ON PROBABILISTIC CONSTRAINED PROGRAMMING

Abstract

the stochastic system as follows min f (x) subject to P (g i (x) ≥ β i , i = 1, . . ., m) ≥ p, a i x ≥ b i , i = 1, . . ., M.(1.2)The nonnegative constraints, if any, are thought to be contained in the system of linear inequalities in (1.2).We introduce the following assumptions.A.1.The functions g 1 (x), . . ., g m (x) are defined in the closure K of an open convex set K. These functions are concave and have continuous derivatives with respect to all variables in K.A.2.The number p is between 0 and 1, 0 < p < 1.A.3.If x ∈ K and x satisfies the constraints in (1.2), then x is an internal point of K, i.e., x ∈ K.A.4.The function f (x) is defined in an open convex set H containing the set of feasible solutions and we suppose that f (x) is convex and has continuous derivatives with respect to all variables in every point of H.A.5.The random variables β 1 , . . ., β m have a continuous joint distribution which has continuous first order derivatives with respect to all variables in any point of the m-dimensional space of the form (g 1 (x), . . ., g m (x)), x∈ K.We consider only such probabilities p, for which the set of feasible solutions is not empty.This set will be denoted by D(p) in the sequel.Let F (z 1 , . . ., z m ) denote the joint probability distribution function of the random variables: β 1 , . . ., β m , i.e.,and let furtherA.6.For every x satisfying the equality G(x) = p, there corresponds a vector y in the set of feasible solutions with the property that 2 ∇G(x)(yx) > 0.(1.3)

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