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Extreme Points of Data Sets
Published in Gerhard X. Ritter, Gonzalo Urcid, Introduction to Lattice Algebra, 2021
Gerhard X. Ritter, Gonzalo Urcid
Various concepts of convex analysis are being employed in such diverse fields as pattern recognition, optimization theory, image analysis, computational geometry, and general data analysis. The subsequent applications of the lattice theory presented thus far will require some basic knowledge of convex set theory. The first part of this chapter recollects the necessary relevant concepts from convex set theory.
Preface
Published in Optimization, 2020
Akhtar A. Khan, Mau Nam Nguyen, Christiane Tammer, Bao Truong
The era of variational and convex analysis started in the 1960s with the pioneering works of Jean Jacques Moreau and R. Tyrrell Rockafellar, who initiated the study of generalized differentiation for convex functions and sets. Convex analysis serves as the mathematical foundation for convex optimization, a field with great impacts in many applied areas nowadays. The 1970s witnessed great effort in developing generalized differentiation theory for nonsmooth functions that are not necessarily convex. Professor Mordukhovich is now considered as one of the founders of modern variational analysis due to his generalized differentiation theory for nonsmooth functions and set-valued mappings that goes beyond convexity and is independent of convexity. Based on a geometric dual approach, in the mid of 1970s, Professor Mordukhovich, who was in the Soviet Union, introduced the concept of limiting/Mordukhovich subgradients for extended-real-valued functions. The generalized differentiation calculus for limiting subgradients relies on another novel notion called the extremal system for nonconvex sets along with the extremal principle, which is as important as the separation theorem in the convex case. In the early of 1980s, Professor Mordukhovich applied his revolutionary idea to the case of set-valued mappings by introducing the concept of coderivatives for set-valued mappings and developed their comprehensive calculus as well as applications to optimization and optimal control.
Abstract convexity of set-valued topical functions with application in DC-type optimization
Published in Applicable Analysis, 2021
Chaoli Yao, Chunlei Tang, Jiawei Chen
Convexity has always been an important structure in optimization, so that the theory of convex analysis is a powerful tool in the study of optimization problems. Various extensions of convexity enable us to deal with more nonconvex problems by virtue of the ideas from classical convex analysis. That is why these generalized convexities are also of great significance. In this paper, along with our previous work [10], we further studied the abstract convex framework of the set-valued topical function, especially on the subdifferentials. Then, we presented some dual characterizations and sufficient conditions of the set-valued topical function. Finally, optimality and duality of set-valued DC-type optimization problems were established.