Mathematical Sciences: Global Optimization for Multidimensional Scaling

小行星9622749 多维尺度数值算法（MDS）通过尝试解决特定的优化问题，从不同的数据构造几何配置。 长期以来，人们普遍认为，大多数MDS问题的困扰是存在的地方解决方案，而不是全球解决方案，并已花费了相当大的努力来解决这一困难。 最近，许多研究人员提出了通过模拟退火、隧道和连续等“现成”的全局优化方法来搜索全局解。 这些方法在计算上是相当昂贵的，主要研究者的研究表明，它们可能在很大程度上是不必要的。 这项研究涉及到适当重新制定的结构，许多重要的MDS问题，使当地的搜索，总是找到全球性的解决方案，可以利用。 本研究涉及多维尺度的计算方法，收集的统计技术，数学构造的几何配置的对象，从这些对象之间的距离信息。 例如，化学家可能想要“构建”具有一定原子间距离的分子，或者心理学家可能想要使用人类受试者在成对刺激之间感知到的差异的数据来几何地表示一组刺激。 这些问题带来了巨大的计算挑战。 本研究的重点是开发更有效的方法来计算这些问题的最优解。

9622749 Trosset Numerical algorithms for multidimensional scaling (MDS) construct geometric configurations from dissimilarity data by attempting to solve specific optimization problems. It has long been widely believed that most MDS problems are plagued by the existence of local solutions that are not global solutions, and considerable effort has been expended addressing this difficulty. Recently, various researchers have proposed searching for global solutions by such "off-the-shelf" global optimization methods as simulated annealing, tunneling, and continuation. These methods are quite computationally expensive and research by the principal investigator suggests that they may be largely unnecessary. This research involves properly reformulating the structure of many important MDS problems such that local searches, which invariably find global solutions, can be exploited. This research concerns computational methods for multidimensional scaling, a collection of statistical techniques for mathematically constructing geometric configurations of objects from information about the distances between those objects. For example, a chemist might want to "construct" a molecule with certain interatomic distances, or a psychologist might want to geometrically represent a set of stimuli using data about the differences that human subjects perceived between pairs of stimuli. Such problems pose formidable computational challenges. The focus of this research is on developing more efficient methods for computing optimal solutions of these problems.