Random projections for linear programming

Random projections are random linear maps, sampled from appropriate distributions, that approx- imately preserve certain geometrical invariants so that the approximation improves as the dimension of the space grows. The well-known Johnson-Lindenstrauss lemma states that there are random ma- trices with surprisingly few rows that approximately preserve pairwise Euclidean distances among a set of points. This is commonly used to speed up algorithms based on Euclidean distances. We prove that these matrices also preserve other quantities, such as the distance to a cone. We exploit this result to devise a probabilistic algorithm to solve linear programs approximately. We show that this algorithm can approximately solve very large randomly generated LP instances. We also showcase its application to an error correction coding problem.Comment: 26 pages, 1 figur

New error measures and methods for realizing protein graphs from distance data

The interval Distance Geometry Problem (iDGP) consists in finding a realization in $\mathbb{R}^K$ of a simple undirected graph $G=(V,E)$ with nonnegative intervals assigned to the edges in such a way that, for each edge, the Euclidean distance between the realization of the adjacent vertices is within the edge interval bounds. In this paper, we focus on the application to the conformation of proteins in space, which is a basic step in determining protein function: given interval estimations of some of the inter-atomic distances, find their shape. Among different families of methods for accomplishing this task, we look at mathematical programming based methods, which are well suited for dealing with intervals. The basic question we want to answer is: what is the best such method for the problem? The most meaningful error measure for evaluating solution quality is the coordinate root mean square deviation. We first introduce a new error measure which addresses a particular feature of protein backbones, i.e. many partial reflections also yield acceptable backbones. We then present a set of new and existing quadratic and semidefinite programming formulations of this problem, and a set of new and existing methods for solving these formulations. Finally, we perform a computational evaluation of all the feasible solver$+$formulation combinations according to new and existing error measures, finding that the best methodology is a new heuristic method based on multiplicative weights updates

An Overview of Methods using Reduced-Ordered Transformation Matrices for Fault-Tolerant Control of 5-phase Machines with an Open Phase

This paper studies control strategies using modified transformation matrices when five-phase machines operate in oneopen-phase faults. The basic idea of these methods is to maintain the rotating field under asymmetrical conditions as the same as in healthy condition by determining new transformation matrices. The dimension of the new matrices is equal to the number of remaining healthy phases in post-fault conditions. There have been different ways to determine the new transformation matrices applied for different types of five-phase machines in recent decades. In this study, an overview and analyses on these methods will be presented. In addition, advantages and drawbacks of these control strategies are clarified by numerical results

Eliminations of Low-frequency Current Harmonics for Five-phase Open-end Winding Non-sinusoidal Machine Drives applying Neural Networks

This study aims at eliminating unwanted harmonics in current control of a five-phase non-sinusoidal permanent magnet synchronous machine (PMSM) in an open-end winding configuration. The machine is supplied by two voltage source inverters (VSIs) using a single DC-bus voltage. High-frequency harmonics, caused by the zero-sequence current with the inverter switching frequency, have been significantly reduced by using a proper pulse width modulation (PWM) strategy. Meanwhile, low-frequency current harmonics are generated by unwanted harmonics of the back electromotive force (back-EMF) and by the inverter nonlinearity. In this study, the low-frequency current harmonics are nullified by simple adaptive linear neural networks (ADALINEs) in rotor reference frames combined with the back-EMF compensation. As a result, the quality of current control is improved. The effectiveness of the proposed strategies is verified by numerical resultsThis work has been achieved within the framework of CE2I project. CE2I is co-financed by European Union with the financial support of European Regional Development Fund (ERDF), French State and the French Region of Hauts-de-France