Projection onto the capped simplex

TL;DR

This paper introduces a straightforward and efficient algorithm for projecting points onto the capped simplex, which is a simplex with uniform bounds on each coordinate, supported by implementations in MATLAB and C++.

Contribution

The paper presents a novel, simple algorithm for Euclidean projection onto the capped simplex, along with an elementary proof of correctness.

Findings

Algorithm is efficient and easy to implement.

Provides downloadable MATLAB and C++ code.

Applicable to optimization problems with bounded simplex constraints.

Abstract

We provide a simple and efficient algorithm for computing the Euclidean projection of a point onto the capped simplex---a simplex with an additional uniform bound on each coordinate---together with an elementary proof. Both the MATLAB and C++ implementations of the proposed algorithm can be downloaded at https://eng.ucmerced.edu/people/wwang5.