  A fundamental problem in numerical analysis and approximation theory is
approximating smooth functions by polynomials. A much harder version under
recent consideration is to enforce bounds constraints on the approximating
polynomial. In this paper, we consider the problem of approximating functions
by polynomials whose Bernstein coefficients with respect to a given degree
satisfy such bounds, which implies such bounds on the approximant. We frame the
problem as an inequality-constrained optimization problem and give an algorithm
for finding the Bernstein coefficients of the exact solution. Additionally, our
method can be modified slightly to include equality constraints such as mass
preservation. It also extends naturally to multivariate polynomials over a
simplex.
