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Saddle point and duality in the optimization theory of convex set functions

Published online by Cambridge University Press:  17 February 2009

Hang-Chin Lai  and
Shu-Shih Yang
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Abstract

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For a set function G on an atomless finite measure space (X, , m), we define the subgradient, conjugate set of and conjugate functional of G. It is proved that a minimization problem of set function G has an optimal solution if and only if the Lagrangian on × L1(X, , m)has a saddle point (Ω0, f0) such that

where f0 is an element of the conjugate set (for the definition, see the later context).

Type
Research Article
Information
The ANZIAM Journal , Volume 24 , Issue 2 , October 1982 , pp. 130 - 137
Copyright
Copyright © Australian Mathematical Society 1982

References

[1]Begis, D. and Glowinski, R., “Application de la inéthode des éléments finis à l'approximation d'un problème de domaine optimal, Méthodes de résolution des problèmes approchés,”, Appl. Math. Optim. 2 (1975), 130–169.CrossRefGoogle Scholar
[2]Cea, J., Gioan, A. and Michel, J., “Quelques résultats sur l'identification de domaines”, Calcolo 10 (1973), 133–145.CrossRefGoogle Scholar
[3]Lai, H. C., Yang, S. S. and Hwang, G. R., “Duality in mathematical programming of set functions: On Fenchel duality theorem”, J. Math. Anal. Appl. (in press).Google Scholar
[4]Luenberger, D. G., Optimization by vector space methods (Wiley, New York, 1969).Google Scholar
[5]Morris, R. J. T., “Optimal constrained selection of a measurable subset”, J. Math. Anal. Appl. 70 (1979), 546–562.CrossRefGoogle Scholar
[6]Morris, R. J. T., “Optimization problems involving set functions”, Ph.D. Dissertation, University of California, Los Angeles, 03 1978; also available as UCLA Tech. Report UCLA -ENG7815, March 1978.Google Scholar
[7]Rockafellar, R. T., “Convex integral functionals and duality”, in Contributions to nonlinear functional analysis (ed. Zarantonello, E.), (Academic, New York, 1971), 215–234.CrossRefGoogle Scholar
[8]Scott, C. H. and Jefferson, T. R., “Characterizations of optimality for continuous convex mathematical programs. Part 1. Linear constraints”, J. A ustral. Math. Soc. B 21 (1979), 37–44.CrossRefGoogle Scholar
[9]Scott, C. H. and Jefferson, T. R., “A generalization of geometric programming with an application to information theory”, Inform. Sci. 12 (1977), 263–269.CrossRefGoogle Scholar
[10]Scott, C. H. and Jefferson, T. R., “Duality in infinite-dimensional mathematical programming: convex integral functionals”, J. Math. Anal. Appl. 61 (1977), 251–261.CrossRefGoogle Scholar
[11]Wang, P. K. C., “On a class of optimization problems involving domain variations”, in International symposium on new trends in system analysis, Versailles, France, (12. 1976), Lecture Notes in Control and Information Sciences, No. 2, (Springer-Verlag, Berlin, 1977).Google Scholar