我们考虑一个具有四次目标函数的非凸最小化问题，该问题产生于在给定嵌入维数为 d 的欧氏距离矩阵 EDM(P) 时，对包含 n 个点的配置矩阵 P∈R^{nd} 进行精确恢复。文献中一个尚未解决的问题是，是否存在某些条件使该最小化问题具有局部非全局极小点（LNGMP）。我们证明，当 n≤d+1 时，所有二阶驻点都是全局极小点。此外，对于 d=1 且 n≥7>d+1 的情形，我们给出一个例子，并能以解析方式展示其中存在一个局部非全局极小点。对于更一般的情形，我们通过数值方法找到一个二阶驻点，随后证明该四次非凸最小化问题在其附近确实存在一个 LNGM。因此，我们对此前关于其存在性的开放问题给出了肯定回答。我们寻找 LNGM 的方法具有新颖性：首先利用平移和旋转不变性消除 Hessian 矩阵的奇异性，并将问题规模从 P 中的 nd 个变量缩减为 (n−1)d−d(d−1)/2 个变量。这使得 Newton 方法得以稳定，并能够找到满足严格二阶充分最优性条件的例子。能够寻找全局极小点的动机在于，即使在数据含噪和/或不完整的情况下，也能获得配置矩阵的精确恢复，而无需借助凸（半定规划）松弛得到近似解。在研究过程中，我们还针对光滑应力函数的 LNGMP 何时存在、何时不存在提出了新的见解。
英文摘要：
We consider the nonconvex minimization problem, with quartic objective function, that arises in the exact recovery of a configuration matrix $P\in \R^{nd}$ of $n$ points when a Euclidean distance matrix, \EDMp, is given with embedding dimension $d$. It is an open question in the literature whether there are conditions such that the minimization problem admits a local nonglobal minimizer, \lngmp. We prove that all second-order stationary points are global minimizers whenever $n \leq d + 1$. {And, for $d=1$ and $n\geq 7>d+1$, we present an example where we can analytically exhibit a local nonglobal minimizer. For more general cases,} we numerically find a second-order stationary point and then prove that there indeed exists a nearby \lngm for the quartic nonconvex minimization problem. Thus, we answer the previously open question about their existence in the affirmative. Our approach to finding the \lngm is novel in that we first exploit the translation and rotation invariance to remove the singularities of the Hessian, and reduce the size of the problem from $nd$ variables in $P$ to $(n-1)d - d(d-1)/2$ variables. This allows for stabilizing Newton's method, and for finding examples that satisfy the strict second order sufficient optimality conditions.
The motivation for being able to find global minima is to obtain \emph{exact recovery} of the configuration matrix, even in the cases where the data is noisy and/or incomplete, without resorting to approximating solutions from convex (semidefinite programming) relaxations. In the process of our work we present new insights into when \lngmp s of the smooth stress function do and do not exist.
A Tight SDP Relaxation for the Cubic-Quartic Regularization Problem
Xiangtan University（湘潭大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）；University of California San Diego（加州大学圣地亚哥分校）；Hong Kong Baptist University（香港浸会大学）
Sensor network localization has a benign landscape after low-dimensional relaxation
The Wharton School, University of Pennsylvania（宾夕法尼亚大学沃顿商学院）；CERMICS, ENPC, Institut Polytechnique de Paris, CNRS（巴黎理工学院高等学院国家科学研究中心计算数学与控制系统系）；Institute of Mathematics, EPFL（洛桑联邦理工学院数学研究所）
