论文标题
正面最大奇异值的可拖动半限度界限
Tractable semidefinite bounds of positive maximal singular values
论文作者
论文摘要
我们专注于计算给定矩阵的正极最大奇异值（PMSV）的上限。 PMSV问题归结为最大化单位球体和非负轨道的交点上的二次多项式。我们提供了可拖动的半决赛弛豫的层次结构，以尽可能接近后一种多项式优化问题的值。该层次结构基于Pólya代表定理的扩展。这样做，可以将阳性多项式分解为$ s $ nomials的平方的加权总和，其中$ s $可以是先验固定的（$ s = 1 $对应于单元，$ s = 2 $对应于二项式等）。反过来，这使我们能够控制产生的半决赛松弛的大小。
We focus on computing certified upper bounds for the positive maximal singular value (PMSV) of a given matrix. The PMSV problem boils down to maximizing a quadratic polynomial on the intersection of the unit sphere and the nonnegative orthant. We provide a hierarchy of tractable semidefinite relaxations to approximate the value of the latter polynomial optimization problem as closely as desired. This hierarchy is based on an extension of Pólya's representation theorem. Doing so, positive polynomials can be decomposed as weighted sums of squares of $s$-nomials, where $s$ can be a priori fixed ($s=1$ corresponds to monomials, $s=2$ corresponds to binomials, etc.). This in turn allows us to control the size of the resulting semidefinite relaxations.
