Application of the second-order optimization methods to the stochastic programming problems with probability function


DOI: 10.34759/trd-2021-121-17

Аuthors

Torishniy R. O.

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

e-mail: arenas-26@yandex.ru

Abstract

The author considers the application of the second-order optimization algorithms for stochastic optimization problems with the probability function as the objective or/and constraint function. The approximation of the probability function is based on the replacement of the Heaviside function with its smooth analog — the sigmoid function. It has been shown previously that such approximation and its first order derivatives with respect to the elements of the control vector converge to the exact ones. Moreover, the replacement of the probability function with its smooth approximation within the stochastic optimization problem leads to a good approximation of the optimal control vector and the optimal value of the target function. The smooth approximation of the derivatives allows us to use the first-order optimization algorithms. Now the direct formulas for the second order derivatives of the approximated probability function with respect to the elements of the control vector are provided. The proof of convergence of the second-order derivatives is not considered in this research. Possible applications of such approximations include the development of the new numerical algorithms for solving stochastic optimization problems, and new algorithms to determine the surface level of the probability function.

Some numerical examples are considered in this article. For the cases of linear, quadratic, and logarithmic loss functions it was shown that the values of the smooth approximation of the probability function and their derivatives tend to exact values as the parameter in the sigmoid function tends to infinity. Also, an example of the constrained stochastic optimization problem with the logarithmic loss function and the probability function as the target was considered. The modification of Newton’s method is used to solve this problem to determine an optimal investment portfolio with three possible assets.

Keywords:

phase detector, noise immunity, radio pulse, narrowband noise

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