ABOUT APPROACH TO OPTIMIZATION WITH PROBABILISTIC UNCERTAINTY USING ORDERING OF RANDOM VARIABLES - Barbolina, T. M. (Computer Science and Applied Mathematics, 2016)
The problems with uncertainty, including probabilistic, attract the attention of researchers recently. Different ways have been proposed to construct constraints and criterions of stochastic optimization problems. For instance wide class of the stochastic models may be written in form of minimization of mathematical expectation of objective function on some domain. At the same time such approach doesn’t allow to compare different solutions if corresponding value of objective function has equal mathematical expectations. Specifically such problems arise in combinatorial optimization. In the paper an example of problem is given. One possible approach for formalization of optimization problems under interval, fuzzy and probabilistic uncertainty would be to use a certain order on the set of relevant variables. In particular an order on a set of discrete random variables proposed earlier is defined through the comparison of the mathematical expectations, dispersions, possible values and associated probabilities. However in many practical problems probabilistic distribution is unknown. Therefore such order cannot be used. In this paper author proposes ordering through the comparison of numerical characteristics of random variables. Let the characteristic vector of the random variable be a vector which components are numerical characteristics of the random variable. We mainly consider characteristic vector which components are initial moments of a random variable, i.e. mathematical expectation of its n-s power. Random variables with equal characteristic vectors are called H-equivalent. H-equivalence relation on a set of random variables is an equivalence relation. Let the characteristic vector of equivalence class is a characteristic vector of its representative. We will call two equivalence classes organized in ascending (nondecreasing) order, if characteristic vector of the first class lexicographically less (less or equal) then characteristic vector of the second one. From properties of lexicographic order it follows that introduced order is linear one. We define sum of two equivalence classes to be the class which contains sum of its representatives. We prove that this definition is correct. If random variables are independent and components of characteristic vectors are initial moments of a random variable then introduced order possesses property important for some practical problems. Ordering of two equivalency classes remained at addition to the left and right part of a relation of the same equivalence classes. Also we prove some corollaries of this property. Consider finite subset Ω of the set of independent random variables. Using introduced linear order, let us order the elements of the factor set of Ω relative to H- equivalence. The first random variable in this ordered list is the minimum value and the last one is the maximum value. The definition of the minimum and maximum allows setting the optimization problem for finding the extreme elements in the given conditions. Statement of optimization problems on factor set of Ω relative to H- equivalence in the general case requires to define operations with equivalence classes. At the same time if operation with classes defines through operation with their representatives then result may depend from choose of classes’ elements. Since statement of optimization problems in the general case is difficult author formulate one class of problems. This statement uses sum of equivalence classes and multiplication class on real number. Also we assume that components of characteristic vector are initial moments of a random variable. If mathematical expectations of objective function and constraint are different under all feasible points then statements of optimization problems proposed in the paper is equivalent to problem of optimization of mathematical expectation under statistical constraints. Studying of optimization problems on linear ordered sets allows to use branch and bound method for its solving. Subsequent studies suggest further study of the properties of the considered problems.