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52
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 ..."
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Cited by 267 (18 self)
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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.
A Robust Optimization Perspective Of Stochastic Programming
, 2005
"... In this paper, we introduce an approach for constructing uncertainty sets for robust optimization using new deviation measures for bounded random variables known as the forward and backward deviations. These deviation measures capture distributional asymmetry and lead to better approximations of c ..."
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Cited by 48 (12 self)
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In this paper, we introduce an approach for constructing uncertainty sets for robust optimization using new deviation measures for bounded random variables known as the forward and backward deviations. These deviation measures capture distributional asymmetry and lead to better approximations of chance constraints. We also propose a tractable robust optimization approach for obtaining robust solutions to a class of stochastic linear optimization problems where the risk of infeasibility can be tolerated as a tradeoff to improve upon the objective value. An attractive feature of the framework is the computational scalability to multiperiod models. We show an application of the framework for solving a project management problem with uncertain activity completion time.
STABILITY OF MULTISTAGE STOCHASTIC PROGRAMS
 SIAM J. OPTIM.
, 2006
"... Quantitative stability of linear multistage stochastic programs is studied. It is shown that the infima of such programs behave (locally) Lipschitz continuous with respect to the sum of an L_rdistance and of a distance measure for the filtrations of the original and approximate stochastic (input) ..."
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Cited by 46 (11 self)
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Quantitative stability of linear multistage stochastic programs is studied. It is shown that the infima of such programs behave (locally) Lipschitz continuous with respect to the sum of an L_rdistance and of a distance measure for the filtrations of the original and approximate stochastic (input) processes. Various issues of the result are discussed and an illustrative example is given. Consequences for the reduction of scenario trees are also discussed.
Analysis of Stochastic Dual Dynamic Programming Method
"... Abstract. In this paper we discuss statistical properties and rates of convergence of the Stochastic Dual Dynamic Programming (SDDP) method applied to multistage linear stochastic programming problems. We assume that the underline data process is stagewise independent and consider the framework wher ..."
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Cited by 36 (2 self)
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Abstract. In this paper we discuss statistical properties and rates of convergence of the Stochastic Dual Dynamic Programming (SDDP) method applied to multistage linear stochastic programming problems. We assume that the underline data process is stagewise independent and consider the framework where at first a random sample from the original (true) distribution is generated and consequently the SDDP algorithm is applied to the constructed Sample Average Approximation (SAA) problem.
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 ..."
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Cited by 33 (6 self)
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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.
A Linear DecisionBased Approximation Approach to Stochastic Programming
, 2008
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Distributionally robust optimization and its tractable approximations
 Operations Research
"... In this paper, we focus on a linear optimization problem with uncertainties, having expectations in the objective and in the set of constraints. We present a modular framework to obtain an approximate solution to the problem that is distributionally robust, and more flexible than the standard techni ..."
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Cited by 25 (4 self)
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In this paper, we focus on a linear optimization problem with uncertainties, having expectations in the objective and in the set of constraints. We present a modular framework to obtain an approximate solution to the problem that is distributionally robust, and more flexible than the standard technique of using linear rules. Our framework begins by firstly affinelyextending the set of primitive uncertainties to generate new linear decision rules of larger dimensions, and are therefore more flexible. Next, we develop new piecewiselinear decision rules which allow a more flexible reformulation of the original problem. The reformulated problem will generally contain terms with expectations on the positive parts of the recourse variables. Finally, we convert the uncertain linear program into a deterministic convex program by constructing distributionally robust bounds on these expectations. These bounds are constructed by first using different pieces of information on the distribution of the underlying uncertainties to develop separate bounds, and next integrating them into a combined bound that is better than each of the individual bounds.
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 ..."
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Cited by 24 (1 self)
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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
Validation Analysis of Mirror Descent Stochastic Approximation Method
 MATHEMATICAL PROGRAMMING
"... The main goal of this paper is to develop accuracy estimates for stochastic programming problems by employing stochastic approximation (SA) type algorithms. To this end we show that while running a Mirror Descent Stochastic Approximation procedure one can compute, with a small additional effort, low ..."
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Cited by 18 (5 self)
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The main goal of this paper is to develop accuracy estimates for stochastic programming problems by employing stochastic approximation (SA) type algorithms. To this end we show that while running a Mirror Descent Stochastic Approximation procedure one can compute, with a small additional effort, lower and upper statistical bounds for the optimal objective value. We demonstrate that for a certain class of convex stochastic programs these bounds are comparable in quality with similar bounds computed by the sample average approximation method, while their computational cost is considerably smaller.