# Computation of the Epsilon-Subdifferential of Convex Piecewise-Defined Functions in Optimal Worst-Case Time

## TL;DR

This paper introduces efficient algorithms for computing the epsilon-subdifferential of convex piecewise-defined functions, reducing the complexity from linear to logarithmic time and enabling rapid graph construction.

## Contribution

It presents novel algorithms that compute the epsilon-subdifferential in logarithmic time and the entire graph in linear time for convex piecewise functions, improving computational efficiency.

## Key findings

- Epsilon-subdifferential can be computed in logarithmic worst-case time.
- The entire graph of the epsilon-subdifferential can be constructed in linear time.
- Algorithms are applicable to a broad class of convex piecewise-defined functions.

## Abstract

The $\epsilon$-subdifferential of convex univariate piecewise linear-quadratic functions can be computed in linear worst-case time complexity as the level-set of a convex function. Using dichotomic search, we show how the computation can be performed in logarithmic worst-case time. Furthermore, a new algorithm to compute the entire graph of the $\epsilon$-subdifferential in linear time is presented. Both algorithms are not limited to convex PLQ functions but are also applicable to any convex piecewise-defined function with little restrictions.