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A generalization of Ekeland’s variational principle. (English) Zbl 0817.90098
Summary: Ekeland’s variational principle is a very deep assertion about the existence of an optimal solution of a slightly perturbed optimization problem in a neighborhood of an $$\varepsilon$$-optimal solution of the original problem. It plays an important role for developing necessary optimality conditions and other theoretical results. In this paper we give an extension of Ekeland’s theorem for vector optimization problems in general spaces where we use a concept of $$\varepsilon$$-efficiency which can be considered as a generalization of the concept of Loridan introduced for the finite-dimensional case. Our results differ essentially from other known generalizations of Ekeland’s theorem given by Loridan, Nemeth and Khanh in the assumptions, in the assertion and in the proof.

MSC:
 90C29 Multi-objective and goal programming 90C26 Nonconvex programming, global optimization 65K10 Numerical optimization and variational techniques
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References:
 [1] DOI: 10.1090/S0273-0979-1979-14595-6 · Zbl 0441.49011 [2] Gerth Chr., Seminarberichte der Sektion Mathematik 90 pp 67– (1987) [3] Gerth Chr., Wissenschaftliche Schriftenreihe (1989) [4] DOI: 10.1007/BF00940478 · Zbl 0692.90063 [5] Jahn J., Mathematical vector optimization in partially ordered linear spaces (1986) · Zbl 0578.90048 [6] Khanh P., Caristi-Kirk’s theorem and Ekeland’s variational principle for Pareto extrema (1986) · Zbl 0767.47039 [7] Kird W.A., Bull l’Acad Polom Soc 23 pp 891– (1975) [8] DOI: 10.1007/BF00936165 · Zbl 0517.90074 [9] Luc D.T., Lecture Notes in Economics and Mathematical Systems 319 (1989) [10] DOI: 10.1016/0362-546X(86)90126-4 · Zbl 0613.49007 [11] Staib T., J. Optim. Theory Appl 59 pp 289– (1988) [12] DOI: 10.1007/BF00941179 · Zbl 0616.90076 [13] Weidner P., Wissenschaftliche Zeitschrift der TH Ilmenau 31 pp 133– (1985) [14] Aubin J.P., Applied nonlinear analysis (1984)
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