Characterization of the monotone polar of subdifferentials

TL;DR

This paper characterizes the monotone polar of subdifferentials of lower semicontinuous functions by linking solutions of the Minty variational inequality to increasing behavior along rays, providing a new geometric insight.

Contribution

It offers a novel characterization of the monotone polar of subdifferentials based solely on the function's behavior, connecting variational inequalities to monotonicity properties.

Findings

Solution points correspond to functions increasing along rays from that point.

The monotone polar is a common subset of subdifferential graphs.

Provides a geometric interpretation of subdifferential monotonicity.

Abstract

We show that a point is solution of the Minty variational inequality of subdifferential type for a given function if and only if the function is increasing along rays starting from that point. This provides a characterization of the monotone polar of subdifferentials of lower semicontinuous functions, which happens to be a common subset of their graphs depending only on the function.