Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-20T17:25:59.343Z Has data issue: false hasContentIssue false
  • English
  • Français

Lipschitz and differentiable dependence of solutions on a parameter in a scalarization method

Published online by Cambridge University Press:  09 April 2009

Alicia Sterna-Karwat
Alicia Sterna-Karwat
Affiliation:
Department of MathematicsMonash UniversityClayton, Victoria 3168, Australia
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper is concerned with a vector optimization problem set in a normed space where optimality is defined through a convex cone. The vector problem can be solved using a parametrized scalar problem. Under some convexity assumptions, it is shown that dependence of optimal solutions on the parameter is Lipschitz continuous. Hence differentiable dependence on the solutions on the parameter is derived.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 42 , Issue 3 , June 1987 , pp. 353 - 364
Copyright
Copyright © Australian Mathematical Society 1987

References

[1]Aronszajn, N., ‘Differentiability of Lipschitzian mappings between Banach spaces’, Studia Math. 57 (1976), 147–190.CrossRefGoogle Scholar
[2]Berge, C., Topological spaces (Macmillan, New York, 1963).Google Scholar
[3]Clarke, F., Optimization and nonsmooth analysis (Wiley-Interscience, New York, 1983).Google Scholar
[4]Craven, B. D., ‘Vector-valued optimization’, Generalized concavity in optimization and economics eds. Schaible, S. and Ziemba, W. T., pp. 661–687 (Academic Press, London, 1981).Google Scholar
[5]Dalahaye, J. P. and Denel, J., ‘The continuities of the point-to-set maps, definitions and equivalences’, Math. Programming Study 10 (1979), 8–13.CrossRefGoogle Scholar
[6]Dugundji, J. and Granas, A., Fixed point theory, Vol. I (PWN-Polish Scientific Publishers, Warszawa, 1982).Google Scholar
[7]Henig, M. I., ‘Existence and characterization of efficient decisions with respect to cones’, Math. Programming 23 (1982), 111–116.CrossRefGoogle Scholar
[8]Holmes, R. B., Geometric functional analysis and its applications, (Springer, New York, New York, 1975).CrossRefGoogle Scholar
[9]Kuratowski, K., Topology (Academic Press, New York and Polish Scientific Publishers, Warszawa, 1966).Google Scholar
[10]Lin, J. L., ‘Maximal vectors and multiobjective optimization’, J. Optim. Theory Appl. 18 (1976), 41–64.CrossRefGoogle Scholar
[11]Lebourg, G., ‘Generic differentiability of Lipschitzian functions’, Trans. Amer. Math. Soc. 256 (1979), 12–144.CrossRefGoogle Scholar
[12]Mankiewicz, P., ‘On the differentiability of Lipschitz mappings in Fréchet spaces’, Studia Math. 45 (1973), 15–29.CrossRefGoogle Scholar
[13]Mankiewicz, P., ‘On topological, Lipschitz, and uniform classification of LF-spaces’, Studia Math. 52 (1975), 109–142.CrossRefGoogle Scholar
[14]Nijenhuis, A., ‘Strong derivatives and inverse mappings’, Amer. Math. Monthly 81 (1974), 969–980.CrossRefGoogle Scholar
[15]Pascoletti, A. and Serafini, P., ‘Scalarizing vector optimization problems’, J. Optim. Theory Appl., 42 (1984), 499–524.CrossRefGoogle Scholar
[16]Penot, J. P. and Sterna-Karwat, A., ‘Parametrized multicriteria optimization; continuity and closedness of the optimal multifunctions’, J. Math. Anal. Appl., to appear.Google Scholar
[17]Penot, J. P. and Sterna-Karwat, A., ‘Parameterized multicriteria optimization; order continuity of the marginal multifunctions’, J. Math. Anal. Appl., to appear.Google Scholar
[18]Pourciau, B. H., ‘Analysis and optimization of Lipschitz continuous mappings’, J. Optim. Theory Appl. 22 (1977), 311–351.CrossRefGoogle Scholar
[19]Simon, L., Lectures on geometric measure theory (Centre of Mathematical Analysis, Australian National University, 1984).Google Scholar
[20]Sterna-Karwat, A., ‘Continuous dependence of solutions on a parameter in a scalarization method’, J. Optim. Theory Appl., to appear.Google Scholar
[21]Yamamuro, S., Differential calculus in topological linear spaces, (Springer-Verlag, Berlin, 1974).CrossRefGoogle Scholar
[22]Yu, P. L., ‘Cone convexity, cone extreme points and nondominated solutions in decision problems with multiobjectives’, J. Optim. Theory Appl. 14 (1974), 319–317.CrossRefGoogle Scholar