# An Active-Set Algorithmic Framework for Non-Convex Optimization Problems over the Simplex

## TL;DR

This paper introduces a novel active-set algorithmic framework for non-convex optimization over the simplex, with proven convergence and practical efficiency demonstrated through numerical experiments.

## Contribution

It presents a new active-set framework with a nonorthogonality condition, guaranteeing convergence and linear rates for non-convex problems over the simplex.

## Key findings

- Global convergence to stationary points
- Linear convergence under certain conditions
- Numerical experiments confirm effectiveness

## Abstract

In this paper, we describe a new active-set algorithmic framework for minimizing a non-convex function over the unit simplex. At each iteration, the method makes use of a rule for identifying active variables (i.e., variables that are zero at a stationary point) and specific directions (that we name active-set gradient related directions) satisfying a new "nonorthogonality" type of condition. We prove global convergence to stationary points when using an Armijo line search in the given framework. We further describe three different examples of active-set gradient related directions that guarantee linear convergence rate (under suitable assumptions). Finally, we report numerical experiments showing the effectiveness of the approach.