Visualization of the {\epsilon}-Subdifferential of Piecewise Linear-Quadratic Functions

TL;DR

This paper presents a method for visualizing the {psilon}-subdifferential of piecewise linear-quadratic functions, combining theoretical insights with practical numerical examples and visualizations in primal, dual, and subdifferential views.

Contribution

It introduces a practical algorithm for computing and visualizing the {psilon}-subdifferential of convex univariate piecewise linear-quadratic functions, including visualization of the Brndsted-Rockafellar Theorem.

Findings

Effective visualization of {psilon}-subdifferentials in multiple views.

Application of the algorithm to practical numerical examples.

Visualization of the Brndsted-Rockafellar Theorem.

Abstract

Computing explicitly the {\epsilon}-subdifferential of a proper function amounts to computing the level set of a convex function namely the conjugate minus a linear function. The resulting theoretical algorithm is applied to the the class of (convex univariate) piecewise linear-quadratic functions for which existing numerical libraries allow practical computations. We visualize the results in a primal, dual, and subdifferential views through several numerical examples. We also provide a visualization of the Br{\o}ndsted-Rockafellar Theorem.