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.
翻译:我们考虑对等客观功能的优化,即在一些连续的州空间中对等客观功能,即 $H(\ mathbf{x}) = H(x_1,\ldots,x_N) = H(x_1,\leq i < j\leqN}H ⁇ ij}(x_i,x_j)$_i美元,在某种连续的州空间中对等目标功能的优化。在这种环境下,全球优化一般都由于可能存在虚假的本地迷你和由于维度的诅咒而不可能进行全球搜索而混乱。在本文中,我们通过图形模型中考虑的边际多功能的松软化来处理这些问题,以多尺度的方式利用成本功能的平滑性。我们从理论上表明,与现有的方法相比,这种方法即使在简单的传感器网络本地化环境中也是有利的。我们成功地将我们的方法应用于SNEL问题,特别是高噪声的例子。我们还验证了Lend-Jones 优化我们接近MR 的配置能够有效地展示我们许多MRM 的配置。