Convex Stochastic Duality and the ÔBiting LemmaÔ

The paper analyzes dynamic problems of stochastic optimization in discrete time. The problems under consideration are concerned with maximizing concave functionals on convex sets of feasible strategies (programs). Feasibility is defined in terms of linear inequality constraints in L∞ holding almost surely. The focus of the work is the existence of dual variables – stochastic Lagrange multipliers in L1 – relaxing the constraints. Such Lagrange multipliers are important in various applications. In particular, they play key roles in the analysis of stability and sensitivity of solutions to stochastic optimization problems, as well as in the design of algorithms for computing these solutions [14]. Also, such multipliers often have clear interpretations, especially in models related to economics, which sheds additional light on the issues under study [1].