On properties of the probabilistic contrained linear programming problem and its dual (Q1821690)

The two problems inf\(\{\) inf\(\{\) cx: \(x\in R^ n\), \(A_ 1x\geq y\), \(A_ 2x\geq b\}:\) \(y\in \sup p F\subset R^ m\), F(y)\(\geq p\}\) and sup\(\{\) inf\(\{\) uy: \(y\in \sup p F\subset R^ m\), \(F(y)\geq p\}+vb:\) \(uA_ 1+vA_ 2=c\), (u,v)\(\geq 0\}\) are investigated, where \(A_ 1\), \(A_ 2\), b, c are given matrices and vectors of finite dimension, F is the joint probability distribution of the random variables \(\beta _ 1,...,\beta _ m\), 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: \(P(A_ 1x\geq \beta)\geq p\), \(A_ 2x\geq b\}\). Properties of the sets and the functions involved in the two problems and regularity conditions of optimality are discussed.