Statistical approximations for recourse constrained stochastic programs

The two stage stochastic program with recourse is known to have numerous applications in financial planning, energy modeling, telecommunications systems etc. Notwithstanding its applicability, the two stage stochastic program is limited in its ability to incorporate a decision maker's attitudes towards risk. In this paper we present an extension via the inclusion of a recourse constraint. This results in a convex integrated chance constraint (ICC), which inherits the convexity properties of two stage programs. However, it also inherits some of the difficulties associated with the evaluation of recourse functions. This motivates our study of conditions that may be applicable to algorithms using statistical approximations of such ICC. We present a set of sufficient conditions that these approximations may satisfy in order to assure convergence. Our conditions are satisfied by a wide range of statistical approximations, and we demonstrate that these approximations can be generated within standard algorithmic procedures.

[1]  R. Wets,et al.  L-SHAPED LINEAR PROGRAMS WITH APPLICATIONS TO OPTIMAL CONTROL AND STOCHASTIC PROGRAMMING. , 1969 .

[2]  George B. Dantzig,et al.  Parallel processors for planning under uncertainty , 1990 .

[3]  R. Wets,et al.  Epi‐consistency of convex stochastic programs , 1991 .

[4]  Julia L. Higle,et al.  Stochastic Decomposition: An Algorithm for Two-Stage Linear Programs with Recourse , 1991, Math. Oper. Res..

[5]  Diana Schadl. Yakowitz.  Two-stage stochastic linear programming: Stochastic decomposition approaches. , 1991 .

[6]  W. Rudin.  Principles of mathematical analysis , 1964 .

[7]  Yuri Ermoliev,et al.  Stochastic quasigradient methods. Numerical techniques for stochastic optimization , 1988 .

[8]  Yuri Ermoliev,et al.  Numerical techniques for stochastic optimization , 1988 .

[9]  A. Charnes,et al.  Chance-Constrained Programming , 1959 .

[10]  Gerd Infanger,et al.  Monte Carlo (importance) sampling within a benders decomposition algorithm for stochastic linear programs , 1991, Ann. Oper. Res..

[11]  Linus Schrage,et al.  OR Practice - A Scenario Approach to Capacity Planning , 1989, Oper. Res..

[12]  Julia L. Higle,et al.  On the Convergence of Algorithms with Implications for Stochastic and Nondifferentiable Optimization , 1992, Math. Oper. Res..

[13]  András Prékopa,et al.  Numerical Solution of Probabilistic Constrained Programming Problems , 1988 .

[14]  R. Horst,et al.  Global Optimization: Deterministic Approaches , 1992 .

[15]  E. Beale.  ON MINIMIZING A CONVEX FUNCTION SUBJECT TO LINEAR INEQUALITIES , 1955 .

[16]  Julia L. Higle,et al.  Conditional Stochastic Decomposition: An Algorithmic Interface for Optimization and Simulation , 1994, Oper. Res..

[17]  W. K. Haneveld.  Duality in Stochastic Linear and Dynamic Programming , 1986 .

[18]  András Prékopa,et al.  Contributions to the theory of stochastic programming , 1973, Math. Program..