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Two-stage stochastic optimization problems with stochastic ordering constraints on the recourse. (English) Zbl 1244.90174
Summary: We consider two-stage risk-averse stochastic optimization problems with a stochastic ordering constraint on the recourse function. Two new characterizations of the increasing convex order relation are provided. They are based on conditional expectations and on integrated quantile functions: a counterpart of the Lorenz function. We propose two decomposition methods to solve the problems and prove their convergence. Our methods exploit the decomposition structure of the risk-neutral two-stage problems and construct successive approximations of the stochastic ordering constraints. Numerical results confirm the efficiency of the methods.

90C15 Stochastic programming
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[1] Birge, J.R.; Louveaux, F., Introduction to stochastic programming, (1997), Springer-Verlag New York, NY · Zbl 0892.90142
[2] Dentcheva, D.; Ruszczyński, A., Optimization with stochastic dominance constraints, SIAM journal on optimization, 14, 548-566, (2003) · Zbl 1055.90055
[3] Dentcheva, D.; Ruszczyński, A., Optimality and duality theory for stochastic optimization problems with nonlinear dominance constraints, Mathematical programming, 99, 329-350, (2004) · Zbl 1098.90044
[4] Dentcheva, D.; Ruszczyński, A., Inverse stochastic dominance constraints and rank dependent expected utility theory, Mathematical programming, 108, 297-311, (2006) · Zbl 1130.91327
[5] Dentcheva, D.; Ruszczyński, A., Inverse cutting plane methods for optimization problems with second-order stochastic dominance constraints, Optimization, 59, 323-338, (2010) · Zbl 1218.90143
[6] D. Drapkin, R. Schultz, An algorithm for stochastic programs with first order dominance constraints induced by linear recourse. Preprint Series, Department of Mathematics, University of Duisburg-Essen, No. 653-2007, 2007. · Zbl 1185.90160
[7] Fábián, C.I.; Mitra, G.; Roman, D., Processing second-order stochastic dominance models using cutting-plane representations, Mathematical programming series A, 130, 33-57, (2011) · Zbl 1229.90108
[8] R. Gollmer, U. Gotzes, F. Neise, R. Schultz, Risk modeling via stochastic dominance in power systems with dispersed generation, Technical report. Department of Mathematics, University of Duisburg-Essen. · Zbl 1405.91482
[9] Gollmer, R.; Neise, F.; Schultz, R., Stochastic programs with first-order dominance constraints induced by mixed-integer linear recourse, SIAM journal on optimization, 19, 552-571, (2008) · Zbl 1173.90490
[10] Lorenz, M., Methods of measuring concentration of wealth, Journal of the American statistical association, 9, 209-219, (1905)
[11] Müller, A.; Stoyan, D., Comparison methods for stochastic models and risks, (2002), John Wiley & Sons Chichester · Zbl 0999.60002
[12] Noyan, N.; Ruszczyński, A., Valid inequalities and restrictions for stochastic programming problems with first order stochastic dominance constraints, Mathematical programming, 115, 249-275, (2008) · Zbl 1145.90046
[13] Ogryczak, W.; Ruszczyński, A., Dual stochastic dominance and related Mean-risk models, SIAM journal on optimization, 13, 60-78, (2002) · Zbl 1022.91017
[14] Rockafellar, R.T., Convex analysis, (1970), Princeton University Press Princeton · Zbl 0229.90020
[15] Rudolf, G.; Ruszczyński, A., Optimization problems with second order stochastic dominance constraints: duality, compact formulations, and cut generation methods, SIAM journal on optimization, 19, 1326-1343, (2008) · Zbl 1198.90308
[16] Ruszczyński, A., A regularized decomposition method for minimizing a sum of polyhedral functions, Mathematical programming, 35, 309-333, (1986) · Zbl 0599.90103
[17] ()
[18] Shaked, M.; Shanthikumar, J.G., Stochastic orders and their applications, (1994), Academic Press Boston · Zbl 0806.62009
[19] A. Shapiro, D. Dentcheva, A. Ruszczyński, Lectures on Stochastic Programming, MPS-SIAM Series on Optimization, 2009.
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