Abstract
Two stochastic programming decision models are presented. In the first one, we use probabilistic constraints, and constraints involving conditional expectations further incorporate penalties into the objective. The probabilistic constraint prescribes a lower bound for the probability of simultaneous occurrence of events, the number of which can be infinite in which case stochastic processes are involved. The second one is a variant of the model: two-stage programming under uncertainty, where we require the solvability of the second stage problem only with a prescribed (high) probability. The theory presented in this paper is based to a large extent on recent results of the author concerning logarithmic concave measures.
Similar content being viewed by others
References
E.M.L. Beale, “On minimizing a convex function subject to linear inequalities”,Journal of the Royal Statistical Society Ser. B 17 (1955) 173–184.
E.F. Beckenbach and R. Bellman,Inequalities (Springer, Berlin, 1961).
A. Charnes, W.W. Cooper and G.H. Symonds, “Cost horizons and certainty equivalents: an approach to stochastic programming of heating oil production”,Management Science 4 (1958) 236–263.
G.B. Dantzig, “Linear programming under uncertainty”,Management Sciénce 1 (1955) 197–206.
G.B. Dantzig and A. Madansky, “On the solution of two stage linear programs under uncertainty”, in:Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability (University of California Press, Berkeley, Calif., 1960) 165–176.
J.L. Doob,Stochastic processes (Wiley, New York, 1953).
A.V. Fiacco and G.P. McCormick,Nonlinear programming: Sequential unconstrained minimization technique (Wiley, New York, 1968).
S.S. Gupta, “Probability integrals of multivariate normal and multivariatet”,Annals of Mathematical Statistics 34 (1963) 829–838.
A. Prékopa, “On probabilistic constrained programming”, in:Proceedings of the Princeton Symposium on Mathematical Programming (Princeton University Press, Princeton, N.J., 1970) 113–138.
A. Prékopa, “Logarithmic concave measures with application to stochastic programming”,Acta Scientiarum Mathematicarum (Szeged) 32 (1971) 301–316.
A. Prékopa, “On logarithmic concave measures and functions”,Acta Scientiarum Mathematicarum (Szeged), to appear.
R. Wets, “Programming under uncertainty: the solution set”,SIAM Journal of Applied Mathematics 14 (1966) 1143–1151.
S.S. Wilks,Mathematical statistics (Wiley, New York, 1962).
G. Zoutendijk,Methods of feasible directions (Elsevier, Amsterdam, 1960).
Author information
Authors and Affiliations
Additional information
This work was supported in part by the Institute of Economic Planning, Budapest.
Rights and permissions
About this article
Cite this article
Prékopa, A. Contributions to the theory of stochastic programming. Mathematical Programming 4, 202–221 (1973). https://doi.org/10.1007/BF01584661
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01584661