On an ε-Solution of Minimax Problem in Stochastic Programming

Let (Ω, S, P) be a probability space; ξ = ξ(ω) be an s-dimensional random vector defined on (Ω, S, P) with the distribution function F ξ(z), z ∈ E s; E Fξ denote the corresponding operator of the mathematical expectation; g(x, z) be real-valued, continuous function defined on E n × E s; X ⊂ E n be a nonempty set (E n, n ≥ 1 denotes the n-dimensional Euclidean space). The stochastic programming problem find $$\underset{E}{\textup f } \ E_F \xi g(x,\xi(\omega))$$ belongs to the problems depending on a probability measure. If F ξ(·) is known, then (1) is a problem of the deterministic optimization. Since such assumption is satisfied only rarely, very often it is necessary to seek for a suitable approach. Of course, if some statistical estimate of F ξ(·) is known, then it can replace the theoretical F ξ(·) to obtain at least estimates of the ptimal value and the optimal solution (see e.g. [5], [9]). If it is possible to assume that F ξ(·) belongs to some class of the distribution functions, then the minimax approach can be employed (see e. g. [3], [11]). In this note we shall deal with the second case. To this end we assume: 1. there exists a compact set Z ⊂ E s such that P {ω: ξ(ω) G Z} = 1, 2. there exist real-valued (continuous) functions g j(z), z ∈ E s, j = 1,2, …, l and constants $$\underline c = (\underline c_1,...\underline c_l),\bar c = (\underline c_1,...\underline c_l), \underline c_j,\bar c\epsilon E_1, \underline c_j\leq \bar c_j, j = 1, 2,..., l$$ such that $$E_F\xi gi(\xi()\omega ) \epsilon \left \langle \underline c_j, \underline c_j, \right \rangle j = 1,2,..., l$$ . Surely, in such situation it is reasonable to consider the problem: Find $$\underset{x \in X}{\textup f } \ \underset{F \in \mathcal{F}(\underline c, \overline c)}{\textup{max}} \ E_Fg(x, \xi(\omega))$$ , $$\mathcal{F}(\underline c, \overline c)= \{F: F \ \textup an \ s-\textup{dimensional distribution function such that the conditions 1. and 2. are fulfilled}\}$$ .