Abstract
The problem of constructing the confidence absorbing set for the analysis of static stochastic systems is considered. The confidence absorbing set is understood as the set of initial positions for which at a terminal time instant a system will not leave an admissible domain with a given probability. Some properties of the confidence absorbing set, in particular, convexity, are established. An algorithm for constructing an inner approximation of the confidence absorbing set based on the confidence method is proposed. The properties of this approximation are established. The results obtained are used for predicting wind speed in the vicinity of a landing airfield. Calculations for a numerical experiment are presented.
Similar content being viewed by others
References
Liptser, R.S. and Shiryaev, A.N., Statistics of Random Processes I,II, Applications of Mathematics, vols. 5,6, Berlin: Springer-Verlag, 2001.
Kibzun, A.I. and Kan, Y.S., Stochastic Programming Problems with Probability and Quantile Functions, Chichester: Wiley, 1996.
Kibzun, A.I. and Kan, Yu.S., Zadachi stokhasticheskogo programmirovaniya s veroyatnostnymi kriteriyami (Stochastic Programming Problems with Probability Criteria), Moscow: Fizmatlit, 2009.
Prékopa, A., Stochastic Programming, Dordrecht: Kluwer, 1995.
Shapiro, A., Dentcheva, D., and Ruszczyński, A., Lectures on Stochastic Programming. Modeling and Theory, Philadelphia: SIAM, 2009.
Tamm, E., On Quasiconvexity of Probability Functions and Quantiles, Izv. Akad. Nauk. Est. SSR, Fiz.-Mat., 1976, vol. 25, no. 2, pp. 141–144.
Kan, Yu.S. and Kibzun, A.I., Convexity Properties of Probability Functions and Quantiles in Optimization Problems, Autom. Remote Control, 1996, vol. 57, no. 3, pp. 368–383.
Van Ackooij, W., Eventual Convexity of Chance Constrained Feasible Sets, Optimization (J. Math. Programm. Oper. Res.), 2015, vol. 64, no. 5, pp. 1263–1284.
Prékopa, A., Logarithmic Concave Measures with Application to Stochastic Programming, Acta Sci. Math. (Szeged), 1971, vol. 32, pp. 301–316.
Prékopa, A., On Logarithmic Concave Measures and Functions, Acta Sci. Math. (Szeged), 1973, vol. 34, pp. 335–343.
Borell, C., Convex Set Functions in d-Space, Period. Math. Hung., 1975, vol. 6, no. 2, pp. 111–136.
Norkin, V.I. and Roenko, N.V., α-Concave Functions and Measures and Their Applications, Cybern. Syst. Anal., 1991, vol. 27, no. 6, pp. 860–869.
Henrion, R., On the Connectedness of Probabilistic Constraint Sets, J. Optim. Theory Appl., 2002, vol. 112, no. 3, pp. 657–663.
Prékopa, A., Dual Method for the Solution of a One-Stage Stochastic Programming Problem with Random RHS Obeying a Discrete Probability Distribution, ZOR, 1990, vol. 34, pp. 441–461.
Lejeune, M. and Noyan, N., Mathematical Programming Approaches for Generating p-Efficient Points, Eur. J. Oper. Res., 2010, vol. 207, pp. 590–600.
Dentcheva, D., Prékopa, A., and Ruszczyński, A., On Convex Probabilistic Programming with Discrete Distribution, Nonlin. Anal., 2001, vol. 47, pp. 1997–2009.
Van Ackooij, W., Berge, V., de Oliveira, W., and Sagastizábal, C., Probabilistic Optimization via Approximate p-Efficient Points and Bundle Methods, Comput. Oper. Res., 2017, vol. 77, pp. 177–193.
Lejeune, M.A. and Prékopa, A., Relaxations for Probabilistically Constrained Stochastic Programming Problems: Review and Extensions, Ann. Oper. Res., 2018. https://doi.org/10.1007/s10479-018-2934-8
Vasil’eva, S.N. and Kan, Yu.S., A Method for Solving Quantile Optimization Problems with a Bilinear Loss function, Autom. Remote Control, 2015, vol. 76, no. 9, pp. 1582–1597.
Vasil’eva, S.N. and Kan, Yu.S., A Visualization Algorithm for the Plane Probability Measure Kernel, Informat. Primen., 2018, vol. 12, no. 2, pp. 60–68.
Kibzun, A.I. and Malyshev, V.V., A Generalized Minimax Approach to the Solution of Problems with Probability Constraints, Izv. Akad. Nauk SSSR, Tekh. Kibern., 1984, no. 1, pp. 20–29.
Author information
Authors and Affiliations
Corresponding authors
Additional information
This paper was recommended for publication by E.Ya. Rubinovich, a member of the Editorial Board
Russian Text © The Author(s), 2020, published in Avtomatika i Telemekhanika, 2020, No. 4, pp. 21–36.
Rights and permissions
About this article
Cite this article
Kibzun, A.I., Ivanov, S.V. & Stepanova, A.S. Construction of Confidence Absorbing Set for Analysis of Static Stochastic Systems. Autom Remote Control 81, 589–601 (2020). https://doi.org/10.1134/S0005117920040025
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0005117920040025