平方和程序的抽样algebraic varieties
Sampling algebraic varieties for sum of squares programs
摘要：
研究了多项式函数在集合$V\cap R^n$上的平方和松弛优化问题，其中$V$是一个复代数簇.我们提出了一种新的方法，而不是依赖于一些代数描述，表示$V$与一个脱殊集的复杂样本。这种方法只依赖于$V$的几何形状，避免了表示问题，如多重性和选择的发电机。它还利用了坐标环的结构，以减少相应的半定程序（SDP）的大小。此外，输入可以作为直线程序给出。我们的方法是特别有吸引力的品种，很容易采样，但其定义方程是复杂的，如$SO(n)$，格拉斯曼或秩$k$张量。对于任意的簇，我们可以通过使用数值代数几何的工具来获得所需的样本。通过这种方式，我们连接SOS优化和数值代数几何领域。
英文摘要：
We study sum of squares (SOS) relaxations to optimize polynomial functions over a set $V\cap R^n$, where $V$ is a complex algebraic variety. We propose a new methodology that, rather than relying on some algebraic description, represents $V$ with a generic set of complex samples. This approach depends only on the geometry of $V$, avoiding representation issues such as multiplicity and choice of generators. It also takes advantage of the coordinate ring structure to reduce the size of the corresponding semidefinite program (SDP). In addition, the input can be given as a straight-line program. Our methods are particularly appealing for varieties that are easy to sample from but for which the defining equations are complicated, such as $SO(n)$, Grassmannians or rank $k$ tensors. For arbitrary varieties we can obtain the required samples by using the tools of numerical algebraic geometry. In this way we connect the areas of SOS optimization and numerical algebraic geometry.
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