Abstract
In this paper we present a general concept for the formulation of the dual program which is based on generalized convexity. This is done in a purely algebraic way where no topological assumptions are made. Moreover all proofs are presented in an extreme simple way. A complete presentation of this subject can be found in the book of D. Pallaschke and S. Rolewicz [14].
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References
Balder EJ (1977) An extension of duality — stability relations to nonconvex optimization problems. SIAM Jour. Contr. Optim. 15:329–343
Clarke FH (1989) Methods of dynamic and nonsmooth optimization. SIAM Regional Conference Series in Applied Mathematics, Capital City Press, Montpellier, Vermont
Dolecki S (1978) Semicontinuity in constrained optimization I. Metric spaces. Control and Cybernetics 7(2):5–16
Dolecki S (1978b) Semicontinuity in constrained optimization Ib. Normed spaces. Control and Cybernetics 7(3):17–25
Dolecki S (1978c) Semicontinuity in constrained optimization II. Control and Cybernetics 7(4):51–68
Dolecki S, Kurcyusz S (1978) On Φ-convexity in extremal problems. SIAM Jour. Control and Optim. 16:277–300
Dolecki S, Rolewicz S (1979b) Exact penalties for local minima. SIAM Jour. Contr. Optim. 17:596–606
Elster KH, Nehse R (1974) Zur Theorie der Polarfunktionale. Math. Operationsforsch. und Stat. Ser. Optimization 5:3–21
Gill PhE, Murray W, Wright MH (1981) Practical Optimization. Academic Press, London, New York, Sidney
Kurcyusz S (1976) Some remarks on generalized Lagrangians. Proc. 7-th IFIP Conference on Optimization Technique, Nice, September 1975, Springer-Verlag
Kurcyusz S (1976b) On existence and nonexistence of Lagrange multipliers in Banach spaces. Jour. Optim. Theory and Appl. 20:81–110
Kutateladze SS, Rubinov AM (1971) Some classes of H-convex functions and sets. Soviet Math. Dokl. 12:665–668
Neumann K, Morlock M (2002) Operations research, 2nd Edition (in German). Carl Hanser Verlag, München
Pallaschke D, Rolewicz S (1997) Foundations of Mathematical Optimization. Mathematics and its Applications. Kluwer Acad. Publ., Dordrecht
Pallaschke D, Rolewicz S (1998) Penalty and augmented Lagrangian in general optimization problems. Charlemagne and his heritage. 1200 years of civilization and science in Europe, Vol. 2 (Aachen, 1995), Brepols, Turnhout, pp. 423–437
Rubinov A (2000) Abstract convexity and global optimization. Non-convex Optimization and its Applications, 44. Kluwer Academic Publishers, Dordrecht
Singer I (1997) Abstract convex analysis. With a foreword by A. M. Rubinov, Canadian Mathematical Society Series of Monographs and Advanced Texts. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York
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Dedicated to Klaus Neumann
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© 2006 Deutscher Universitäts-Verlag/GWV Fachverlage GmbH, Wiesbaden
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Pallaschke, D. (2006). A Remark on the Formulation of Dual Programs Based on Generalized Convexity. In: Morlock, M., Schwindt, C., Trautmann, N., Zimmermann, J. (eds) Perspectives on Operations Research. DUV. https://doi.org/10.1007/978-3-8350-9064-4_6
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DOI: https://doi.org/10.1007/978-3-8350-9064-4_6
Publisher Name: DUV
Print ISBN: 978-3-8350-0234-0
Online ISBN: 978-3-8350-9064-4
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