Abstract
In this paper a regularized stochastic decomposition algorithm with master programs of finite size is described for solving two-stage stochastic linear programming problems with recourse. In a deterministic setting cut dropping schemes in decomposition based algorithms have been used routinely. However, when only estimates of the objective function are available such schemes can only be properly justified if convergence results are not sacrificed. It is shown that almost surely every accumulation point in an identified subsequence of iterates produced by the algorithm, which includes a cut dropping scheme, is an optimal solution. The results are obtained by including a quadratic proximal term in the master program. In addition to the cut dropping scheme, other enhancements to the existing methodology are described. These include (i) a new updating rule for the retained cuts and (ii) an adaptive rule to determine when additional reestimation of the cut associated with the current solution is needed. The algorithm is tested on problems from the literature assuming both descrete and continuous random variables.
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 and C.M. Shetty, Nonlinear Programming, Theory and Algorithms, John Wiley & Sons, Inc.: New York, NY, 1979.
J.R. Birge, “Decomposition and partitioning methods for multi-stage stochastic linear programs,” Oper. Res. vol. 33 pp. 989–1007, 1985.
B.C. Eaves and W.I. Zangwill, “Generalized cutting plane algorithms,” SIAM J. Control vol. 9 pp. 529–542, 1971.
H.I. Gassmann, “MSLiP: A computer code for the multistage stochastic linear programming problem,” Math. Prog. vol. 47 pp. 407–423, 1990.
J.L. Higle and S. Sen, “Stochastic decomposition: An algorithm for two stage linear programs with recourse,” Math. Oper. Res. vol. 16 pp. 650–669, 1991.
J.K. Ho and E. Loute, “A set of staircase linear programming test problems,” Math. Prog. vol. 20 pp. 245–250, 1981.
K.C. Kiwiel, “Methods of descent for nondifferentiable optimization,” Lecture Notes in Mathematics no. 1133, Springer-Verlag: Berlin, 1985.
F.V. Louveaux and Y. Smeers, “Optimal investment for electricity generation: A stochastic model and a test problem,” in Numerical Techniques for Stochastic Optimization, Y. Ermoliev and R.J.-B. Wets, Eds., Springer-Verlag: Berlin, 1988.
R.E. Marsten, XMP Technical Reference Manual, Department of Management Information Systems, College of Business and Public Administration: University of Arizona, Tucson, AZ, 1987.
R. Mifflin, “An algorithm for constrained optimization with semismooth functions,” Math. Oper. Res. vol. 2 pp. 191–207, 1977.
R. Mifflin, “A modification and an extension of Lemarechal's algorithm for nonsmooth minimization,” in Nondifferential and Variational Techniques in Optimization, D.C. Sorensen and R.J.-B Wets, Eds., Math. Prog. Study vol. 17 pp. 77–90, 1982.
M.J.D. Powell, ZQPCVX Algorithm, Department of Applied Mathematics and Theoretical Physics, University of Cambridge: Cambridge, England, 1986.
R.T. Rockafellar and R.J.-B. Wets, “A Lagrangian finite generation technique for solving linear-quadratic problems in stochastic programming,” Math. Prog. Study vol. 28 pp. 63–93, 1987.
A. Ruszczynski, “A regularized decomposition method for minimizing a sum of polyhedral functions,” Math. Prog. vol. 35 pp. 309–333, 1986.
A. Ruszczynski, “A linearization method for nonsmooth stochastic programming problems,” Math. Oper. Res. vol. 12 pp. 32–49, 1987.
R. Van Slyke and R.J.-B. Wets, “L-shaped linear programs with applications to optimal control and stochastic programming,” SIAM J. Appl. Math. vol. 17 pp. 638–663, 1969.
R.J.-B. Wets, “Stochastic programming: Solution techniques and approximation schemes,” in Mathematical Programming: The State of the Art, A. Bachem, M. Groetschel, B. Korte, Eds., Springer-Verlag: pp. 506–603, Berlin, 1982.
D.S. Yakowitz, “Two-stage stochastic linear programming: Stochastic decomposition approaches,” Ph.D. Dissertation, University of Arizona, 1991.
D.S. Yakowitz, “An exact penalty algorithm for recourse-constrained stochastic linear programs,” Appl. Math. and Comp. vol. 49 pp. 39–62, 1991.
J.L. Higle and S. Sen, “Finite Master Programs in Regularized Stochastic Decomposition,” Math. Prog. (to appear) 1994.
Author information
Authors and Affiliations
Additional information
A majority of this work is part of the author's Ph.D. dissertation prepared at the University of Arizona in 1990.
Rights and permissions
About this article
Cite this article
Yakowitz, D.S. A regularized stochastic decomposition algorithm for two-stage stochastic linear programs. Comput Optim Applic 3, 59–81 (1994). https://doi.org/10.1007/BF01299391
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01299391