# Symbolic computation with monotone operators

## TL;DR

This paper explores a class of monotone operators suitable for symbolic computation in computer algebra systems, examining their properties and applications in convex analysis and risk measure computations.

## Contribution

It introduces a new class of monotone operators linked to convex functions, enabling symbolic manipulation and computation of various operators and measures.

## Key findings

- Established structural properties of the class
- Demonstrated symbolic computation of proximity operators
- Applied to risk measures and convex penalties

## Abstract

We consider a class of monotone operators which are appropriate for symbolic representation and manipulation within a computer algebra system. Various structural properties of the class (e.g., closure under taking inverses, resolvents) are investigated as well as the role played by maximal monotonicity within the class. In particular, we show that there is a natural correspondence between our class of monotone operators and the subdifferentials of convex functions belonging to a class of convex functions deemed suitable for symbolic computation of Fenchel conjugates which were previously studied by Bauschke & von Mohrenschildt and by Borwein & Hamilton. A number of illustrative examples utilizing the introduced class of operators are provided including computation of proximity operators, recovery of a convex penalty function associated with the hard thresholding operator, and computation of superexpectations, superdistributions and superquantiles with specialization to risk measures.