Abstract
We present alternative methods for verifying the quality of a proposed solution to a two stage stochastic program with recourse. Our methods revolve around implications of a dual problem in which dual multipliers on the nonanticipativity constraints play a critical role. Using randomly sampled observations of the stochastic elements, we introduce notions of statistical dual feasibility and sampled error bounds. Additionally, we use the nonanticipativity multipliers to develop connections to reduced gradient methods. Finally, we propose a statistical test based on directional derivatives. We illustrate the applicability of these tests via some examples.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M.S. Bazaraa, C.M. Shetty and H.D. Sherali,Nonlinear Programming: Theory and Algorithms (Wiley, New York, 1992).
J.F. Benders, “Partitioning procedures for solving mixed variables programming problems,”Numerische Mathematik 4 (1962) 238–252.
J.R. Birge, “Aggregation in stochastic linear programming,”Mathematical Programming 31 (1985) 25–41.
F.H. Clarke,Optimization and Nonsmooth Analysis, (Wiley, New York, 1983).
G.B. Dantzig and P.W. Glynn, “Parallel processors for planning under uncertainty,”Annals of Operations Research 22 (1990) 1–21.
M.A.H. Dempster, “On stochastic programming II: Dynamic problems under risk,”Stochastics 25 (1988) 15–42.
B. Efron, “Bootstrap methods: Another look at the jackknife,”Annals of Statistics 7 (1979) 1–26.
Y. Ernoliev, “Stochastic quasigradient methods,” in: Y. Ermoliev and R.J.-B. Wets, eds.,Numerical Techniques for Stochastic Optimization, (Springer, Berlin, 1988) pp. 141–185.
Y. Ermoliev and R.J.-B. Wets, eds.,Numerical Techniques for Stochastic Optimization (Springer, Berlin, 1988).
K. Frauendorfer and P. Kall, “A solution method for SLP recourse problems with arbitrary multivariate distributions—The independent case,”Problems in Control and Information Theory 17 (1988) 177–205.
J.L. Higle and S. Sen, “Stochastic decomposition: An algorithm for two-stage linear programs with recourse.”Mathematics of Operations Research 16 (1991a) 650–669.
J.L. Higle and S. Sen, “Statistical verification of optimality conditions,”Annals of Operations Research 30 (1991b) 215–240.
J.L. Higle and S. Sen, “On the convergence of algorithms, with implications for stochastic and nondifferentiable optimization,”Mathematics of Operations Research 17 (1992) 112–131.
J.L. Higle and S. Sen, “Finite master programs in regularized stochastic decomposition,”Mathematical Programming 67 (1994) 143–168.
G. Infanger, “Monte Carlo (importance) sampling within a Benders' decomposition for stochastic linear programs,”Annals of Operations Research 39 (1992) 69–95.
G. Infanger,Planning under Uncertainty—Solving Large-scale Stochastic Linear Programs (The Scientific Press Series, Boyd & Fraser, New York, 1994).
F.V. Louveaux and Y. Smeers, “Optimal investment for electricity generation: A stochastic model and a test problem,” in: Y. Ermoliev and R.J.-B. Wets, eds.,Numerical Techniques for Stochastic Optimization (Springer, Berlin, 1988) pp. 445–453.
J.M. Mulvey and A. Ruszczyński, “A new scenario decomposition method for large scale stochastic optimization,”Operations Research 43 (1995) 477–490.
G.C. Pflug, “Stepsize rules, stopping times, and their implementation in stochastic quasigradient algorithms,” in: Y. Ermoliev and R.J.-B. Wets, eds.,Numerical Techniques for Stochastic Optimization, (Springer, Berlin, 1988) pp. 353–372.
R.T. Rockafellar,Convex Analysis (Princeton University Press, Princeton, NJ, 1970).
R.T. Rockafellar and R.J.-B. Wets, “Stochastic convex programming: Basic duality,”Pacific Journal of Mathematics 62 (1976a) 173–195.
R.T. Rockafellar and R.J.-B. Wets, “Stochastic convex programming: Relatively complete recourse and induced feasibility,”SIAM Journal on Control and Optimization 14 (1976b) 574–589.
R.T. Rockafellar and R.J.-B. Wets”Nonanticipativity and L 1-martingales in stochastic optimization problems,”Mathematical Programming Study 6 (1976c) 170–186.
R.T. Rockafellar and R.J.-B. Wets,”Scenarios and policy aggregation in optimization under uncertainty”,Mathematics of Operations Research 16 (1991) 1–29.
R.T. Rockafellar and R.J.-B. Wets, “A dual strategy for the implementation of the aggregation principle in decision making under uncertainty,”Applied Stochastic Models and Data Analysis 8 (1992) 245–255.
A. Ruszczyński, “A regularized decomposition method for minimizing a sum of polyhedral functions,”Methamatical Programming 35 (1986) 309–333.
S. Sen, R.D. Doverspike and S. Cosares, “Network planning with random demand,”Telecommunications Systems 3 (1994) 11–30.
A. Takayama,Mathematical Economics (Dryden Press, Hinsdale, IL, 1974).
R. Van Slyke and R.J.-B. Wets, “L-shaped linear programs with application to optimal control and stochastic programming,”SIAM Journal on Applied Mathematics 17 (1969) 638–663.
R.J.-B. West, “On the relation between stochastic and deterministic optimization,” in: A. Bensoussan and J.L. Lions, eds.,Control Theory, Numerical Methods and Computer Systems Modeling Lecture Notes in Economics and Mathematical Systems, Vol. 107 (Springer, Berlin, 1975) pp. 350–361.
R.J.-B. Wets, “Stochastic programming,” in: G.L. Nemhauser, A.H.G. Rinnooy Kan and M.J. Todd, eds.,Handbooks in Operations Research: Optimization (North-Holland, Amsterdam, 1989) pp. 573–629.
P. Zipkin, “Bounds on the effect of aggregating variables in linear programming,”Operations Research 28 (1980a) 403–418.
P. Zipkin, “Bounds for row aggregation in linear programming,”Operations Research 28 (1980b) 903–918.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported in part by Grant No. NSF-DMI-9414680 from the National Science Foundation
Rights and permissions
About this article
Cite this article
Higle, J.L., Sen, S. Duality and statistical tests of optimality for two stage stochastic programs. Mathematical Programming 75, 257–275 (1996). https://doi.org/10.1007/BF02592155
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02592155