Results 1  10
of
50
An Integer Programming Approach for Linear Programs with Probabilistic Constraints
, 2008
"... Linear programs with joint probabilistic constraints (PCLP) are difficult to solve because the feasible region is not convex. We consider a special case of PCLP in which only the righthand side is random and this random vector has a finite distribution. We give a mixedinteger programming formulati ..."
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Cited by 43 (8 self)
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Linear programs with joint probabilistic constraints (PCLP) are difficult to solve because the feasible region is not convex. We consider a special case of PCLP in which only the righthand side is random and this random vector has a finite distribution. We give a mixedinteger programming formulation for this special case and study the relaxation corresponding to a single row of the probabilistic constraint. We obtain two strengthened formulations. As a byproduct of this analysis, we obtain new results for the previously studied mixing set, subject to an additional knapsack inequality. We present computational results which indicate that by using our strengthened formulations, instances that are considerably larger than have been considered before can be solved to optimality.
Sample Average Approximation Method for Chance Constrained Programming: Theory and Applications
 J OPTIM THEORY APPL (2009) 142: 399–416
, 2009
"... We study sample approximations of chance constrained problems. In particular, we consider the sample average approximation (SAA) approach and discuss the convergence properties of the resulting problem. We discuss how one can use the SAA method to obtain good candidate solutions for chance constrai ..."
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Cited by 31 (1 self)
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We study sample approximations of chance constrained problems. In particular, we consider the sample average approximation (SAA) approach and discuss the convergence properties of the resulting problem. We discuss how one can use the SAA method to obtain good candidate solutions for chance constrained problems. Numerical experiments are performed to correctly tune the parameters involved in the SAA. In addition, we present a method for constructing statistical lower bounds for the optimal value of the considered problem and discuss how one should tune the underlying parameters. We apply the SAA to two chance constrained problems. The first is a linear portfolio selection problem with returns following a multivariate lognormal distribution. The second is a joint chance constrained version of a simple blending problem.
From CVaR to Uncertainty Set: Implications in Joint Chance Constrained Optimization
, 2009
"... We review and develop different tractable approximations to individual chance constrained problems in robust optimization on a varieties of uncertainty sets and show their interesting connections with bounds on the conditionalvalueatrisk (CVaR) measure. We extend the idea to joint chance constrai ..."
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Cited by 14 (1 self)
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We review and develop different tractable approximations to individual chance constrained problems in robust optimization on a varieties of uncertainty sets and show their interesting connections with bounds on the conditionalvalueatrisk (CVaR) measure. We extend the idea to joint chance constrained problems and provide a new formulation that improves upon the standard approach. Our approach builds on a classical worst case bound for order statistics problems and is applicable even if the constraints are correlated. We provide an application of the model on a network resource allocation problem with uncertain demand.
IIS BranchandCut for Joint ChanceConstrained Stochastic Programs and Application to Optimal Vaccine Allocation
 European Journal of Operational Research
"... We present a new method for solving stochastic programs with joint chance constraints with random technology matrices and discretely distributed random data. The problem can be reformulated as a largescale mixed 01 integer program. We derive a new class of optimality cuts called IIS cuts and apply ..."
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Cited by 13 (0 self)
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We present a new method for solving stochastic programs with joint chance constraints with random technology matrices and discretely distributed random data. The problem can be reformulated as a largescale mixed 01 integer program. We derive a new class of optimality cuts called IIS cuts and apply them to our problem. The cuts are based on irreducibly infeasible subsets (IIS) of an LP defined by requiring that all scenarios be satisfied. We propose an efficient method for improving the upper bound of the problem when no cut can be found. We derive and implement a branchandcut algorithm based on IIS cuts, and refer to this algorithm as the IIS BranchandCut algorithm. We report on computational results with several test instances from optimal vaccine allocation and a production planning problem from the literature. The computational results are very promising as the IIS branchandcut algorithm gives significantly better results than a stateoftheart commercial solver.
On the sample complexity of randomized approaches to the analysis and design under uncertainty.
 In the Proceedings of the American Control Conference (ACC),
, 2010
"... AbstractIn this paper, we study the sample complexity of probabilistic methods for control of uncertain systems. In particular, we show the role of the binomial distribution for some problems involving analysis and design of robust controllers with finite families. We also address the particular c ..."
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Cited by 10 (1 self)
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AbstractIn this paper, we study the sample complexity of probabilistic methods for control of uncertain systems. In particular, we show the role of the binomial distribution for some problems involving analysis and design of robust controllers with finite families. We also address the particular case in which the design problem can be formulated as an uncertain convex optimization problem. The results of the paper provide simple explicit sample bounds to guarantee that the obtained solutions meet some prespecified probabilistic specifications.
New formulations for optimization under stochastic dominance constraints
 SIAM Journal on Optimization
"... Stochastic dominance constraints allow a decisionmaker to manage risk in an optimization setting by requiring their decision to yield a random outcome which stochastically dominates a reference random outcome. We present new integer and linear programming formulations for optimization under first ..."
