# Convex envelopes for ray-concave functions

arXiv: https://arxiv.org/abs/2105.03532 · HTML: https://mlsanity.com/paper/2105.03532/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 May 2021.

## Built on by (most cited first, 1 of 1)

- [Tightening convex relaxations of trained neural networks: a unified approach for convex and S-shaped activations](https://mlsanity.com/paper/2410.23362/) (2024)

## Abstract (arXiv)

Convexification based on convex envelopes is ubiquitous in the non-linear optimization literature. Thanks to considerable efforts of the optimization community for decades, we are able to compute the convex envelopes of a considerable number of functions that appear in practice, and thus obtain tight and tractable approximations to challenging problems. We contribute to this line of work by considering a family of functions that, to the best of our knowledge, has not been considered before in the literature. We call this family ray-concave functions. We show sufficient conditions that allow us to easily compute closed-form expressions for the convex envelope of ray-concave functions over arbitrary polytopes. With these tools, we are able to provide new perspectives to previously known convex envelopes and derive a previously unknown convex envelope for a function that arises in probability contexts.