Vector conjugation and subdifferential of vector topical function in complete lattice
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Vector conjugation and subdifferential of vector topical function in complete lattice C. L. Yao1 · J. W. Chen1 Received: 24 March 2019 / Accepted: 19 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In our previous work, we introduced a vector topical function which takes values in a general ordered Banach space. With some weak versions of supremum, a set-valued mapping was proposed to build the envelope for it. In this paper, regarding the image space as a complete lattice, we present a vector mapping as the support to study the abstract convex framework of the vector topical function. This framework provides a dual space consisting of vector mappings. Based on that, we obtain the global Lipschitz of the vector topical mapping, discussing the abstract convex conjugation and subdifferential. Further more, we investigate how the vector topical feature behaves from the dual point of view. Some dual characterizations of vector topical mapping are also established. Keywords Vector topical function · Abstract convexity · Vector support · Complete lattice
1 Introduction Convexity is a significant structure in the research of various areas. Problems with convexity usually have many good features and properties. Meanwhile, convex analysis provides powerful tools and abundant theories for studying these problems. However, there are also plenty of nonconvex models and problems arising out of practice, for which the classical convex theory might be invalid. That is why there are a large number of papers dedicated to the generalizations of the convex structure as well as the corresponding theories, see [6,16,25]. These generalizations open a way to cope with more nonconvex objects, taking advantage of the ideas and techniques from convex
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C. L. Yao [email protected] J. W. Chen [email protected]
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College of Mathematics and Statistics, Southwest University, Chongqing 400715, China
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C. L. Yao, J. W. Chen
analysis. Abstract convexity is a typical extension of the classical one. It can be seen as a nonlinear generalization of the classical convexity from the global perspective. With different collections of support functions, it covers a wide range of nonconvex mappings, see [18,19]. The initial idea of abstract convexity can be found in some early papers such as [1,2,12]. Books like [17] and [21] provided elaborate surveys on both the theories and the applications of abstract convex analysis, especially on the applications in global optimization. It is worth mentioning that the most important area that abstract convexity is instrumental and very successful is the theory of optimal mass transportation ([24]), where the optimal plan is obtained from an abstract maximal monotone set. Nowadays, the abstract convex analysis in scalar case has become a rich theory with wide applications in numerous fields. However, unlike the classical convexity, which has already been extended and studied in the vector or even set-valued situation, there is not much work investigating abstract convexity
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