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].
Key words
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
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
Author information
Authors and Affiliations
Editor information
Additional information
Dedicated to Klaus Neumann
Rights and permissions
Copyright information
© 2006 Deutscher Universitäts-Verlag/GWV Fachverlage GmbH, Wiesbaden
About this chapter
Cite this chapter
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
Download citation
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
eBook Packages: Business and EconomicsBusiness and Management (R0)