Solving Chance-Constrained Stochastic Programs via Sampling and Integer Programming

Ahmed, Shabbir; Shapiro, Alexander

doi:10.1287/educ.1080.0048

Cited by 108 publications

(124 citation statements)

References 31 publications

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“…More recently, to solve chance constrained stochastic programs, scenario approximation approaches (see, e.g., [7,28,31], and [33]) are studied and demonstrated to be computationally tractable and can guarantee to obtain a solution satisfying a chance constraint with high probability. In particular, integer programming (IP) techniques are successfully applied to exactly solve chance constrained stochastic problems (see, e.g., [3,22,24,29], and [27]). Interested readers are also referred to classical textbooks in stochastic programming (see, e.g., [6,21,38], and [42]).…”

Section: Motivation and Literature Reviewmentioning

confidence: 99%

“…In this paper, we focus on a class of DCCs under the density-based confidence set D φ in (3). As compared to the moment-based confidence sets where the first two moments are typically used (see, e.g., [8,44,48,49], and [2]), the density-based confidence set D φ can more accurately depict the profile of the ambiguous probability distribution and so potentially provide a less conservative DCC.…”

Section: Model Settings and Confidence Setsmentioning

confidence: 99%

“…This is because the perturbed risk levels α are lower than their counterpart α in the classical models. As a result, when we apply the sample approximation approaches (see, e.g., [3,22,24,29], and [27]), the mixed-integer program approximation of the DCC model has a smaller number of combinations to consider than that of the classical model. Hence, the DCC model has a smaller branch-and-bound tree which leads to faster convergence.…”

Section: Numerical Experimentsmentioning

confidence: 99%

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Data-driven chance constrained stochastic program

2015

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In this paper, we study data-driven chance constrained stochastic programs, or more specifically, stochastic programs with distributionally robust chance constraints (DCCs) in a data-driven setting to provide robust solutions for the classical chance constrained stochastic program facing ambiguous probability distributions of random parameters. We consider a family of density-based confidence sets based on a general φ-divergence measure, and formulate DCC from the perspective of robust feasibility by allowing the ambiguous distribution to run adversely within its confidence set. We derive an equivalent reformulation for DCC and show that it is equivalent to a classical chance constraint with a perturbed risk level. We also show how to evaluate the perturbed risk level by using a bisection line search algorithm for general φ-divergence measures. In several special cases, our results can be strengthened such that we can derive closed-form expressions for the perturbed risk levels. In addition, we show that the conservatism of DCC vanishes as the size of historical data goes to infinity. Furthermore, we analyze the relationship between the conservatism of DCC An early version of this paper is available online at and the size of historical data, which can help indicate the value of data. Finally, we conduct extensive computational experiments to test the performance of the proposed DCC model and compare various φ-divergence measures based on a capacitated lotsizing problem with a quality-of-service requirement.

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Section: Motivation and Literature Reviewmentioning

confidence: 99%

Section: Model Settings and Confidence Setsmentioning

confidence: 99%

Section: Numerical Experimentsmentioning

confidence: 99%

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Data-driven chance constrained stochastic program

2015

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“…The formulation ( The VaR can also be modeled as a chance-constrained program [1,37]. The corresponding formulation (21) can be related to (22) through the change of variables p i (1−s i ) = (1−ζ )λ i .…”

Section: Quantile Minimizationmentioning

confidence: 99%

On linear programs with linear complementarity constraints

Mitchell

Pang

et al. 2011

J Glob Optim

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The paper is a manifestation of the fundamental importance of the linear program with linear complementarity constraints (LPCC) in disjunctive and hierarchical programming as well as in some novel paradigms of mathematical programming. In addition to providing a unified framework for bilevel and inverse linear optimization, nonconvex piecewise linear programming, indefinite quadratic programs, quantile minimization, and 0 minimization, the LPCC provides a gateway to a mathematical program with equilibrium constraints, which itself is an important class of constrained optimization problems that has broad applications.It is our great pleasure to dedicate this work to Professor Richard W. Cottle on the occasion of his 75th birthday in 2009. Professor Cottle is the father of the linear complementarity problem (LCP) [16]. The linear program with linear complementarity constraints (LPCC) treated in this paper is a natural extension of the LCP; our hope is that the LPCC will one day become as fundamental as the LCP, thereby continuing Professor Cottle's legacy, bringing it to new heights, and extending its breadth. 123 30 J Glob Optim (2012) 53:29-51We describe several approaches for the global resolution of the LPCC, including a logical Benders approach that can be applied to problems that may be infeasible or unbounded.

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“…For continuously distributed random variables, the methods based on supporting hyperplanes and reduced gradients are available. In a case where the underlying distribution is continuous or discrete with many realizations, the sample-approximation techniques and the mixed-integer programming reformulation can help us to solve the problem approximately, see [1,16,17]. With an increased sample size we can even approximate the true solution of the chance-constrained problem.…”

Section: Introductionmentioning

confidence: 99%

Stochastic programming problems with generalized integrated chance constraints

Branda

2012

Optimization

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If the constraints in an optimization problem are dependent on a random parameter, we would like to ensure that they are fulfilled with a high level of reliability. The most natural way is to employ chance constraints. However, the resulting problem is very hard to solve. We propose an alternative formulation of stochastic programs using penalty functions. The expectations of penalties can be left as constraints leading to generalized integrated chance constraints, or incorporated into the objective as a penalty term. We show that the penalty problems are asymptotically equivalent under quite mild conditions. We discuss applications of sample-approximation techniques to the problems with generalized integrated chance constraints and propose rates of convergence for the set of feasible solutions. We will direct our attention to the case when the set of feasible solutions is finite, which can appear in integer programming. The results are then extended to the bounded sets with continuous variables. Additional binary variables are necessary to solve sample-approximated chance-constrained problems, leading to a large mixed-integer non-linear program. On the other hand, the problems with penalties can be solved without adding binary variables; just continuous variables are necessary to model the penalties. The introduced approaches are applied to the blending problem leading to comparably reliable solutions.

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Solving Chance-Constrained Stochastic Programs via Sampling and Integer Programming

Cited by 108 publications

References 31 publications

Data-driven chance constrained stochastic program

Data-driven chance constrained stochastic program

On linear programs with linear complementarity constraints

Stochastic programming problems with generalized integrated chance constraints

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