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Robust stochastic dominance and its application to risk-averse optimization

  • Darinka DentchevaEmail author
  • Andrzej Ruszczyński
Full Length Paper Series B

Abstract

We introduce a new preference relation in the space of random variables, which we call robust stochastic dominance. We consider stochastic optimization problems where risk-aversion is expressed by a robust stochastic dominance constraint. These are composite semi-infinite optimization problems with constraints on compositions of measures of risk and utility functions. We develop necessary and sufficient conditions of optimality for such optimization problems in the convex case. In the nonconvex case, we derive necessary conditions of optimality under additional smoothness assumptions of some mappings involved in the problem.

Keywords

Robust preferences Stochastic order Stochastic dominance constraints Risk constraints Semi-infinite optimization 

Mathematics Subject Classification (2000)

90C15 90C46 90C48 46N10 60E15 

References

  1. 1.
    Artzner P., Delbaen F., Eber J.-M., Heath D.: Coherent measures of risk. Math. Finance 9, 203–228 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bonnans J.F., Cominetti R.: Perturbed optimization in Banach spaces. III. Semi-infinite optimization. SIAM J. Control Optim. 34, 1555–1567 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bonnans J.F., Shapiro A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)zbMATHGoogle Scholar
  4. 4.
    Canovas M.J., Dontchev A.L., Lopez M.A., Parra J.: Metric regularity of semi-infinite constraint systems. Math. Program. 104, 329–346 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Dentcheva D., Ruszczyński A.: Optimization with stochastic dominance constraints. SIAM J. Optim. 14, 548–566 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Dentcheva D., Ruszczyński A.: Optimality and duality theory for stochastic optimization problems with nonlinear dominance constraints. Math. Program. 99, 329–350 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Dentcheva D., Ruszczyński A.: Portfolio optimization with stochastic dominance constraints. J. Banking Finance 30(2), 433–451 (2006)CrossRefGoogle Scholar
  8. 8.
    Dentcheva D., Ruszczyński A.: Semi-infinite probabilistic optimization: first order stochastic dominance constraints. Optimization 53, 583–601 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Dentcheva D., Ruszczyński A.: Inverse stochastic dominance constraints and rank dependent expected utility theory. Math. Program. 108, 297–311 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Dentcheva D., Ruszczyński A.: Optimization with multivariate stochastic dominance constraints. Math. Program. 117, 111–127 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Dentcheva D., Ruszczyński A.: Composite semi-infinite optimization. Control Cybern 36, 633–646 (2007)zbMATHGoogle Scholar
  12. 12.
    Dentcheva D., Ruszczyński A.: Duality between coherent risk measures and stochastic dominance constraints in risk-averse optimization. Pac. J. Optim. 4, 433–446 (2008)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Dentcheva D., Ruszczyński A.: Stochastic dominance for sequences and implied utility in dynamic optimization. C. R. Acad. Bulgare Sci. 57(1), 15–22 (2008)Google Scholar
  14. 14.
    Dudley R.M.: Real Analysis and Probability. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar
  15. 15.
    Fishburn P.C.: Utility Theory for Decision Making. Wiley, New York (1970)zbMATHGoogle Scholar
  16. 16.
    Föllmer H., Schied A.: Convex measures of risk and trading constraints. Finance Stoch. 6, 429–447 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Gilboa I., Schmeidler D.: Maxmin expected utility with non-unique prior. J. Math. Econ. 18, 141–153 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Goberna M.A., Lopez M.A.: Linear Semi-Infinite Optimization. Wiley, Chichester (1998)zbMATHGoogle Scholar
  19. 19.
    Gollmer R., Neise F., Schultz R.: Stochastic programs with first-order dominance constraints induced by mixed-integer linear recourse. SIAM J. Optim. 19, 552–571 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Hadar J., Russell W.: Rules for ordering uncertain prospects. Am. Econ. Rev. 59, 25–34 (1969)Google Scholar
  21. 21.
    Lehmann E.: Ordered families of distributions. Ann. Math. Stat. 26, 399–419 (1955)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Levin, V.L.: Convex Analysis in Spaces of Measurable Functions and its Applications in Economics, Nauka, Moscow (1985, in Russian)Google Scholar
  23. 23.
    Maccheroni F., Rustichini A., Marinacci M.: Ambiguity aversion, robustness, and the variational representation of preferences. Econometrica 74, 1447–1498 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Mann H.B., Whitney D.R.: On a test of whether one of two random variables is stochastically larger than the other. Ann. Math. Stat. 18, 50–60 (1947)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Mordukhovich B.: Variational Analysis and Generalized Differentiation. Springer, Berlin (2006)Google Scholar
  26. 26.
    Mosler, K., Scarsini, M. (eds.): Stochastic Orders and Decision Under Risk. Institute of Mathematical Statistics, Hayward (1991)zbMATHGoogle Scholar
  27. 27.
    Müller A., Stoyan D.: Comparison Methods for Stochastic Models and Risks. Wiley, Chichester (2002)zbMATHGoogle Scholar
  28. 28.
    Ogryczak W., Ruszczyński A.: From stochastic dominance to mean—risk models: semideviations as risk measures. Eur. J. Oper. Res. 116, 33–50 (1999)zbMATHCrossRefGoogle Scholar
  29. 29.
    Penot J.P.: Optimality conditions in mathematical programming and composite optimization. Math. Program. 67, 225–246 (1994)CrossRefMathSciNetGoogle Scholar
  30. 30.
    Phelps R.R.: Convex Functions, Monotone Operators and Differentiability, Lecture Notes in Mathematics 1364. Springer, Berlin (1989)Google Scholar
  31. 31.
    Quirk J.P., Saposnik R.: Admissibility and measurable utility functions. Rev. Econ. Stud. 29, 140–146 (1962)CrossRefGoogle Scholar
  32. 32.
    Robinson S.M.: Stability theory for systems of inequalities. II: differentiable nonlinear systems. SIAM J. Numer. Anal. 13, 497–513 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Ruszczyński A., Shapiro A.: Optimization of convex risk functions. Math. Oper. Res. 31, 433–452 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Shaked M., Shanthikumar J.G.: Stochastic Orders and Their Applications. Academic Press, Boston (1994)zbMATHGoogle Scholar
  35. 35.
    Studniarski M., Jeyakumar V.: A generalized mean-value theorem and optimality conditions in composite nonsmooth minimization. Nonlinear Anal. 24, 883–894 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Szekli R.: Stochastic Ordering and Dependence in Applied Probability. Springer, New York (1995)zbMATHGoogle Scholar
  37. 37.
    Yang X.Q.: Second-order global optimality conditions for convex composite optimization. Math. Prog. 81, 327–347 (1998)Google Scholar

Copyright information

© Springer and Mathematical Programming Society 2009

Authors and Affiliations

  1. 1.Department of Mathematical SciencesStevens Institute of TechnologyHobokenUSA
  2. 2.Department of Management Science and Information SystemsRutgers UniversityPiscatawayUSA

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