# Projections onto the canonical simplex with additional linear inequalities

## TL;DR

This paper develops efficient algorithms for projecting onto the canonical simplex with extra linear constraints, crucial for distributionally robust optimization, by transforming the problem into finding zeros of well-behaved functions.

## Contribution

It introduces a novel approach to compute projections with additional linear inequalities using zero-finding of convex or monotonic functions, with guaranteed convergence and near-linear complexity.

## Key findings

- Algorithms have guaranteed convergence.
- Observed complexity is nearly linear.
- Applicable to various distance functions in distributionally robust optimization.

## Abstract

We consider the distributionally robust optimization and show that computing the distributional worst-case is equivalent to computing the projection onto the canonical simplex with additional linear inequality. We consider several distance functions to measure the distance of distributions. We write the projections as optimization problems and show that they are equivalent to finding a zero of real-valued functions. We prove that these functions possess nice properties such as monotonicity or convexity. We design optimization methods with guaranteed convergence and derive their theoretical complexity. We demonstrate that our methods have (almost) linear observed complexity.