Scaled relative graphs for pairs of operators beyond classical monotonicity

TL;DR

This paper extends the scaled relative graph (SRG) framework to pairs of operators, offering a geometric visualization tool for analyzing their properties beyond classical monotonicity, with applications to nonlinear resolvents and electrical circuits.

Contribution

It introduces a generalized SRG for pairs of operators, enabling analysis of nonlinear resolvents and complex electrical circuits with nonmonotone components.

Findings

SRG framework extended to pairs of operators.

Applicable to linear operators with monotone mappings.

Enables analysis of multivalued, nonsmooth, nonmonotone circuits.

Abstract

We introduce a generalization of the scaled relative graph (SRG) to pairs of operators, enabling the visualization of their relative incremental properties. This novel SRG framework provides the geometric counterpart for the study of nonlinear resolvents based on paired monotonicity conditions. We demonstrate that these conditions apply to linear operators composed with monotone mappings, a class that notably includes NPN transistors, allowing us to compute the response of multivalued, nonsmooth and highly nonmonotone electrical circuits.