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The standard simplex in R^n, also known as the probability simplex, is theset of nonnegative vectors whose entries sum up to 1. They frequently appear asconstraints in optimization problems that arise in machine learning,statistics, data science, operations research, and beyond. We convert thestandard simplex to the unit sphere and thus transform the correspondingconstrained optimization problem into an optimization problem on a simple,smooth manifold. We show that KKT points and strict-saddle points of theminimization problem on the standard simplex all correspond to those of thetransformed problem, and vice versa. So, solving one problem is equivalent tosolving the other problem. Then, we propose several simple, efficient, andprojection-free algorithms using the manifold structure. The equivalence andthe proposed algorithm can be extended to optimization problems with unitsimplex, weighted probability simplex, or `1-norm sphere constraints. Numericalexperiments between the new algorithms and existing ones show the advantages ofthe new approach

From the simplex to the sphere: Faster constrained optimization using the Hadamard parametrization