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This article is cited in 4 scientific papers (total in 4 papers)
Mathematical Modeling, Numerical Methods
On smooth approximation of probabilistic criteria in stochastic programming problems
V. R. Sobol, R. O. Torishnyi Moscow Aviation Institute (National Research University)
Abstract:
One of the possible variants of smooth approximation of probability criteria in
stochastic programming problems allowing to obtain estimates of the probability function gradient
and the quantile function gradient in the form of a volume integral is considered. The research is
applied to problems of probability function maximization and quantile function minimization for
the loss functional depending on the control vector and one-dimensional absolutely continuous
random variable.
The main idea of the approximation is to replace the discontinuous Heaviside function in the
integral representation of the probability function with a smooth function having such properties
as continuity, smoothness, and easily computable derivatives. An example of such function is
the distribution function of a random variable distributed according to the logistic law with zero
mean and finite dispersion, which is a sigmoid. The value inversely proportional to the root of the
variance is a parameter that provides the proximity of the original function and its approximation.
This replacement allows to obtain a smooth approximation of the probability function, and for this
approximation derivatives by the control vector and by other parameters of the problem can be
easily found.
The main result of the article is the obtained expressions for approximation of the probability
function derivatives by the control vector and by the acceptable loss level, as well as expressions for
approximation of the quantile function gradient in the form of volume integrals. The article proves
the convergence of the probability function approximation obtained by replacing the Heaviside
function with the sigmoidal function to the original probability function, and the error estimate of
such approximation is obtained. The convergence of the approximation of probability function
derivatives to the true derivatives under a number of conditions on the loss functional is also
proved.
Examples are considered to demonstrate the possibility of applying the proposed estimates to
the solution of stochastic programming problems with criteria in the form of a probability function
and a quantile function, including the case of a multidimensional random variable
Keywords:
stochastic programming, probability criteria, quantile criteria, approximation, numerical methods, sigmoidal function.
Received: 05.08.2019
Citation:
V. R. Sobol, R. O. Torishnyi, “On smooth approximation of probabilistic criteria in stochastic programming problems”, Tr. SPIIRAN, 19:1 (2020), 180–217
Linking options:
https://www.mathnet.ru/eng/trspy1096 https://www.mathnet.ru/eng/trspy/v19/i1/p180
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