An integer programming approach for linear programs with probabilistic constraints
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 right-hand side is random and this random vector has a finite distribution. We give a mixed-integer programming formulati...
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| Published in | Mathematical programming Vol. 122; no. 2; pp. 247 - 272 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Berlin/Heidelberg
Springer-Verlag
01.04.2010
Springer Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
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| Abstract | 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 right-hand side is random and this random vector has a finite distribution. We give a mixed-integer 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. |
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| AbstractList | 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 right-hand side is random and this random vector has a finite distribution. We give a mixed-integer 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. 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 right-hand side is random and this random vector has a finite distribution. We give a mixed-integer 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. [PUBLICATION ABSTRACT] |
| Author | Nemhauser, George L. Luedtke, James Ahmed, Shabbir |
| Author_xml | – sequence: 1 givenname: James surname: Luedtke fullname: Luedtke, James email: jrluedt1@wisc.edu organization: Department of Industrial and Systems Engineering, University of Wisconsin-Madison – sequence: 2 givenname: Shabbir surname: Ahmed fullname: Ahmed, Shabbir organization: School of Industrial and Systems Engineering, Georgia Institute of Technology – sequence: 3 givenname: George L. surname: Nemhauser fullname: Nemhauser, George L. organization: School of Industrial and Systems Engineering, Georgia Institute of Technology |
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| Keywords | Integer programming Probabilistic constraints 90C11 Chance constraints 90C15 Mixing set Stochastic programming Probabilistic approach Linear programming Mixed integer programming Random vector Combinatorial optimization Modeling Knapsack problem Case study Mathematical programming |
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| SubjectTerms | Applied sciences Approximation Calculus of Variations and Optimal Control; Optimization Combinatorics Exact sciences and technology Flows in networks. Combinatorial problems Full Length Paper Hierarchies Inequality Integer programming Knapsack problem Linear programming Mathematical and Computational Physics Mathematical Methods in Physics Mathematical programming Mathematics Mathematics and Statistics Mathematics of Computing Numerical Analysis Operational research and scientific management Operational research. Management science Probability Production planning Random variables Stochastic models Studies Systems engineering Theoretical |
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