Scenario cluster Lagrangean decomposition for risk averse in multistage stochastic optimization
•A Multistage scenario Cluster Dualization and Lagrangean Relaxation is presented.•Time Stochastic Dominance (TSD) risk averse measure is considered.•Dualization of the NAC and Relaxation of the cross node constraints are considered.•Three Lagrangean multipliers updating procedures are presented.•A...
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| Vydáno v: | Computers & operations research Ročník 85; s. 154 - 171 |
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| Hlavní autoři: | , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
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New York
Elsevier Ltd
01.09.2017
Pergamon Press Inc |
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| ISSN: | 0305-0548, 1873-765X, 0305-0548 |
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| Abstract | •A Multistage scenario Cluster Dualization and Lagrangean Relaxation is presented.•Time Stochastic Dominance (TSD) risk averse measure is considered.•Dualization of the NAC and Relaxation of the cross node constraints are considered.•Three Lagrangean multipliers updating procedures are presented.•A broad computational comparison between our algorithm and CPLEX is reported.
In this work we present a decomposition approach as a mixture of dualization and Lagrangean Relaxation for obtaining strong lower bounds on large-sized multistage stochastic mixed 0–1 programs with a time stochastic dominance risk averse measure. The objective function to minimize is a composite function of the expected cost along the time horizon over the scenarios and the penalization of the expected cost excess on reaching the set of thresholds under consideration, subject to a bound on the expected cost excess for each threshold and a bound on the failure probability of reaching it. The main differences with some other risk averse strategies are presented. The problem is represented by a mixture of the splitting representation up to a given stage, so-called break stage, and the compact representation for the other stages along the time horizon. The dualization of the nonanticipativity constraints for the node-based and risk averse variables up to the break stage and the Lagrangean Relaxation of the cross node constraints of the risk averse strategy result in a model that can be decomposed into a set of independent scenario cluster submodels. Three Lagrangean multipliers updating schemes as the Subgradient method, the Lagrangean Progressive Hedging algorithm and the Dynamic Constrained Cutting Plane are computationally compared. We have observed in the randomly generated instances we have experimented with that the smaller the number of clusters, the stronger the lower bound provided for the original problem (even, frequently, it is the solution value) obtained with an affordable computing time. |
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| AbstractList | In this work we present a decomposition approach as a mixture of dualization and Lagrangean Relaxation for obtaining strong lower bounds on large-sized multistage stochastic mixed 0-1 programs with a time stochastic dominance risk averse measure. The objective function to minimize is a composite function of the expected cost along the time horizon over the scenarios and the penalization of the expected cost excess on reaching the set of thresholds under consideration, subject to a bound on the expected cost excess for each threshold and a bound on the failure probability of reaching it. The main differences with some other risk averse strategies are presented. The problem is represented by a mixture of the splitting representation up to a given stage, so-called break stage, and the compact representation for the other stages along the time horizon. The dualization of the nonanticipativity constraints for the node-based and risk averse variables up to the break stage and the Lagrangean Relaxation of the cross node constraints of the risk averse strategy result in a model that can be decomposed into a set of independent scenario cluster submodels. Three Lagrangean multipliers updating schemes as the Subgradient method, the Lagrangean Progressive Hedging algorithm and the Dynamic Constrained Cutting Plane are computationally compared. We have observed in the randomly generated instances we have experimented with that the smaller the number of clusters, the stronger the lower bound provided for the original problem (even, frequently, it is the solution value) obtained with an affordable computing time. •A Multistage scenario Cluster Dualization and Lagrangean Relaxation is presented.•Time Stochastic Dominance (TSD) risk averse measure is considered.•Dualization of the NAC and Relaxation of the cross node constraints are considered.•Three Lagrangean multipliers updating procedures are presented.•A broad computational comparison between our algorithm and CPLEX is reported. In this work we present a decomposition approach as a mixture of dualization and Lagrangean Relaxation for obtaining strong lower bounds on large-sized multistage stochastic mixed 0–1 programs with a time stochastic dominance risk averse measure. The objective function to minimize is a composite function of the expected cost along the time horizon over the scenarios and the penalization of the expected cost excess on reaching the set of thresholds under consideration, subject to a bound on the expected cost excess for each threshold and a bound on the failure probability of reaching it. The main differences with some other risk averse strategies are presented. The problem is represented by a mixture of the splitting representation up to a given stage, so-called break stage, and the compact representation for the other stages along the time horizon. The dualization of the nonanticipativity constraints for the node-based and risk averse variables up to the break stage and the Lagrangean Relaxation of the cross node constraints of the risk averse strategy result in a model that can be decomposed into a set of independent scenario cluster submodels. Three Lagrangean multipliers updating schemes as the Subgradient method, the Lagrangean Progressive Hedging algorithm and the Dynamic Constrained Cutting Plane are computationally compared. We have observed in the randomly generated instances we have experimented with that the smaller the number of clusters, the stronger the lower bound provided for the original problem (even, frequently, it is the solution value) obtained with an affordable computing time. |
| Author | Garín, María Araceli Unzueta, Aitziber Escudero, Laureano F. |
| Author_xml | – sequence: 1 givenname: Laureano F. surname: Escudero fullname: Escudero, Laureano F. email: laureano.escudero@urjc.es organization: Dpto. Estadística e Investigación Operativa, Universidad Rey Juan Carlos, Móstoles, Madrid, Spain – sequence: 2 givenname: María Araceli surname: Garín fullname: Garín, María Araceli email: mariaaraceli.garin@ehu.eus organization: Dpto. Economía Aplicada III, Universidad del País Vasco, UPV/EHU, Bilbao, Bizkaia, Spain – sequence: 3 givenname: Aitziber surname: Unzueta fullname: Unzueta, Aitziber email: aitziber.unzueta@ehu.eus organization: Dpto. Matemática Aplicada, Universidad del País Vasco, UPV/EHU, Bilbao, Bizkaia, Spain |
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| Keywords | Time stochastic dominance risk averse measure Dynamic Constrained Cutting Plane algorithm Multistage stochastic mixed 0–1 optimization Cluster Lagrangean problem Subgradient method Progressive Hedging algorithm |
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| Snippet | •A Multistage scenario Cluster Dualization and Lagrangean Relaxation is presented.•Time Stochastic Dominance (TSD) risk averse measure is... In this work we present a decomposition approach as a mixture of dualization and Lagrangean Relaxation for obtaining strong lower bounds on large-sized... |
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| SubjectTerms | Cluster Lagrangean problem Clusters Computing time Decomposition Dominance Dynamic Constrained Cutting Plane algorithm Economic models Lower bounds Multipliers Multistage Multistage stochastic mixed 0–1 optimization Operations research Probability theory Progressive Hedging algorithm Risk Splitting Subgradient method Thresholds Time stochastic dominance risk averse measure Traveling salesman problem |
| Title | Scenario cluster Lagrangean decomposition for risk averse in multistage stochastic optimization |
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