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A generalized Ekeland vector variational principle and its applications in optimization. (English) Zbl 1170.58004
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:
58E17Pareto optimality, etc., applications to economics (variational methods in infinite-dimensional spaces)
58E30Variational principles on infinite-dimensional spaces
46N10Applications of functional analysis in optimization and programming
65K10Optimization techniques (numerical methods)
90C48Programming in abstract spaces