  Polynomial approximations of functions are widely used in scientific
computing. In certain applications, it is often desired to require the
polynomial approximation to be non-negative (resp. non-positive), or bounded
within a given range, due to constraints posed by the underlying physical
problems. Efficient numerical methods are thus needed to enforce such
conditions. In this paper, we discuss effective numerical algorithms for
polynomial approximation under non-negativity constraints. We first formulate
the constrained optimization problem, its primal and dual forms, and then
discuss efficient first-order convex optimization methods, with a particular
focus on high dimensional problems. Numerical examples are provided, for up to
$200$ dimensions, to demonstrate the effectiveness and scalability of the
methods.
