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Zbl 1170.58004
Qiu, Jing-Hui
A generalized Ekeland vector variational principle and its applications in optimization. (English)
[J] Nonlinear Anal., Theory Methods Appl. 71, No. 10, A, 4705-4717 (2009). ISSN 0362-546X

Summary: We give a generalized Ekeland vector variational principle. By using the principle, we extend and improve the related results in sharp efficiency. In the framework of locally convex spaces, we introduce two kinds of generalized sharp efficiencies and prove that they are equivalent. In particular, we show that a sharp efficient solution with respect to an interior point of the ordering cone is also one with respect to every interior point. Moreover, we introduce the generalized Takahashi's condition and the generalized Hamel's condition for vector-valued functions. From the generalized Ekeland principle we deduce that the two conditions are equivalent. From this, we discuss the relationship between the `distance' of $f(x)$ from $E(f(X))$ and the distance of $x$ from $E(f)$, where $E(f(X))$ denotes the efficient point set of $f(X)$ and $E(f)$ denotes the efficient solution set.

MSC 2000:: *58E17 Pareto optimality, etc., appl. to economics
58E30 Variational principles on infinite-dimensional spaces
46N10 Appl. of functional analysis in optimization and math. programming
65K10 Optimization techniques (numerical methods)
90C48 Programming in abstract spaces

Keywords: Ekeland vector variational principle; locally convex space; ordering cone; sharp efficient solution

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Zentralblatt MATH Berlin [Germany]