based approach to stochastic discrete optimization problems. The basic idea of such methods is that a random sample is generated and consequently the expected value function is approximated by the corresponding sample average function. The obtained sample average optimization problem is solved, and the procedure is repeated several times until a stopping criterion is satisfied. We discuss convergence rates and stopping rules of this procedure and present a numerical example of the stochastic knapsack problem. Key words. Stochastic programming, discrete optimization, Monte Carlo sampling, Law of Large Numbers, Large Deviations theory, sample average approximation, stopping rules, stochastic knapsack problem AMS subject classifications. 90C10, 90C15
Optimization under uncertainty: Stateoftheart and opportunities
 Computers and Chemical Engineering
, 2004
"... A large number of problems in production planning and scheduling, location, transportation, finance, and engineering design require that decisions be made in the presence of uncertainty. Uncertainty, for instance, governs the prices of fuels, the availability of electricity, and the demand for chemi ..."
Abstract

Cited by 86 (0 self)
 Add to MetaCart
A large number of problems in production planning and scheduling, location, transportation, finance, and engineering design require that decisions be made in the presence of uncertainty. Uncertainty, for instance, governs the prices of fuels, the availability of electricity, and the demand for chemicals. A key difficulty in optimization under uncertainty is in dealing with an uncertainty space that is huge and frequently leads to very largescale optimization models. Decisionmaking under uncertainty is often further complicated by the presence of integer decision variables to model logical and other discrete decisions in a multiperiod or multistage setting. This paper reviews theory and methodology that have been developed to cope with the complexity of optimization problems under uncertainty. We discuss and contrast the classical recoursebased stochastic programming, robust stochastic programming, probabilistic (chanceconstraint) programming, fuzzy programming, and stochastic dynamic programming. The advantages and shortcomings of these models are reviewed and illustrated through examples. Applications and the stateoftheart in computations are also reviewed. Finally, we discuss several main areas for future development in this field. These include development of polynomialtime approximation schemes for multistage stochastic programs and the application of global optimization algorithms to twostage and chanceconstraint formulations.
A stochastic programming approach for supply chain network design under uncertainty
, 2003
"... ..."
On complexity of multistage stochastic programs
 Operations Research Letters
, 2006
"... 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 dimensio ..."
Abstract

Cited by 44 (6 self)
 Add to MetaCart
(Show Context)
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.
Stochastic mathematical programs with equilibrium constraints, modeling and . . .
 SCHOOL OF INDUSTRIAL AND SYSTEM ENGINEERING, GEORGIA INSTITUTE OF TECHNOLOGY
, 2005
"... In this paper, we discuss the sample average approximation (SAA) method applied to a class of stochastic mathematical programs with variational (equilibrium) constraints. To this end, we briefly investigate the structure of both – the lower level equilibrium solution and objective integrand. We sho ..."
Abstract

Cited by 34 (6 self)
 Add to MetaCart
(Show Context)
In this paper, we discuss the sample average approximation (SAA) method applied to a class of stochastic mathematical programs with variational (equilibrium) constraints. To this end, we briefly investigate the structure of both – the lower level equilibrium solution and objective integrand. We show almost sure convergence of optimal values, optimal solutions (both local and global) and generalized KarushKuhnTucker points of the SAA program to their true counterparts. We also study uniform exponential convergence of the sample average approximations, and as a consequence derive estimates of the sample size required to solve the true problem with a given accuracy. Finally we present some preliminary numerical test results.
An Approximation Scheme for Stochastic Linear Programming and its Application to Stochastic Integer Programs
"... Stochastic optimization problems attempt to model uncertainty in the data by assuming that the input is specified by a probability distribution. We consider the wellstudied paradigm of 2stage models with recourse: first, given only distributional information about (some of) the data one commits on ..."
Abstract

Cited by 33 (6 self)
 Add to MetaCart
Stochastic optimization problems attempt to model uncertainty in the data by assuming that the input is specified by a probability distribution. We consider the wellstudied paradigm of 2stage 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 2stage 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 secondstage scenarios may occur. In conjunction with a suitable rounding scheme, this yields the first approximation algorithms for 2stage 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 constantfactor loss, provably nearoptimal 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.
Approximation algorithms for 2stage stochastic optimization problems
 SIGACT News
, 2006
"... 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 p ..."
Abstract

Cited by 23 (1 self)
 Add to MetaCart
(Show Context)
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 — 2stage 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, primaldual algorithms, and the role of randomization more generally. 1