Abstract
Optimal value functions in parametric programming are studied as compositions of objective functions and point-to-set mappings which define the constrained sets. Sufficiency conditions for the common regularity properties of optimal value functions are derived.
Zusammenfassung
Der Optimalwert eines parametrischen Programms wird aufgefaßt als Verknüpfung der Zielfunktion mit einer mengenwertigen Abbildung, die den zulässigen Bereich definiert. Es werden hinreichende Bedingungen für einige häufig benutzte Regularitätseigenschaften der Optimalwertfunktion hergeleitet.
Similar content being viewed by others
References
Berge, C. Topological Spaces. Macmillan, New York 1963.
Dantzig, G.B., J. Folkman, andN. Shapiro: On the Continuity of the Minimum Set of Continuous Functions. Journal of Mathematical Analysis and Applications, Vol.17, 1967, 519–548.
Elton, E.J., andM.J. Gruber: Modern Portfolio Theory and Investment Analysis. John Wiley and Sons, Inc., 1981.
Evans, J.P., andF.J. Gould: Stability in Nonlinear Programming. Operations Research, Vol.18, 1970, 107–118.
Gal, T., andJ. Nedoma: Multiparametric Linear Programming. Management Science, Vol.18, 1972, 406–422.
Hogan, W.W.: Point-to-Set Maps in Mathematical Programming. SIAM Review, Vol.15, 1973, 591–603.
Huang, C.C., D.A. Wehrung, andW.T. Ziemba: A Homogeneous Distribution Problem with Applications to Finance. Management Science, Vol.23, 1976, 297–304.
Mangasarian, O.L.: Convexity, Pseudo-Convexity and Quasi-Convexity of Composite Functions. Cahiers du Centre d'Etudes de Recherche Opérationelle, Vol.12, 1970, 114–122.
Oettli, W.: Optimality Conditions Involving Generalized Convex Mappings. In: S. Schaible and W.T. Ziemba, eds., Generalized Concavity in Optimization and Economics, Academic Press, New York 1981.
Ponstein, J.: Seven Kinds of Convexity. Siam Review, Vol.9, 1967, 115–119.
Robinson, S.M.: Regularity and Stability for Convex Multivalued Functions. Mathematics of Operations Research, Vol.1, 1976, 130–142.
Zang, I., andM. Avriel: On Functions Whose Local Minima Are Global. Journal of Optimization Theory and Applications, Vol.16, 1975, 183–190.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Choo, E.U., Chew, K.P. Optimal value functions in parametric programming. Zeitschrift für Operations Research 29, 47–57 (1985). https://doi.org/10.1007/BF01920495
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01920495