Relatively Monotone and Relatively Generalized Monotone Maps in Nonsmooth Setting and Relative r-Monotonicity. (17th February 2023)

• Record Type: Journal Article
• Authors: Fajri, Youness Al Habib
• Is Part Of: Numerical functional analysis and optimization. Volume 44:Number 3(2023)
• Page Start: 161
• Page End: 178
• Publication Date: 2023-02-17
• Subjects: Nonsmooth analysis -- relatively monotone and generalized monotone maps -- relatively r-monotone operators; 47H05 -- 49J52; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2022.2119248
• Languages: English
• ISSNs: 0163-0563

Abstract

Abstract: The aim of this article is twofold. First, we consider relatively monotone, relatively pseudomonotone, and relatively quasimonotone maps in nonsmooth setting. For continuous maps, we describe ( α, β ) -monotonicity when the relative parameters are different and prove that ( α, β ) -quasimonotonicity implies ( α, α ) -quasimonotonicity as well as ( α, β ) -pseudomonotonicity implies ( α, α ) -pseudomonotonicity. Then, under local Lipschitzianity assumption, we establish first-order criteria for ( α, α ) -monotonicity, ( α, α ) -pseudomonotonicity, and ( α, α ) -quasimonotonicity. Second, we introduce and study the new notion of relatively r -monotone operators. We provide characterizations and properties. We give in particular criteria of first order for relative r -monotonicity, in smooth and nonsmooth locally Lipschitz settings, when the relative parameters are equal.