Abstract
We analyze nonlinear stochastic optimization problems with probabilistic constraints on nonlinear inequalities with random right hand sides. We develop two numerical methods with regularization for their numerical solution. The methods are based on first order optimality conditions and successive inner approximations of the feasible set by progressive generation of p-efficient points. The algorithms yield an optimal solution for problems involving α-concave probability distributions. For arbitrary distributions, the algorithms solve the convex hull problem and provide upper and lower bounds for the optimal value and nearly optimal solutions. The methods are compared numerically to two cutting plane methods.
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References
Bank B., Guddat J., Klatte D., Kummer B., Tammer K.: Nonlinear Parametric Optimization. Akademie-verlag, Berlin (1982)
Berge C.: Espaces Topologiques Functions Multivoques. Dunod, Paris (1959)
Bonnans J.-F., Gilvert J.C., Lemaréchal C., Sagastizábal C.: Numerical Optimization, Theoretical and Practical Aspects. Springer, Berlin (2003)
Dentcheva D., Prékopa A., Ruszczyński A.: Concavity and efficient points of discrete distributions in probabilistic programming. Math. Program. 89, 55–77 (2000)
Dentcheva D., Prékopa A., Ruszczyński A.: Bounds for integer stochastic programs with probabilistic constraints. Discret. Appl. Math. 124, 55–65 (2002)
Dentcheva D., Lai B., Ruszczyński A.: Dual methods for probabilistic optimization problems. Math. Methods Oper. Res. 60, 331–346 (2004)
Dowd K.: Beyond Value at Risk. The Science of Risk Management. Wiley, New York (1997)
Henrion R.: Some remarks on value-at-risk optimization. Manag. Sci. Eng. Manag. 1, 111–118 (2006)
Hiriart-Urruty J.-B., Lemaréchal C.: Convex Analysis and Minimization Algorithms I and II. Springer, Berlin (1993)
Kan Y.: Application of the quantile optimization to bond portfolio selection. In: Marti, K. (ed.) Stochastic optimization techniques. Lecture Notes in Economics and Mathematical Systems, vol. 513, pp. 285–308. Springer, Berlin (2002)
Kiwiel K.C.: Methods of Descent for Non differentiable Optimization, Lecture Notes in Mathematics, vol. 1133. Springer, Berlin (1985)
Lejeune M., Noyan N.: An efficient method for generating p-efficient points. Eur. J. Oper. Res. 20(2), 590–600 (2010)
Luedtke J., Ahmed S., Nemhauser G.: An integer programming approach for linear programming with probabilistic constraints. Math. Program. 122, 247–272 (2010)
Norkin, V.I., Roenko, N.V.: α-concave functions and measures and their applications. Kibernet. Sistem. Anal. 189, 77–88 (1991) (in Russian); translation in: Cybernet. Systems Anal. 27, 860–869 (1991)
Pflug G.C.: Some remarks on the value-at-risk and the conditional value-at-risk. In: Uryasev, S. (ed.) Probabilistic Constrained Optimization—Methodology and Applications., pp. 272–281. Kluwer, Dordrecht (2000)
Prékopa A.: Stochastic Programming. Kluwer, Dordrecht (1995)
Prékopa A., Vizvári B., Badics T. et al.: Programming under probabilistic constraint with discrete random variable. In: Grandinetti, L. et al. (eds.) New Trends in Mathematical Programming, pp. 235–255. Kluwer, Dordrecht (1998)
Rockafellar R.T.: The multiplier method of Hestenes and Powell applied to convex programming. J. Opt. Theory Appl. 12, 555–562 (1973)
Ruszczyński A.: Nonlinear Optimization. Princeton University Press, Princeton (2006)
Ruszczyński, A., Shapiro, A. (eds.): Stochastic Programming. Elsevier Science, Amsterdam (2003)
Shapiro, A., Dentcheva, D., Ruszczyński, A.: Lectures on Stochastic Programming: Modeling and Theory, MPS/SIAM Series on Optimization 9, SIAM, Philadelphia (2009)
Sen S.: Relaxations for the probabilistically constrained programs with discrete random variables. Oper. Res. Lett. 11, 81–86 (1992)
Szántai, T.: Calculation of the multivariate probability distribution function values and their gradient vectors, IIASA Working Paper, WP-87-82 (1987)
Szegö G.: Measures of risk. J. Banking Finance 26, 1253–1272 (2002)
Tretyakov N.V.: The method of penalty estimates for problems of convex programming. Econ. Math. Methods 9, 525–540 (1973)
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This research is supported by the NSF award 0604060 and 0965702.
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Dentcheva, D., Martinez, G. Regularization methods for optimization problems with probabilistic constraints. Math. Program. 138, 223–251 (2013). https://doi.org/10.1007/s10107-012-0539-6
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DOI: https://doi.org/10.1007/s10107-012-0539-6