Given multiple univariate or multivariate real polynomials $f_1,\dots ,f_m$ and a desired real zero $\mathbf{z}$, this paper studies the problem of calculating a real polynomial $\tilde{f}$ such that $\tilde{f}\left(\mathbf{z}\right)=0$ and the distance between $\tilde{f}$ and $f_1,\dots ,f_m$ is minimal, where the distance is defined by a norm pair $({\ell }_r,{\ell }_s)$. Though geometrical and numerical algorithms have been studied in previous works, only few norm pairs can be dealt with and the algorithms are really difficult to implement and generalize. In this paper, we propose a unified algorithmic framework from an optimization viewpoint. We first convert the problem into a constrained ${\ell }_{r,s}$-norm minimization problem. Then for all the cases of $(r,s)$, we develop a unified solving algorithm based on the classical ADMM with guaranteed convergence. Moreover, we detailedly discuss the optimal solutions to the ${\ell }_r∕{\ell }_s$-norm subproblem. Finally, a representative example is shown to demonstrate the effectiveness of the proposed approach.

给定多个单变量或多元实多项式 $F_{\mathrm{1个}}，\dots ，F_{米}$ 和期望的实零 $\mathbf{ž}$，研究了计算实多项式的问题 $\stackrel{〜}{F}$ 这样 $\stackrel{〜}{F}（\mathbf{ž}）=0$ 和之间的距离 $\stackrel{〜}{F}$ 和 $F_{\mathrm{1个}}，\dots ，F_{米}$ 最小，其中距离由范数对定义 $（{\ell }_{\mathrm{[R}}，{\ell }_s）$。尽管在先前的工作中已经研究了几何和数值算法，但是只能处理很少的范数对，并且这些算法确实难以实现和推广。本文从优化的角度提出了一个统一的算法框架。我们首先将问题转化为约束${\ell }_{\mathrm{[R}，s}$-规范最小化问题。那么对于所有情况$（\mathrm{[R}，s）$，我们开发了基于经典ADMM且具有保证收敛性的统一求解算法。此外，我们详细讨论了针对${\ell }_{\mathrm{[R}}∕{\ell }_s$-norm子问题。最后，通过一个有代表性的例子来说明所提出方法的有效性。