Maximal Monotone Operators and Saddle Functions I

Abstract

We investigate the monotone operator $T_K\subseteq E\times E^{\ast },f\in T_{K}x:=[–f,f]\in \partial K(x,x)$, which is defined via the subdifferential of a concave-convex saddle function $K$. Our considerations are motivated by the fact that each maximal monotone operator $A$ possesses a representation of the form $A=T_K$. We show that $T_K$ is maximal monotone if and only if $K$ is in a relaxed form skew-symmetric. This allows a generalization of results obtained previously.