Abstract
Probability function maximization and quantile function minimization problems are approximated starting from the weak convergence of discrete measures with increasing dimension. It is assumed that solutions of both problems depend on a random parameter, i.e., solutions are sought as decision rules from the class of bounded measurable functions L ∞. Both problems are approximated by sequences of finite dimensional extremum problems with discrete measures and with increasing dimensions. Convergence conditions for optimal values and solutions of both problems are presented.
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References
Charnes, A. and W.W. Cooper (1959). Chance-constrained programming. Management Science, 6, No 1, 73–79.
Raik, E. (1971) On stochastic programming problems with probability and quan-tile Junctionals. Proc. Acad. Sci. Estonian SSR. Phys.-Math., 21, 142–148 (in Russian).
Kibzun, A. and Y. Kan (1996) Stochastic Programming Probles with Probability and Quantile Functions. John Wiley and Sons, Chichester, New York, e.t.c, 1996, 300 p.
Kan, Y. and A. Mistryukov (1998) On the equivalence in stochastic programming with probability and quantile objectives. In: “Stochastic Programming. Methods and Technical Applications.” K. Marti and P. Kail, eds. Lecture Notes in Economic and Mathematical Systems, 458, 145–153.
Vainikko, G. (1971) On the convergence of quadrature formulae method for integral equations with discontinuous kernels. Siberian Math. Journal, 12, 40–53.
Birge, J. and R. Wets (1986) Designing approximation schemes for stochastic optimization problems, in particular for stochastic programs with recourse. Mathematical Programming Study, 27, 54–102.
Lepp, R. (1988) Discrete approximation conditions of the space of essentially bounded measurable functions. Proc. Acad. Sci. Estonian SSR. Phys.-Math. 38, 204–208.
Lepp, R. (1994) Projection and discretization methods in stochastic programming. J. Comput. Appl. Math., 56, 55–64.
Lepp, R. (1996) On approximation of optimal controls with discontinuous strategies. Proc. Estonian Acad. Sci. Phys.-Math., 45, No-s 2/3, 193–200.
Lepp, R. (1998) Discrete approximation of Hammerstein integral equations with discontinuous kernels. Numer. Funct. Anal, and Optim., 19, 7/8, 835–848.
Kan, Y. (1996) A quasigradient algorithm for minimization of the quantile function. Theory and Systems of Control, No 2, 81–86 (in Russian).
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© 2000 Springer Science+Business Media Dordrecht
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Lepp, R. (2000). Approximation of Value-at-Risk Problems with Decision Rules. In: Uryasev, S.P. (eds) Probabilistic Constrained Optimization. Nonconvex Optimization and Its Applications, vol 49. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3150-7_10
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DOI: https://doi.org/10.1007/978-1-4757-3150-7_10
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4419-4840-3
Online ISBN: 978-1-4757-3150-7
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