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Cited by 10 (1 self)
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Stochastic dominance constraints allow a decisionmaker to manage risk in an optimization setting by requiring their decision to yield a random outcome which stochastically dominates a reference random outcome. We present new integer and linear programming formulations for optimization under first and secondorder stochastic dominance constraints, respectively. These formulations are more compact than existing formulations, and relaxing integrality in the firstorder formulation yields a secondorder formulation, demonstrating the tightness of this formulation. We also present a specialized branching strategy and heuristics which can be used with the new firstorder formulation. Computational tests illustrate the potential benefits of the new formulations.
A BranchandCut Decomposition Algorithm for Solving ChanceConstrained Mathematical Programs with Finite Support
, 2013
"... We present a new approach for exactly solving chanceconstrained mathematical programs having discrete distributions with finite support and random polyhedral constraints. Such problems have been notoriously difficult to solve due to nonconvexity of the feasible region, and most available methods ar ..."
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Cited by 9 (2 self)
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We present a new approach for exactly solving chanceconstrained mathematical programs having discrete distributions with finite support and random polyhedral constraints. Such problems have been notoriously difficult to solve due to nonconvexity of the feasible region, and most available methods are only able to find provably good solutions in certain very special cases. Our approach uses both decomposition, to enable processing subproblems corresponding to one possible outcome at a time, and integer programming techniques, to combine the results of these subproblems to yield strong valid inequalities. Computational results on a chanceconstrained formulation of a resource planning problem inspired by a call center staffing application indicate the approach works significantly better than both an existing mixedinteger programming formulation and a simple decomposition approach that does not use strong valid inequalities. We also demonstrate how the approach can be used to efficiently solve for a sequence of risk levels, as would be done when solving for the efficient frontier of risk and cost.
Solving chanceconstrained stochastic programs via sampling and integer programming
 2008 TUTORIALS IN OPERATIONS RESEARCH: STATEOFTHEART DECISIONMAKING TOOLS IN THE INFORMATIONINTENSIVE AGE
, 2008
"... Various applications in reliability and risk management give rise to optimization problems with constraints involving random parameters, which are required to be satisfied with a prespecified probability threshold. There are two main difficulties with such chanceconstrained problems. First, checki ..."
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Cited by 9 (0 self)
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Various applications in reliability and risk management give rise to optimization problems with constraints involving random parameters, which are required to be satisfied with a prespecified probability threshold. There are two main difficulties with such chanceconstrained problems. First, checking feasibility of a given candidate solution exactly is, in general, impossible since this requires evaluating quantiles of random functions. Second, the feasible region induced by chance constraints is, in general, nonconvex leading to severe optimization challenges. In this tutorial we discuss an approach based on solving approximating problems using Monte Carlo samples of the random data. This scheme can be used to yield both feasible solutions and statistical optimality bounds with high confidence using modest sample sizes. The approximating problem is itself a chanceconstrained problem, albeit with a finite distribution of modest support, and is an NPhard combinatorial optimization problem. We adopt integer programming based methods for its solution. In particular, we discuss a family valid inequalities for a integer programming formulations for a special but large class of chanceconstraint problems that have demonstrated significant computational advantages.
Staffing Call Centers with Uncertain Demand Forecasts: A ChanceConstrained Optimization Approach
, 2010
"... We consider the problem of staffing call centers with multiple customer classes and agent types operating under qualityofservice (QoS) constraints and demand rate uncertainty. We introduce a formulation of the staffing problem that requires that the QoS constraints are met with high probability wi ..."
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Cited by 8 (1 self)
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We consider the problem of staffing call centers with multiple customer classes and agent types operating under qualityofservice (QoS) constraints and demand rate uncertainty. We introduce a formulation of the staffing problem that requires that the QoS constraints are met with high probability with respect to the uncertainty in the demand rate. We contrast this chanceconstrained formulation with the averageperformance constraints that have been used so far in the literature. We then propose a twostep solution for the staffing problem under chance constraints. In the first step, we introduce a random static planning problem (RSPP) and discuss how it can be solved using two different methods. The RSPP provides us with a firstorder (or fluid) approximation for the true optimal staffing levels and a staffing frontier. In the second step, we solve a finite number of staffing problems with known arrival rates—the arrival rates on the optimal staffing frontier. Hence, our formulation and solution approach has the important property that it translates the problem with uncertain demand rates to one with known arrival rates. The output of our procedure is a solution that is feasible with respect to the chance constraint and nearly optimal for large call centers.
Convex relaxations of chance constrained optimization problems
, 2011
"... In this paper we develop convex relaxations of chance constrained optimization problems in order to obtain lower bounds on the optimal value. Unlike existing statistical lower bounding techniques, our approach is designed to provide deterministic lower bounds. We show that a version of the proposed ..."
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Cited by 3 (0 self)
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In this paper we develop convex relaxations of chance constrained optimization problems in order to obtain lower bounds on the optimal value. Unlike existing statistical lower bounding techniques, our approach is designed to provide deterministic lower bounds. We show that a version of the proposed scheme leads to a tractable convex relaxation when the chance constraint function is affine with respect to the underlying random vector and the random vector has independent components. We also propose an iterative improvement scheme for refining the bounds. 1