# 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.

## References

- Ahmed S (2006) Convexity and decomposition of mean-risk stochastic programs. Math Program, Ser A 106(3):443–446 CrossRefGoogle Scholar
- Arroyo JM, Conejo AJ (2000) Optimal response of a thermal unit to an electricity spot market. IEEE Trans Power Syst 15(3):1098–1104 CrossRefGoogle Scholar
- Artzner P, Delbaen F, Eber JM, Heath D (1999) Coherent measures of risk. Math Finance 9(3):203–228 CrossRefMATHMathSciNetGoogle Scholar
- Birge JR, Louveaux F (1997) Introduction to stochastic programming. Springer, New York MATHGoogle Scholar
- Carrión M, Arroyo JM, Conejo AJ (2009) A bilevel stochastic programming approach for retailer futures market trading. IEEE Trans Power Syst 24(3):1446–1456 CrossRefGoogle Scholar
- Conejo AJ, Nogales FJ, Arroyo JM, García-Bertrand R (2004) Risk-constrained self-scheduling of a thermal power producer. IEEE Trans Power Syst 19(3):1569–1574 CrossRefGoogle Scholar
- Conejo AJ, García-Bertrand R, Carrión M, Caballero A, de Andrés A (2008) Optimal involvement in futures markets of a power producer. IEEE Trans Power Syst 23(2):703–711 CrossRefGoogle Scholar
- Dupačová J, Consigli G, Wallace SW (2000) Scenarios for multistage stochastic programs. Ann Oper Res 100(1):25–53 CrossRefMATHMathSciNetGoogle Scholar
- Dupačová J, Gröwe-Kuska N, Römisch W (2003) Scenario reduction in stochastic programming. An approach using probability metrics. Math Program, Ser A 95(3):493–511 CrossRefMATHGoogle Scholar
- Eichhorn A, Römisch W, Wegner I (2005) Mean-risk optimization of electricity portfolios using multiperiod polyhedral risk measures. In: IEEE St. Petersburg PowerTech proceedings Google Scholar
- Fábián CI (2008) Handlingn CVaR objectives and constraints in two-stage stochastic models. Eur J Oper Res 191(3):888–911 CrossRefMATHGoogle Scholar
- Fábián CI, Vesprémi A (2008) Algorithms for handling CVaR constraints in dynamic stochastic programming models with applications to finance. J Risk 10(3):111–131 Google Scholar
- GAMS Development Corporation (2011) GAMS—the solver manuals. Washington, DC Google Scholar
- Gröwe-Kuska N, Heitsch H, Römisch W (2003) Scenario reduction and scenario tree construction for power management problems. In: 2003 IEEE Bologna power tech conference proceedings, vol 3 CrossRefGoogle Scholar
- Hatami A, Seifi H, Eslami MK S-E (2011) A stochastic-based decision-making framework for an electricity retailer: time-of-use pricing and electricity portfolio optimization. IEEE Trans Power Syst 26(4):1808–1816 CrossRefGoogle Scholar
- Heitsch H, Römisch W (2003) Scenario reduction algorithms in stochastic programming. Comput Optim Appl 24(2–3):187–206 CrossRefMATHMathSciNetGoogle Scholar
- Heitsch H, Römisch W (2007) A note on scenario reduction for two-stage stochastic programs. Oper Res Lett 35(6):731–738 CrossRefMATHMathSciNetGoogle Scholar
- Heitsch H, Römisch W (2009a) Scenario tree modeling for multistage stochastic programs. Math Program 118(2):371–406 CrossRefMATHMathSciNetGoogle Scholar
- Heitsch H, Römisch W (2009b) Scenario tree reduction for multistage stochastic programs. Comput Manag Sci 6(2):117–133 CrossRefMATHMathSciNetGoogle Scholar
- Heitsch H, Römisch W, Strugarek C (2006) Stability of multistage stochastic programs. SIAM J Optim 17(2):511–525 CrossRefMATHMathSciNetGoogle Scholar
- Henrion R, Küchler C, Römisch W (2008) Discrepancy distances and scenario reduction in two-stage stochastic mixed-integer programming. J Ind Manag Optim 4(2):363–384 CrossRefMATHMathSciNetGoogle Scholar
- Henrion R, Küchler C, Römisch W (2009) Scenario reduction in stochastic programming with respect to discrepancy distances. Comput Optim Appl 43(1):67–93 CrossRefMATHMathSciNetGoogle Scholar
- Høyland K, Wallace SW (2001) Generating scenario trees for multistage decision problems. Manag Sci 47(2):295–307 CrossRefGoogle Scholar
- Høyland K, Kaut M, Wallace SW (2003) A heuristic for moment-matching scenario generation. Comput Optim Appl 24(2–3):169–185 CrossRefMathSciNetGoogle Scholar
- Jabr RA (2005) Robust self-scheduling under price uncertainty using conditional value-at-risk. IEEE Trans Power Syst 20(4):1852–1858 CrossRefGoogle Scholar
- Krokhmal P, Palmquist J, Uryasev S (2002) Portfolio optimization with conditional value-at-risk objective and constraints. J Risk 4(2):11–27 Google Scholar
- Künzi-Bay A, Mayer J (2006) Computation aspects of minimizing conditional value-at risk. Comput Manag Sci 3(1):3–27 CrossRefMATHMathSciNetGoogle Scholar
- Lim C, Sherali HD, Uryasev S (2010) Portfolio optimization by minimizing conditional value-at-risk via nondifferentiable optimization. Comput Optim Appl 46(3):391–415 CrossRefMATHMathSciNetGoogle Scholar
- Mínguez R, Conejo AJ, García-Bertrand R (2011) Reliability and decomposition techniques to solve certain class of stochastic programming problems. Reliab Eng Syst Safety 96(2):314–323 CrossRefGoogle Scholar
- Olsson M, Soder L (2004) Generation of regulating power price scenarios. In: 2004 international conference on probabilistic methods applied to power systems, pp 26–31 Google Scholar
- Pflug GC (2001) Scenario tree generation for multiperiod financial optimization by optimal discretization. Math Program 89(2):251–271 CrossRefMATHMathSciNetGoogle Scholar
- Pineda S, Conejo AJ (2010) Scenario reduction for risk-averse electricity trading. IET Gener Transm Distrib 4(6):694–705 CrossRefGoogle Scholar
- Rockafellar RT, Uryasev S (2000) Optimization of conditional value-at risk. J Risk 2(3):21–41 Google Scholar
- Rockafellar RT, Uryasev S (2002) Conditional value-at risk for general loss distributions. J Bank Finance 26(7):1443–1471 CrossRefGoogle Scholar
- Rosenthal RE (2008) GAMS—a user’s guide. GAMS Development Corporation, Washington Google Scholar
- Zhang W, Wang X (2009) Hedge contract characterization and risk-constrained electricity procurement. IEEE Trans Power Syst 24(3):1547–1558 CrossRefGoogle Scholar