Abstract
A new approach to constrained optimization, which has appeared recently under various forms and in several contexts, is presented in a general and unifying setting. This approach is then employed to establish some new conditions for the existence of the minimum of a constrained minimum problem.
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Elster, K. H., andNehse, R.,Optimality Conditions for Some Nonconvex Problems, Optimization Techniques, Vol. 2, Edited by K. Iraki, K. Malanowski, and S. Walukiewicz, Springer-Verlag, New York, New York, pp. 1–9, 1980.
Giannessi, F.,Theorems of the Alternative, Quadratic Programming, and Complementarity Problems, Variational Inequalities and Complementarity Problems, Edited by R. W. Cottle, F. Giannessi, and J. L. Lions, John Wiley and Sons, New York, New York, pp. 151–185, 1980.
Giannessi, F.,Theorems of the Alternative and Optimality Conditions, Journal of Optiimization Theory and Applications, Vol. 42, pp. 331–365, 1984.
Hestenes, M. R.,Optimization Theory: The Finite Dimensional Case, Wiley-Interscience, New York, New York, 1975.
Nehse, R.,Some General Separation Theorems, Mathematische Nachrichten, Vol. 84, pp. 316–327, 1978.
Pourciau, B. H.,Multiplier Rules, American Mathematical Monthly, Vol. 87, pp. 443–452, 1980.
Pourciau, B. H.,Multiplier Rules and the Separation of Convex Sets, Journal of Optimization Theory and Applications, Vol. 40, pp. 321–331, 1983.
Ioffe, A. D.,Necessary Conditions in Nonsmooth Optimization, Mathematics of Operation Research, Vol. 9, pp. 159–189, 1984.
Martein, L.,Regularity Conditions for Constrained Extremum Problems, Journal of Optimization Theory and Applications, Vol. 47, pp. 217–233, 1985.
Hayashi, M., andKomiya, H.,Perfect Duality for Convexlike Programs, Journal of Optimization Theory and Applications, Vol. 38, pp. 177–189, 1982.
Tardella, F.,Some Topological Properties in Optimization Theory, Journal of Optimization Theory and Applications, Vol. 60, pp. 105–116, 1988.
Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1969.
Kelley, J. L., andNamioka, I.,Linear Topological Spaces, Van Nostrand, Princeton, New Jersey, 1963.
Naccasche, P. H.,Connectedness of the Set of Nondominated Outcomes in Multicritieria Optimization, Journal of Optimization Theory and Applications, Vol. 25, pp. 459–467, 1978.
Walkzak, S.,Some Properties of Cones in Normal Spaces and Their Applications to Investigating Extremal Problems, Journal of Optimization Theory and Applications, Vol. 42, pp. 561–582, 1984.
Wilansky, A.,Topology for Analysis, Ginn, Waltham, Massachusetts, 1970.
Corley, H. W.,An Existence Result for Maximization with Respect to Cones, Journal of Optimization Theory and Applications, Vol. 31, pp. 277–281, 1980.
Calligaris, O., andOliva, P.,Necessary and Sufficient Conditions for Pareto Problems, Bollettino dell'Unione Matematica Italiana, Vol. 18B, pp. 177–216, 1981.
Harteley, R.,On Cone-Efficiency, Cone-Convexity, and Cone-Compactness, SIAM Journal on Applied Mathematics, Vol. 34, pp. 211–222, 1978.
Aubin, J. P.,Mathematical Methods of Game and Economic Theory, North-Holland, New York, New York, 1979.
Cesari, L., andSuryanarayana, M. B.,An Existence Theorem for Pareto Optimization: Multivalued and Banach Space Valued Functionals, Transactions of the American Mathematical Society, Vol. 224, pp. 37–65, 1978.
Yu, P. L.,Cone-Convexity, Cone Extreme Points, and Nondominated Solutions in Decision Problems with Multiobjectives, Journal of Optimization Theory and Applications, Vol. 14, pp. 319–377, 1974.
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Tardella, F. On the image of a constrained extremum problem and some applications to the existence of a minimum. J Optim Theory Appl 60, 93–104 (1989). https://doi.org/10.1007/BF00938802
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DOI: https://doi.org/10.1007/BF00938802