Bounds-constrained polynomial approximation using the Bernstein basis

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 constructioning such approximations using polynomials in the Bernstein basis. We consider a family of inequality-constrained quadratic programs. In the univariate case, a quadratic cone constraint allows us to search over all nonnegative polynomials of a given degree. In both the univariate and multivariate cases, we consider approximate problems with linear inequality constraints. Additionally, our method can be modified slightly to include equality constraints such as mass preservation.