基于张量分解的全局优化

Arthur Marmin, M. Castella, J. Pesquet
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引用次数: 1

摘要

虽然全局优化在非凸环境中是一个具有挑战性的话题,但最近一种优化多项式的方法将问题重新表述为度量上的等效问题,即矩问题。然后将其松弛为一个凸半定规划问题,其解给出支持最优点的测度的首矩。然而,从这些矩中提取多项式问题的全局解仍然是困难的,特别是当后者的估计很差时。在本文中,我们解决了提取最优点的问题,并将其解释为张量分解问题。通过利用为噪声张量分解开发的工具,我们提出了一种方法,从其相应矩问题的解的噪声估计中找到多项式优化问题的全局解。最后,通过详细的实例分析,说明了张量分解方法在全局多项式优化中的应用价值。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Globally Optimizing Owing to Tensor Decomposition
While global optimization is a challenging topic in the nonconvex setting, a recent approach for optimizing polynomials reformulates the problem as an equivalent problem on measures, which is called a moment problem. It is then relaxed into a convex semidefinite programming problem whose solution gives the first moments of a measure supporting the optimal points. However, extracting the global solutions to the polynomial problem from those moments is still difficult, especially if the latter are poorly estimated. In this paper, we address the issue of extracting optimal points and interpret it as a tensor decomposition problem. By leveraging tools developed for noisy tensor decomposition, we propose a method to find the global solutions to a polynomial optimization problem from a noisy estimation of the solution of its corresponding moment problem. Finally, the interest of tensor decomposition methods for global polynomial optimization is shown through a detailed case study.
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