A Class of Stochastic Programming Decision Problems

A class of probabilistic constrained programming problems are considered where the probabilistic constraint is of the form P{gi(x, ξ) ≥ 0, i = 1, . . . , r} ≥ p and the functions gi, i = 1, . . . , r are concave. It is shown that the x-function on the left hand side is logarithmic concave provided ξ has a logarithmic concave density. Special cases are mentioned and algorithmic solution of problems containing such constraint is discussed. 1 Remarks on logarithmic concave measures The notion of logarithmic concave probability measure was introduced in [3]. Let P be a probability measure defined on the measurable subsets of Rn. It is called logarithmic concave if for every pair of convex sets A, B ⊂ Rn and for every 0 < λ < 1 the following inequality holds P{λA+ (1− λ)B} ≥ (P{A})λ(P{B})1−λ. (1.1) It is proved in [3] that if P is a continuous probability measure the density of which is of the following form f(x) = e−Q(x), x ∈ R, (1.2) where Q(x) is a convex function in the entire n-dimensional space, then P is a logarithmic concave measure. The value +∞ is also allowed for the function Q. It follows from (1.1) that if D is an arbitrary convex subset of Rn then the function P{D + x} = ∫ D+x f(z) dz, x ∈ R (1.3) is logarithmic concave in the entire space (see [3]).