两两结构定位问题的多尺度半定规划方法
Multiscale semidefinite programming approach to positioning problems with pairwise structure
摘要：
我们考虑成对目标函数的优化，即，对于某些连续状态空间$\mathcal{X}_i$中的$x_i$，形式为$H(\mathbf{x}) = H(x_1,\ldots,x_N) = \sum_{1\leq i<j \leq N} H_{ij}(x_i,x_j)$的目标函数。在这种情况下，全局优化通常会被可能存在的伪局部最小值和由于维数灾难而不可能进行全局搜索所混淆。在本文中，我们接近这样的问题，通过凸松弛的边缘多面体考虑图形建模，进行多尺度的方式，利用光滑的费用函数。我们从理论上表明，与现有的方法相比，这样的方法是有利的，即使在简单的设置传感器网络定位（SNL）。我们成功地将我们的方法应用于SNL问题，特别是高噪声的困难情况。我们还验证了性能的优化的Lennard-Jones潜力，这是困扰着许多接近最佳的配置的存在。我们证明，在MMR允许我们有效地探索这些配置。
英文摘要：
We consider the optimization of pairwise objective functions, i.e., objective functions of the form $H(\mathbf{x}) = H(x_1,\ldots,x_N) = \sum_{1\leq i<j \leq N} H_{ij}(x_i,x_j)$ for $x_i$ in some continuous state spaces $\mathcal{X}_i$. Global optimization in this setting is generally confounded by the possible existence of spurious local minima and the impossibility of global search due to the curse of dimensionality. In this paper, we approach such problems via convex relaxation of the marginal polytope considered in graphical modeling, proceeding in a multiscale fashion which exploits the smoothness of the cost function. We show theoretically that, compared with existing methods, such an approach is advantageous even in simple settings for sensor network localization (SNL). We successfully apply our method to SNL problems, particularly difficult instances with high noise. We also validate performance on the optimization of the Lennard-Jones potential, which is plagued by the existence of many near-optimal configurations. We demonstrate that in MMR allows us to effectively explore these configurations.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
