Monte Carlo (importance) sampling within a benders decomposition algorithm for stochastic linear programs

This paper focuses on Benders decomposition techniques and Monte Carlo sampling (importance sampling) for solving two-stage stochastic linear programs with recourse, a method first introduced by Dantzig and Glynn [7]. The algorithm is discussed and further developed. The paper gives a complete presentation of the method as it is currently implemented. Numerical results from test problems of different areas are presented. Using small test problems, we compare the solutions obtained by the algorithm with universe solutions. We present the solutions of large-scale problems with numerous stochastic parameters, which in the deterministic formulation would have billions of constraints. The problems concern expansion planning of electric utilities with uncertainty in the availabilities of generators and transmission lines and portfolio management with uncertainty in the future returns.

[1]  J. F. Benders.  Partitioning procedures for solving mixed-variables programming problems , 1962 .

[2]  J. Hammersley,et al.  Monte Carlo Methods , 1965 .

[3]  R. Wets.  Programming Under Uncertainty: The Equivalent Convex Program , 1966 .

[4]  Philip Rabinowitz,et al.  Methods of Numerical Integration , 1985 .

[5]  Arthur M. Geoffrion,et al.  Elements of large-scale mathematical programming , 1969 .

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

[7]  Arthur M. Geoffrion,et al.  Elements of Large-Scale Mathematical Programming Part I: Concepts , 1970 .

[8]  P. Kall.  Computational methods for solving two-stage stochastic linear programming problems , 1979 .

[9]  P. Hall.  Fast rates of convergence in the central limit theorem , 1983 .

[10]  P. Hall.  Rates of convergence in the central limit theorem , 1983 .

[11]  Y. Ermoliev.  Stochastic quasigradient methods and their application to system optimization , 1983 .

[12]  John R. Birge,et al.  Decomposition and Partitioning Methods for Multistage Stochastic Linear Programs , 1985, Oper. Res..

[13]  R. Wets,et al.  Designing approximation schemes for stochastic optimization problems, in particular for stochastic programs with recourse , 1986 .

[14]  J. L. Nazareth,et al.  Algorithms for stochastic programs: The case of nonstochastic tenders , 1986 .

[15]  Andrzej Ruszczynski,et al.  A regularized decomposition method for minimizing a sum of polyhedral functions , 1986, Math. Program..

[16]  Y. Smeers,et al.  Optimal Investments for Electricity Generation: A Stochastic Model and a Test-Problem , 1988 .

[17]  Karl Frauendorfer,et al.  Solving SLP Recourse Problems with Arbitrary Multivariate Distributions - The Dependent Case , 1988, Math. Oper. Res..

[18]  J. Birge,et al.  A separable piecewise linear upper bound for stochastic linear programs , 1988 .

[19]  Peter Kall,et al.  Approximation Techniques in Stochastic Programming , 1988 .

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

[21]  Donald L. Iglehart,et al.  Importance sampling for stochastic simulations , 1989 .

[22]  George B. Dantzig,et al.  Decomposition techniques for multi-area generation and transmission planning under uncertainty: Final report , 1989 .

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

[24]  R. Tyrrell Rockafellar,et al.  Scenarios and Policy Aggregation in Optimization Under Uncertainty , 1991, Math. Oper. Res..

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