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应用广泛。在某些应用中，由于底层物理问题的约束条件，通常要求多项式逼近满足非负性（或非正性），或在给定范围内有界。因此需要有效的数值方法来实施此类约束。本文讨论了非负性约束下多项式逼近的有效数值算法。我们首先构建了约束优化问题及其原形式与对偶形式，随后重点针对高维问题探讨了高效的一阶凸优化方法。通过高达$200$维的数值算例，验证了所提方法的有效性和可扩展性。