Probabilistic constraints in primal and dual linear programs

We present two pairs of dually related probabilistic constrained problems as extensions of the linear programming duality concept. In the first pair, a bilinear function appears in the objectives and each objective directly depends on the feasibility set of the other problem, as in the game theoretical formulation of dual linear programs. In the second pair, we reformulate the objectives and eliminate their direct dependence on the feasibility set of the other problem. We develop conditions under which the dually related problems have no duality gap and conditions under which the two pairs of problems are equivalent as far as their optimality sets are concerned.