Relatively Monotone and Relatively Generalized Monotone Maps in Nonsmooth Setting and Relative r-Monotonicity. (17th February 2023) - Fajri, Youness Al Habib
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.