Abstract
It is shown that duality in mathematical programming can be treated as a purely order theoretic concept which leads to some applications in economics. Conditions for strong duality results are given. Furthermore the underlying sets are endowed with (semi-)linear structures, and the perturbation function of arising linear and integer problems, which include bottleneck problems and extremal problems (in the sense of K. Zimmermann), is investigated.
Zusammenfassung
In dieser Arbeit wird aufgezeigt, daß Dualitätskonzepte der mathematischen Optimierung in ordnungstheoretischem Rahmen beschrieben werden können. Dies führt u.a. auf neue Anwendungen in der Ökonomie. Ferner werden Bedingungen hergeleitet, unter denen starke Dualitätsaussagen gelten. Sodann werden die zugrundeliegenden Mengen mit algebraischen Strukturen versehen und es werden Dualitätssätze für lineare und ganzzahlige Programme über diesen Mengen bewiesen. Darunter fallen nicht nur die klassischen linearen und ganzzahligen Programme, sondern auch Probleme mit Engpaßzielfunktion und „extremale Probleme“ im Sinne von K. Zimmermann.
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References
Ben-Tal, A., andA. Ben-Israel:F-convex Functions: Properties and Applications. Generalised Concavity in Optimization and Economics. Ed. by S. Schaible and W.C. Ziemba. New York 1981, 301–334.
Blair, C.E., andR.G. Jeroslow: The Value Function of an Integer Program. Mathematical Programming23, 1982, 237–273.
Burkard, R.E., andU. Zimmermann: Combinatorial Optimization in Linearly Ordered Semimodules: A Survey. Modern Applied Mathematics: Optimization and Operations Research. Ed. by B. Korte. Amsterdam 1982, 391–436.
Dolecki, S., andS. Kurcyusz: On Φ-Convexity in Extremal Problems. SIAM Journal on Control and Optimization16, 1978, 277–300.
Elster, K.H., andR. Nehse: Zur Theorie der Polarfunktionale. Mathematische Operationsforschung und Statistik5, 1974, 3–21.
Everett, H.: Generalized Lagrange Multiplier Method for Solving Problems of Optimum Allocation of Resources. Operations Research11, 1963, 399–417.
Evers, J.M.M., andH. van Maaren: Duality Principles in Mathematics and Their Relations to Conjugate Functions. Memorandum No 336, Department of Applied Mathematics, Twente University of Technology, Enschede 1981.
Gould, F.J.: Extensions of Lagrange Multipliers in Nonlinear Programming. SIAM Journal on Applied Mathematics17, 1969, 1280–1297.
—: Nonlinear Duality Theorems. Cahiers du Centre d'Etudes de Recherche Opérationelle14, 1972, 196–212.
Hoffman, A.J.: On Abstract Dual Linear Programs. Naval Research Logistics Quarterly10, 1963, 369–373.
Jeroslow, R.G.: Cutting Plane Theory: Algebraic Methods. Discrete Mathematics23, 1978, 121–150.
Johnson, E.L.: Cyclic Groups, Cutting Planes and Shortest Paths. Mathematical Programming. Ed. by T.C. Hu and S. Robinson. New York 1973, 185–211.
Koopmans, T.C.: Concepts of Optimality and Their Uses. Nobel Memorial Lecture, December 1975. Mathematical Programming11, 1976, 212–228.
Luce, R.D., andH. Raiffa: Games and Decisions. New York 1957.
Meyer, R.R.. On the Existence of Optimal Solutions to Integer and Mixed Integer Programming Problems. Mathematical Programming7, 1974, 223–235.
Rockafellar, R.T.: Convex Analysis. Princeton, 1970.
—: Augmented Lagrange Multiplier Functions and Duality in Nonconvex Programming. SIAM Journal on Control12, 1974, 268–283.
Tind, J.: On Duality in Nonconvex and Integer Programming. Operations Research Verfahren32, 1979, 193–201.
Tind, J., andL.A. Wolsey: An Elementary Survey of General Duality Theory in Mathematical Programming. Mathematical Programming21, 1981, 241–261.
Wolsey, L.A.: Integer Programming Duality: Price Functions and Sensitivity Analysis. Mathematical Programming20, 1981, 173–194.
Zimmermann, K.: Conjugate Optimization Problems and Algorithms in the Extremal Vector Space. Ekonomicko-mathematický obzor10, 1974, 428–439.
Zimmermann, U.: On Some Extremal Optimization Problems. Ekonomicko-mathematický obzor15, 1979, 438–442.
—: Duality for Algebraic Linear Programming. Linear Algebra and Its Applications32, 1980, 9–31.
-: Linear and Combinatorial Optimization in Ordered Algebraic Structures. Annals of Discrete Mathematics 10. Amsterdam 1981.
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This paper was partially supported by the NATO Research Grants Programme under SRG 8.
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Burkard, R.E., Hamacher, H. & Tind, J. On abstract duality in mathematical programming. Zeitschrift für Operations Research 26, 197–209 (1982). https://doi.org/10.1007/BF01917114
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DOI: https://doi.org/10.1007/BF01917114