Volume 15, no. 4Pages 115 - 122

Smooth Approximation of the Quantile Function Derivatives

V.R. Sobol, R.O. Torishnyy
In this paper, a smooth approximation of the second-order derivatives of quantile function is provided. The convergence of approximations of the first and second order derivatives of quantile function is studied in cases when there exists a deterministic equivalent for the corresponding stochastic programming problem. The quantile function is one of common criteria in stochastic programming problems. The first-order derivative of quantile function can be represented as a ratio of partial derivatives of probability function. Using smooth approximation of probability function and its derivatives we obtain approximations of these derivatives in the form of volume integrals. Approximation of the second-order derivative is obtained directly as derivative of the first-order derivative. A numerical example is provided to evaluate the accuracy of the presented approximations.
Full text
stochastic programming; probability function; quantile function and its derivatives.
1. Kibzun A.I., Kan Yu.S. Stochastic Programming Problems with Probability and Quantile Functions. London, John Wiley and Sons, 1996.
2. Raik E. The Differentiability in the Parameter of the Probability Function and Optimization of the Probability Function via the Stochastic Pseudogradient Method. Proceedings of Academy of Sciences of the Estonian SSR. Physics. Mathematics, 1975, vol. 24, no. 1, pp. 3-9.
3. Uryas'ev S. Derivatives of Probability Functions and Some Applications. Annals of Operations Research, 1995, vol. 56, pp. 287-311. DOI:10.1007/BF02031712
4. Henrion R. Gradient Estimates for Gaussian Distribution Functions: Application to Probabilistically Constrained Optimization Problems. Numerical Algebra, Control and Optimization, 2012, vol. 2, no. 4, pp. 655-668. DOI:10.3934/naco.2012.2.655
5. Pflug G., Weisshaupt H. Probability Gradient Estimation by Set-Valued Calculus and Applications in Network Design. SIAM Journal on Optimization, 2005, vol. 15, no. 3, pp. 898-914. DOI:10.1137/S1052623403431639
6. Sobol V.R., Torishnyi R.O. On Smooth Approximation of Probabilistic Criteria in Stochastic Programming Problems. SPIIRAS Proceedings, 2020, vol. 19, no. 1, pp. 181-217. DOI:10.15622/sp.2020.19.1.7
7. Sobol V., Torishnyi R. Smooth Approximation of Probability and Quantile Functions: Vector Generalization and its Applications. Journal of Physics: Conference Series, 2021, vol. 1925, article ID 012034.
8. Torishyi, R. Application of the Second-Order Optimization Methods to the Stochastic Programming Problems with Probability Function. Trudy MAI, 2021, no. 121, 27 p.
9. Cox D.R., Hinkley D.V. Theoretical Statistics. London, Chapman and Hall, 1979.