Directionally differentiable multiobjective optimization involving discrete inclusions

Journal of Optimization Theory and Applications

Volumn 88, Issue 3, 1996, Pages 585-616

  Ishizuka, Y ^a   Tuan, H D ^b  

^a SOPHIA UNIVERSITY   (Japan)

^b NAGOYA UNIVERSITY   (Japan)

Author keywords

Alternative theorems; Directional derivatives; Discrete inclusions; Efficient solutions; Maximum principle; Multiobjective optimization; Tangent cones

Indexed keywords

CONES; MULTIOBJECTIVE OPTIMIZATION;

ALTERNATIVE THEOREM; CONVEX CONE; DIRECTIONAL DERIVATIVE; DISCRETE INCLUSION; EFFICIENT SOLUTION; MULTI-OBJECTIVES OPTIMIZATION; MULTIOBJECTIVE OPTIMIZATION PROBLEMS; OBJECTIVE FUNCTIONS; OPTIMAL SOLUTIONS; TANGENT CONE;

MAXIMUM PRINCIPLE;

EID: 0030534904     PISSN: 00223239     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF02192200     Document Type: Article

References (28)

1. BOLTYANSKII, V. G., Optimal Control of Discrete Systems, Wiley, New York, New York, 1978.
2. IOFFE, A. D., and TIKHOMIROV, V. M., Theory of Extremal Problems, North Holland, Amsterdam, Holland, 1979.
3. KUSRAEV, V. G., Discrete Maximum Principle, Mathematical Notes, Vol. 34, pp. 617-619, 1984.
4. PROPOI, A. I., The Maximum Principle for Discrete Control Systems, Automation and Remote Control, Vol. 26, pp. 1167-1177, 1965.
5. PSHENICHNYI, B. N., Necessary Conditions for an Extremum, Marcel Dekker, New York, New York, 1971.
6. CLARKE, F. H., Optimization and Nonsmooth Analysis, Wiley Interscience, New York, New York, 1983.
7. FRANKOWSKA, H., Contingent Cone to Reachable Sets of Control Systems, SIAM Journal on Control and Optimization, Vol. 27, pp. 170-198, 1989.
8. KHANH, P. Q., and NUONG, T. H., On Necessary Optimality Conditions in Vector Optimization Problems, Journal of Optimization Theory and Applications, Vol. 58, pp. 63-81, 1988.
9. SAWARAGI, Y., NAKAYAMA, H., and TANINO, T., Theory of Multiobjective Optimization, Academic Press, London, England, 1985.
10. YU, P. L., Multicriteria Decision Making: Concepts, Techniques, and Extensions, Plenum Press, New York, New York, 1985.
11. WHITE, D. J., Optimality and Efficiency, Wiley, New York, New York, 1982.
12. ISHIZUKA, Y., and SHIMIZU, K., Necessary and Sufficient Conditions for the Efficient Solutions of Nondifferentiable Multiobjective Problems, IEEE Transactions on Systems, Man, and Cybernetics, Vol. 14, pp. 625-629, 1984.
13. MINAMI, M., Weak Pareto-Optimal Necessary Conditions in a Nondifferentiable Multiobjective Program on a Banach Space, Journal of Optimization Theory and Applications, Vol. 41, pp. 451-461, 1983.
14. KANNIAPPAN, P., Necessary Conditions for Optimality of Nondifferentiable Convex Multiobjective Programming, Journal of Optimization Theory and Applications, Vol. 40, pp. 167-174, 1983.
15. KHANH, P. Q., Proper Solutions of Vector Optimization Problems, Journal of Optimization Theory and Applications, Vol. 74, pp. 105-130, 1992.
16. ISHIZUKA, Y., Optimality Conditions for Directionally Differentiable Multiobjective Programming Problems, Journal of Optimization Theory and Applications, Vol. 72, pp. 91-111, 1992.
17. HOLMES, R. B., Geometric Functional Analysis and Applications, Springer, New York, New York, 1975.
18. TUY, H., Convex Inequalities and the Hahn-Banach Theorem, Dissertationes Mathematicae, Vol. 107, pp. 1-69, 1972.
19. ROCKAFELLAR, R. T., Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.
20. AUBIN, J. P., and FRANKOWSKA, H., Set-Valued Analysis, Birkhäuser, Boston, Massachusetts, 1990.
21. DEMYANOV, V. F., and RUBINOV, A. M., On Quasi-Differentiable Functionals, Soviet Mathematics Docklady, Vol. 21, pp. 14-17, 1980.
22. DEMYANOV, V. F., and POLYAKOVA, L. N., Minimization of a Quasi-Differentiable Function in a Quasi-Differentiable Set, USSR Computational Mathematics and Mathematical Physics, Vol. 20, pp. 34-43, 1981.
23. ISHIZUKA, Y., Optimality Conditions for Quasi-Differentiable Programs with Application to Two-Level Optimization, SIAM Journal on Control and Optimization, Vol. 26, pp. 1388-1398, 1988.
24. FIACCO, A. V., Optimal Value Continuity and Differential Stability Bounds under the Mangasarian-Fromovitz Constraint Qualification, Mathematical Programming with Data Perturbations, Edited by A. V. Fiacco, Marcel Dekker, New York, New York, Vol. 2, pp. 65-90, 1983.
25. ROCKAFELLAR, R. T., Directional Differentiability of the Optimal Value Function in a Nonlinear Programming Problem, Mathematical Programming Study, Vol. 21, pp. 213-226, 1984.
26. HARTLEY, R., On Cone Efficiency, Cone Convexity, and Cone Compactness, SIAM Journal on Applied Mathematics, Vol. 34, pp. 211-222, 1978.
27. KUHN, H. W., and TUCKER, A. W., Nonlinear Programming, Proceedings of the 2nd Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley, California, pp. 481-492, 1951.
28. AUBIN, J. P., FRANKOWSKA, H., and OLECH, C., Controllability of Convex Processes, SIAM Journal on Control and Optimization, Vol. 24, 1192-1211, 1986.