Abstract
We analyze nonlinear stochastic optimization problems with probabilistic constraints described by continuously differentiable non-convex functions. We describe the tangent and the normal cone to the level sets of the underlying probability function and provide new insight into their structure. Furthermore, we formulate fist order and second order conditions of optimality for these problems based on the notion of p-efficient points. We develop an augmented Lagrangian method for the case of discrete distribution functions. The method is based on progressive inner approximation of the level set of the probability function by generation of p-efficient points. Numerical experience is provided.
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References
Bonnans, J. F., & Shapiro, A. (2000). Perturbation analysis of optimization problems. New York: Springer.
Dentcheva, D., & Martinez, G. (2010, submitted). Regularization in probabilistic optimization.
Dentcheva, D., Prékopa, A., & Ruszczyński, A. (2000). Concavity and efficient points of discrete distributions in probabilistic programming. Mathematical Programming, 89, 55–77.
Dentcheva, D., Prékopa, A., & Ruszczyński, A. (2002). Bounds for integer stochastic programs with probabilistic constraints. Discrete Applied Mathematics, 124, 55–65.
Dentcheva, D., Lai, B., & Ruszczyński, A. (2004). Dual methods for probabilistic optimization problems. Mathematical Methods of Operations Research, 60, 331–346.
Lejeune, M., & Noyan, N. (2010). Mathematical programming generation of p-efficient points (Working paper).
Luedtke, J., Ahmed, S., & Nemhauser, G. (2010). An integer programming approach for linear programming with probabilistic constraints. Mathematical Programming, 122, 247–272.
Norkin, V. I., & Roenko, N. V. (1991). α-concave functions and measures and their applications. Kibernetika I Sistemnyj Analiz, 189, 77–88 (in Russian); translation in: Cybernet. Systems Anal. 27 (1991) 860–869.
Prékopa, A. (1970). On probabilistic constrained programming. In Proceedings of the Princeton symposium on mathematical programming (pp. 113–138). Princeton: Princeton University Press.
Prékopa, A. (1995). Stochastic programming. Dordrecht: Kluwer.
Prékopa, A., Vizvári, B., & Badics, T. (1998). Programming under probabilistic constraint with discrete random variable. In L. Grandinetti et al. (Ed.), New trends in mathematical programming (pp. 235–255). Dordrecht: Kluwer.
Rinott, Y. (1976). On convexity of measures. Annals of Probability, 4, 1020–1026.
Ruszczyński, A. (2002). Probabilistic programming with discrete distributions and precedence constrained knapsack polyhedra. Mathematical Programming, 93, 195–215.
Ruszczyński, A. (2006). Nonlinear optimization. Princeton: Princeton University Press.
Ruszczyński, A., & Shapiro, A. (Eds.) (2003). Stochastic programming. Amsterdam: Elsevier Science.
Shapiro, A., Dentcheva, D., & Ruszczyński, A. (2009). MPS/SIAM Series on Optimization: Vol. 9. Lectures on Stochastic Programming: Modeling and Theory. Philadelphia: SIAM.
Szántai, T. (1987). Calculation of the multivariate probability distribution function values and their gradient vectors (IIASA Working Paper, WP-87-82).
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This paper is dedicated to András Prékopa on the occasion of his 80th birthday in recognition of his fundamental contributions to optimization under uncertainty.
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Dentcheva, D., Martinez, G. Augmented Lagrangian method for probabilistic optimization. Ann Oper Res 200, 109–130 (2012). https://doi.org/10.1007/s10479-011-0884-5
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DOI: https://doi.org/10.1007/s10479-011-0884-5