Translation Mappings of Quasimonotonicity Beyond Smoothness
Oday Hazaimah
TL;DR
This paper addresses when quasimonotonicity implies monotonicity for nonsmooth, set-valued maps on convex domains within real topological vector spaces, using translation maps $F+\omega$ and without differentiability assumptions. The authors extend Schaible's finite-dimensional results to infinite-dimensional, nonsmooth contexts by proving an equivalence: there exists a line $L$ not orthogonal to $\Omega$ such that $F+\omega$ is quasimonotone for all $\omega\in L$ if and only if the Clarke subdifferential $\partial F(x)$ is positive semidefinite for every $x\in\Omega$ (positive definiteness yields a stronger form). They also show that the condition on $L$ is essential and derive a corollary for pseudomonotone translations. Overall, the work broadens the theory of generalized monotone maps to nonsmooth, topological vector spaces, with implications for variational and equality problems in economics and optimization.
Abstract
Monotonicity of a mapping implies its pseudomonotonicity and hence quasimonotonocity, the converse is not true. In this note we intend to study the situations under which quasimono tonicity of a mapping implies its monotonicity. Thus we generalize some results in the literature related to the connection between monotonocity and its generalized classes for multi-valued mappings via translation maps in real topological spaces. No differentiability assumption is required but continuity assumption is imposed.