# Iterative scenario based reduction technique for stochastic optimization using conditional value-at-risk

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## Abstract

In the last decades, several tools for managing risks in competitive markets, such as the conditional value-at-risk, have been developed. These techniques are applied in stochastic programming models primarily based on scenarios and/or finite sampling, which in case of large-scale models increase considerably their size according to the number of scenarios, sometimes resulting in intractable problems. This shortcoming is solved in the literature using (i) scenario reduction methods, and/or (ii) speeding up optimization techniques. However, when reducing the number of scenarios, part of the stochastic information is lost. In this paper, an iterative scheme is proposed to get the solution of a stochastic problem representing the stochastic processes via a set of scenarios and/or finite sampling, and modeling risk via conditional value-at-risk. This iterative approach relies on the fact that the solution of a stochastic programming problem optimizing the conditional value-at risk only depends on the scenarios on the upper tail of the loss distribution. Thus, the solution of the stochastic problem is obtained by solving, within an iterative scheme, problems with a reduced number of scenarios (subproblems). This strategy results in an important reduction in the computational burden for large-scale problems, while keeping all the stochastic information embedded in the original set of scenarios. In addition, each subproblem can be solved using speeding-up optimization techniques. The proposed method is very easy to implement and, as numerical results show, the reduction in computing time can be dramatic, and more pronounced as the number of initial scenarios or samples increases.

## Keywords

Risk management Decision making Scenario reduction Stochastic programming Conditional value-at-risk## Notes

### Acknowledgements

R. García-Bertrand is partly supported by the Ministry of Science and Innovation of Spain through CICYT Project ENE2009-07836. R. Mínguez is partly supported by the Spanish Ministry of Science and Innovation through the “Ramon y Cajal” program (RYC-2008-03207) and project “AMVAR” (CTM2010-15009) from Spanish Ministry of Science and Innovation.

We also thank the editor and referees for their very helpful comments and suggestions, which have led to an improved manuscript.

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