Abstract
It is proved a sufficient condition that the optimal value of a linear program be a continuous function of the coefficients. The condition isessential, in the sense that, if it is not imposed, then examples with discontinuous optimal-value function may be found. It is shown that certain classes of linear programs important in applications satisfy this condition. Using the relation between parametric linear programming and the distribution problem in stochastic programming, a necessary and sufficient condition is given that such a program has optimal value. Stable stochastic linear programs are introduced, and a sufficient condition of such stability, important in computation problems, is established.
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References
Dantzig, G. B., Folkman, J., andShapiro, N.,On the Continuity of the Minimum Set of a Continuous Function, Journal of Mathematical Analysis and Applications, Vol. 17, pp. 519–548, 1967.
Gale, D.,The Theory of Linear Economic Models, McGraw-Hill Book Company, New York, New York, 1960.
Dantzig, G. B.,Linear Programming and Extensions, Princeton University Press, Princeton, New Jersey, 1963.
Stoer, J., andWitzgall, C.,Convexity and Optimization in Finite Dimensions, I, Springer-Verlag, Berlin, Germany, 1970.
Nicolescu, M.,Analiză Matematică, Vol. 2 (in Romanian), Editura Technica, Bucharest, Romania, 1958.
Bereanu, B.,The Distribution Problem in Stochastic Linear Programming. The Cartesian Integration Method, Academy of the Romanian Socialist Republic, Center of Mathematical Statistics, Report No. 71-03, 1971.
Karlin, S.,Mathematical Methods and Theory in Games, Programming, and Economics, Vol. 1, Addison-Wesley Publishing Company, Cambridge, Massachusetts, 1959.
Temelt, W.,On the Space of Parameters of the General Problem of Linear Programming (in Russian), Siberian Mathematical Journal, Vol. 8, No. 3, 1967.
Williams, A. C.,Marginal Values in Linear Programming, SIAM Journal on Applied Mathematics, Vol. 11, No. 1, 1963.
Van Cutsem, B.,Elements Aléatoires a Valeurs Convexes Compactes, Université Scientifique et Medicale de Grenoble, PhD Thesis, 1971.
Dantzig, G. B.,Linear Programming under Uncertainty, Management Science, Vol. 1, pp. 197–206, 1955.
Charnes, A., andCooper, W. W.,Chance-Constrained Programming, Management Science, Vol. 6, No. 1, 1959.
Tintner, G.,Stochastic Linear Programming with Applications to Agricultural Economics, Proceedings of the Second Symposium in Linear Programming, Vol. 1, Edited by H. A. Antosiewicz, National Bureau of Standards and Directorate of Management Analysis, DCS/Comptroller, USAF, Washington, D.C., 1955.
Cramer, H.,Random Variables and Probability Distributions, Cambridge University Press, Cambridge, Massachusetts, 1970.
Bereanu, B.,On Stochastic Linear Programming, II, Distribution Problems: Nonstochastic Technology Matrix, Revue Roumaine des Mathématiques Pures et Appliquées, Vol. 11, No. 6, 1966.
Bereanu, B.,On Stochastic Linear Programming, Distribution Problems: Stochastic Technology Matrix, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, Vol. 8, pp. 148–152, 1967.
Bereanu, B.,Renewal Processes and Some Stochastic Programming Problems in Economics, SIAM Journal on Applied Mathematics, Vol. 19, No. 2, 1970.
Babbar, M. M.,Distribution of Solutions of a Set of Linear Equations (with an Application to Linear Programming), Journal of the American Statistical Association, Vol. 50, pp. 854–869, 1955.
Dempster, M. A. H.,Distribution in Intervals and Linear Programming, Interval Analysis, Proceedings of the Oxford University Computing Laboratory Symposium, Edited by E. Hansen, Oxford University Press, London, England, 1969.
Wets, R.,Programming Under Uncertainty: the Solution Set, SIAM Journal on Applied Mathematics, Vol. 14, No. 5, 1966.
Prekopa, A.,On the Probability Distribution of the Optimum of a Random Linear Program, SIAM Journal on Control, Vol. 4, No. 1, 1966.
Wilks, S. S.,Mathematical Statistics, John Wiley and Sons, New York, New York, 1962.
Bereanu, B.,The Cartesian Integration Method in Stochastic Linear Programming, Numerische Methoden bei Optimierungsaufgaben, Edited by L. Collatz and W. Wetterling, Birkhäuser Verlag, Stuttgart, Germany, 1973.
Rockafellar, R. T.,Duality and Stability in Extremum Problems Involving Convex Functions, Pacific Journal of Mathematics, Vol. 21, pp. 167–187, 1967.
Evans, J. P., andGould, F. J.,Stability in Nonlinear Programming, Operations Research, Vol. 18, No. 1, 1970.
Krabs, W.,Zur Stetigen Abhängigkeit des Extremalwertes Eines Konvexen Optimierungsproblems von Einer Stetigen Änderung des Problems, Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 52, pp. 359–368, 1972.
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Communicated by S. Karamardian
This note is a slightly modified version of a paper presented at the Institute of Econometrics and Operations Research of the University of Bonn, Bonn, Germany, 1972.
The author is grateful to G. B. Dantzig and S. Karamardian for useful comments on an earlier draft of this paper. In particular, S. Karamardian proposed modifications which made clearer the proof of Lemma 2.1.
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Bereanu, B. The continuity of the optimum in parametric programming and applications to stochastic programming. J Optim Theory Appl 18, 319–333 (1976). https://doi.org/10.1007/BF00933815
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DOI: https://doi.org/10.1007/BF00933815