Faces and Support Functions for the Values of Maximal Monotone Operators
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Faces and Support Functions for the Values of Maximal Monotone Operators Bao Tran Nguyen1,2 · Pham Duy Khanh3,4 Received: 7 October 2019 / Accepted: 8 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Representation formulas for faces and support functions of the values of maximal monotone operators are established in two cases: either the operators are defined on reflexive and locally uniformly convex real Banach spaces with locally uniformly convex duals, or their domains have nonempty interiors on real Banach spaces. Faces and support functions are characterized by the limit values of the minimal-norm selections of maximal monotone operators in the first case while in the second case they are represented by the limit values of any selection of maximal monotone operators. These obtained formulas are applied to study the structure of maximal monotone operators: the local unique determination from their minimal-norm selections, the local and global decompositions, and the unique determination on dense subsets of their domains. Keywords Maximal monotone operator · Face · Support function · Minimal-norm selection · Yosida approximation · Strong convergence · Weak convergence Mathematics Subject Classification 26B25 · 47B48 · 47H04 · 47H05 · 54C60
Communicated by Constantin Z˘alinescu.
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Pham Duy Khanh [email protected]; [email protected] Bao Tran Nguyen [email protected] ; [email protected]
1
Universidad de O’Higgins, Rancagua, Chile
2
Quy Nhon University, Quy Nhon, Vietnam
3
Department of Mathematics, HCMC University of Education, Ho Chi Minh, Vietnam
4
Center for Mathematical Modeling, Universidad de Chile, Santiago, Chile
123
Journal of Optimization Theory and Applications
1 Introduction Faces and support functions are important tools in representation and analysis of closed convex sets (see [1, Chapter V]). For a closed convex set, a face is the set of points on the given set which maximizes some (nonzero) linear form while the support function is the signed distance from the origin point to the supporting planes of that set. The face associated with a given direction can be defined via the value of the support function at this direction [2, Definition 3.1.3, p. 220]. Recently, this notion has been defined and studied for the values of maximal monotone operators in [2, Sect. 3]. In this paper, the authors provided some characterizations for the boundary and faces of the values of maximal monotones operators in Hilbert spaces. Their work is motivated by the applications of these characterizations to the stability issues of semi-infinite linear programming problems. Motivated by the study of the structure of maximal monotone operators, our paper will investigate the faces and support functions for the values of maximal monotone operators in real Banach spaces. We aim to establish some representation formulas for the faces and support functions in two cases regarding the reflexivity and local uniform convexity of the given spaces and theirs d
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