盒上多项式和有理函数的线性最优化
Linear Optimization of Polynomials and Rational Functions over Boxes
摘要：
本文研究了盒上多元多项式的紧线性下界函数的求法问题。这些函数是由多项式展开成伯恩斯坦形式，并使用线性最小二乘函数。证明了给定多项式在提高次数、盒宽和细分时收敛到其下界的性质。随后，我们提供了一个新的方法来构建一个仿射下界函数的多元有理函数的基础上的伯恩斯坦控制点，凸包的非正多项式$s$和度提升。最后给出了与著名的Bernstein常数下界函数的数值比较。
英文摘要：
In this paper, we investigate the problem of finding tight linear lower bounding functions for multivariate polynomials over boxes. These functions are obtained by the expansion of polynomials into Bernstein form and using the linear least squares function. Convergence properties of the given polynomials to their lower bounds are shown with respect to raising the degree, width of the box and subdivision. Subsequently, we provide a new method for constructing an affine lower bounding function for a multivariate rational function based on the Bernstein control points, the convex hull of a non-positive polynomial $s$ and degree elevation. Numerical comparisons with the well known Bernstein constant lower bounding function are finally given.
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