On properties of the probabilistic constrained linear programming problem and its dual

In this paper, the two problems inf{inf{cx:x Rⁿ,A₁xy,A₂xb}:y suppF R^m,F(y)p} and sup{inf{uy:y suppF R^m,F(y)p}+vb:uA₁+vA₂=c, (u,v0} are investigated, whereA₁,A₂,b,c are given matrices and vectors of finite dimension,F is the joint probability distribution of the random variables ₁,...,_m, and 0<p<1. The first problem was introduced as the deterministic equivalent and the second problem was introduced as the dual of the probabilistic constrained linear programming problem inf{cx:P(A₁x)p,A₂xb}.b}. Properties of the sets and the functions involved in the two problems and regularity conditions of optimality are discussed.