A semismooth newton method for the nearest Euclidean distance matrix problem

The Nearest Euclidean distance matrix problem (NEDM) is a fundamentalcomputational problem in applications such asmultidimensional scaling and molecularconformation from nuclear magnetic resonance data in computational chemistry.Especially in the latter application, the problem is often large scale with the number ofatoms ranging from a few hundreds to a few thousands.In this paper, we introduce asemismooth Newton method that solves the dual problem of (NEDM). We prove that themethod is quadratically convergent.We then present an application of the Newton method to NEDM with $H$-weights.We demonstrate the superior performance of the Newton method over existing methodsincluding the latest quadratic semi-definite programming solver.This research also opens a new avenue towards efficient solution methods for the molecularembedding problem

Exact Solution Methods for the kk-item Quadratic Knapsack Problem

The purpose of this paper is to solve the 0-1 $k$-item quadratic knapsack problem $(kQKP)$, a problem of maximizing a quadratic function subject to two linear constraints. We propose an exact method based on semidefinite optimization. The semidefinite relaxation used in our approach includes simple rank one constraints, which can be handled efficiently by interior point methods. Furthermore, we strengthen the relaxation by polyhedral constraints and obtain approximate solutions to this semidefinite problem by applying a bundle method. We review other exact solution methods and compare all these approaches by experimenting with instances of various sizes and densities.Comment: 12 page

A Fresh Variational-Analysis Look at the Positive Semidefinite Matrices World

International audienceEngineering sciences and applications of mathematics show unambiguously that positive semidefiniteness of matrices is the most important generalization of non-negative real num- bers. This notion of non-negativity for matrices has been well-studied in the literature; it has been the subject of review papers and entire chapters of books. This paper reviews some of the nice, useful properties of positive (semi)definite matrices, and insists in particular on (i) characterizations of positive (semi)definiteness and (ii) the geometrical properties of the set of positive semidefinite matrices. Some properties that turn out to be less well-known have here a special treatment. The use of these properties in optimization, as well as various references to applications, are spread all the way through. The "raison d'être" of this paper is essentially pedagogical; it adopts the viewpoint of variational analysis, shedding new light on the topic. Important, fruitful, and subtle, the positive semidefinite world is a good place to start with this domain of applied mathematics

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2015-2016 > Academic research: refereed > Publication in refereed journalAccepted ManuscriptSelf-fundedPublishe

Supersymmetry signals of supercritical string cosmology at the Large Hadron Collider

We investigate the minimal supergravity (mSUGRA) signals at the LHC in the context of supercritical string cosmology (SSC). In this theory, the presence of a time dependent dilaton provides us with a smoothly evolving dark energy and modifies the dark matter allowed region of the mSUGRA model with standard cosmology. Such a dilaton dilutes the supersymmetric dark matter density (of neutralinos) by a factor O(10) and consequently the regions with too much dark matter in the standard scenario are allowed in the SSC. The final states expected at the LHC in this scenario, unlike the standard scenario, consist of Z bosons, Higgs bosons, and/or high energy taus. We show how to characterize these final states and determine the model parameters. Using these parameters, we determine the dark matter content and the neutralino-proton cross section. All these techniques can also be applied to determine model parameters in SSC models with different SUSY breaking scenarios.Comment: 26 pages, 21 figures, minor changes and references adde