Euclidean distance geometry and applications

Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance. This is useful in several applications where the input data consists of an incomplete set of distances, and the output is a set of points in Euclidean space that realizes the given distances. We survey some of the theory of Euclidean distance geometry and some of the most important applications: molecular conformation, localization of sensor networks and statics.Comment: 64 pages, 21 figure

PRISMA: PRoximal Iterative SMoothing Algorithm

Motivated by learning problems including max-norm regularized matrix completion and clustering, robust PCA and sparse inverse covariance selection, we propose a novel optimization algorithm for minimizing a convex objective which decomposes into three parts: a smooth part, a simple non-smooth Lipschitz part, and a simple non-smooth non-Lipschitz part. We use a time variant smoothing strategy that allows us to obtain a guarantee that does not depend on knowing in advance the total number of iterations nor a bound on the domain

Oracle-Based Primal-Dual Algorithms for Packing and Covering Semidefinite Programs

Packing and covering semidefinite programs (SDPs) appear in natural relaxations of many combinatorial optimization problems as well as a number of other applications. Recently, several techniques were proposed, that utilize the particular structure of this class of problems, to obtain more efficient algorithms than those offered by general SDP solvers. For certain applications, such as those described in this paper, it maybe required to deal with SDP\u27s with exponentially or infinitely many constraints, which are accessible only via an oracle. In this paper, we give an efficient primal-dual algorithm to solve the problem in this case, which is an extension of a logarithmic-potential based algorithm of Grigoriadis, Khachiyan, Porkolab and Villavicencio (SIAM Journal of Optimization 41 (2001)) for packing/covering linear programs