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.

翻译：函数的多项式逼近在科学计算中广泛应用。在某些应用中，由于底层物理问题带来的约束，常常要求多项式逼近具有非负性（或非正性），或限定在给定范围内。因此需要高效的数值方法来强制实施此类条件。本文讨论了非负约束下多项式逼近的有效数值算法。我们首先阐述约束优化问题及其原始形式与对偶形式，随后探讨高效的一阶凸优化方法，特别关注高维问题。文中提供了高达$200$维的数值算例，以展示方法的有效性与可扩展性。