Closedness under addition for families of quasimonotone operators

Abstract

In the last two decades, several properties of operators that are weaker than monotonicity have received attention by researchers from many areas including mathematical economics, with the goal to develop new tools applicable in convex analysis and related topics. This paper puts in perspective notions that are extensions of monotoniticity but not beyond quasimonotonicity like pseudomonotonicity, semistrict quasimonotonicity, strict quasimonotonicity and proper quasimonotonicity, and discusses systematically when the sum of two operators satisfying one of those properties, inherits the same property. The case of properly quasimonotone operators deserves a special attention since this notion, being stronger than quasimonotonicity, suffices to obtain many results, including the solvability of variational inequality problems. Several examples showing the optimality in some sense of our results, are presented.

Keywords:

• Quasiconvex function
• quasimonotone operator
• quasiconvex optimization

Mathematics Subject Classifications:

• 26A51
• 26B25
• 90C26

Acknowledgments

Part of the work of the third author was carried out when he was visiting the Department of Mathematical Engineering, Universidad de Concepción (Chile). The author wishes to thank the Department for its hospitality.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research for the first author was supported in part by ANID-Chile through FONDECYT [1212004 and BASAL FB210005].