  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 strict-saddle 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
projection-free algorithms using the manifold structure. The equivalence and
the proposed algorithm can be extended to optimization problems with unit
simplex, weighted probability simplex, or `1-norm sphere constraints. Numerical
experiments between the new algorithms and existing ones show the advantages of
the new approach
