From the simplex to the sphere: Faster constrained optimization using the Hadamard parametrization
Abstract
The standard simplex in R^n, also known as the probability simplex, is the set of nonnegative vectors whose entries sum up to 1. They frequently appear as constraints in optimization problems that arise in machine learning, statistics, data science, operations research, and beyond. We convert the standard simplex to the unit sphere and thus transform the corresponding constrained optimization problem into an optimization problem on a simple, smooth manifold. We show that KKT points and strictsaddle points of the minimization problem on the standard simplex all correspond to those of the transformed problem, and vice versa. So, solving one problem is equivalent to solving the other problem. Then, we propose several simple, efficient, and projectionfree algorithms using the manifold structure. The equivalence and the proposed algorithm can be extended to optimization problems with unit simplex, weighted probability simplex, or `1norm sphere constraints. Numerical experiments between the new algorithms and existing ones show the advantages of the new approach
 Publication:

arXiv eprints
 Pub Date:
 December 2021
 arXiv:
 arXiv:2112.05273
 Bibcode:
 2021arXiv211205273L
 Keywords:

 Mathematics  Optimization and Control
 EPrint:
 Version 2