Alternative methods for representing the inverse of linear programming basis matrices

Methods for representing the inverse of Linear Programming (LP) basis matrices are closely related to techniques for solving a system of sparse unsymmetric linear equations by direct methods. It is now well accepted that for these problems the static process of reordering the matrix in the lower block triangular (LBT) form constitutes the initial step. We introduce a combined static and dynamic factorisation of a basis matrix and derive its inverse which we call the partial elimination form of the inverse (PEFI). This factorization takes advantage of the LBT structure and produces a sparser representation of the inverse than the elimination form of the inverse (EFI). In this we make use of the original columns (of the constraint matrix) which are in the basis. To represent the factored inverse it is, however, necessary to introduce special data structures which are used in the forward and the backward transformations (the two major algorithmic steps) of the simplex method. These correspond to solving a system of equations and solving a system of equations with the transposed matrix respectively. In this paper we compare the nonzero build up of PEFI with that of EFI. We have also investigated alternative methods for updating the basis inverse in the PEFI representation. The results of our experimental investigation are presented in this pape

Application of Linear Programming to Analyze Profit of Flour Factory, in the Case of Sanate Flour Factory, at Robe Town

The purpose of this study was to analyze the total production and profit of Sanate flour factory located in Ethiopia, Oromia regional state, Bale zone, Robe town, by applying linear programming. A factory is situated within Robe town about 430 KM, from Addis Ababa (Capital of Ethiopia). Today linear programming was the most popular method of manipulating a large amount of data. Hence, Different studies bring out the necessity of using quantitative techniques for utilization in the factory. So, in this paper to analyze the production and profit of this factory, the study incorporates different steps; the first step is collecting data. A data collecting formats prepared and circulated among factory staff to executive managers, co-managers, sellers, machine operators, and technicians to determine the production, sales, and profit during five months of November 30, 2018- June 18, 2019. In the second step, a collected data is modeled to mathematical form, particularly modeled to linear program. In the third step the mathematical modeled data was solved (analyzed). Finally, depending on the empirical results (the solution of a modeled data) some problem facing the factory was indicated and the solution for the problem has been recommended

Baechi: Fast Device Placement of Machine Learning Graphs

Machine Learning graphs (or models) can be challenging or impossible to train when either devices have limited memory, or models are large. To split the model across devices, learning-based approaches are still popular. While these result in model placements that train fast on data (i.e., low step times), learning-based model-parallelism is time-consuming, taking many hours or days to create a placement plan of operators on devices. We present the Baechi system, the first to adopt an algorithmic approach to the placement problem for running machine learning training graphs on small clusters of memory-constrained devices. We integrate our implementation of Baechi into two popular open-source learning frameworks: TensorFlow and PyTorch. Our experimental results using GPUs show that: (i) Baechi generates placement plans 654 X - 206K X faster than state-of-the-art learning-based approaches, and (ii) Baechi-placed model's step (training) time is comparable to expert placements in PyTorch, and only up to 6.2% worse than expert placements in TensorFlow. We prove mathematically that our two algorithms are within a constant factor of the optimal. Our work shows that compared to learning-based approaches, algorithmic approaches can face different challenges for adaptation to Machine learning systems, but also they offer proven bounds, and significant performance benefits.Comment: Extended version of SoCC 2020 paper: https://dl.acm.org/doi/10.1145/3419111.342130

Development of free software for modeling the propagation of 2D elastic waves by the finite element method

El proyecto de investigación es el “desarrollo de un software libre para modelado por elementos finitos de la propagación de ondas sísmicas en 2d para aplicaciones de prospección de yacimientos de petróleo” y su objetivo principal como su nombre lo indica es la generación de un software libre para el modelado de ondas su autor es Peter Escamilla Mahecha con la ayuda de su director de tesis PhD. Sebastián Roa. En este documento el autor presenta al público el desarrollo de la aplicación EWaveFem para realizar el simulado de ondas por medio de elementos finitos, este proyecto utiliza los elementos más simples los cuales son elementos triangulares CST, el software presentado tiene configuración de parámetros iniciales, procesamiento y generación de resultados dentro del mismo aplicativo. También se muestra el estado actual del software que trabaja elementos finitos y simulaciones de onda como es el caso de SpecFem2D, EcoElast2D y Elmer, y se hace una comparación entre las particularidades de cada uno de estos aplicativos y sus pros y contras para el caso de estudio que es propagación de ondas en el subsuelo. Finalmente dentro del libro de tesis también se realiza una descripción de la formulación del algoritmo de elementos finitos desde la definición de la formula a discretizar, pasando por la definición de los elementos, como obtener tanto su matriz de rigidez como de masa a finalmente obtener el desplazamiento, la velocidad y la aceleración para cada uno de los nodos en cada uno de los diferentes tiempos.Universitat Oberta de Catalunya UOCIntroducción 1 1. Marco teórico 2 1.1. Software libre 2 1.1.1. Git. 3 1.1.2. Github (GitHub, 2018). 3 1.1.3. Licenciamiento software. 3 1.2.1. C. 4 1.2.2. C++. 5 1.2.3. .Net. 5 1.2.4. Java. 6 1.2.5. Fortran. 6 1.2.6. Ruby. 7 1.2.7. Python. 7 1.3.1. Elmer. 8 1.3.2. EcoElast2D. 9 1.3.3. 3Dhp90. 9 1.3.4. SpecFem2D. 10 1.3.5. Gmsh. 11 1.3.6. Cubit. 11 1.3.7. GiD. 12 1.4.1. Métodos directos. 13 1.4.2. Método de elementos finitos.. 13 1.4.3. Definición del Algoritmo. 13 1.4.4. Discretización del Dominio. 13 2. Método de investigación 23 2.1. Sistema de carga de datos 23 2.2. Vector de fuerza 24 2.3. Generar archivo 24 2.4. Desarrollo de la interfaz gráfica 25 2.5. Publicación del software en comunidad open source 25 3. Resultados 26 3.1. algoritmo modelación ondas sismicas usando elementos finitos 26 3.2. Comunidad y repositorio 26 3.3. Análisis a modo comparativo 26 3.4. Algoritmo implementado en el código 27 3.5. interfaz de usuario generada 29 3.6. Programas existentes y aspectos diferenciadores 42 4. Conclusiones 44 5. Trabajos futuros 45 Bibliografía 46 Anexos 48MaestríaThe investigation project is titled “Development of an open source software to model waves propagation in 2D by using the finite element method to apply in the prospecting of oil deposits” and the main objective as its name implies is the creation of a free software for modeling waves the author is Peter Escamilla Mahecha with the help of his director Phd. Sebastian Roa. In this document the author presents to the public the development of the software EWaveFem for making waves simulations by using the finite element method, this project uses the simplest elements which are the triangular elements CST, the presented software receives some initial parameters, makes the processing and generates the results with the same application. This document also shows the actual state of the finite element and wave simulation software like SpecFem2, EcoElast2D and Elmer, and compares the different aspects of these applications, its pros and cons, for the study case which is the wave propagation in the subsoil. Finally in this document the formulation of the finite element method algorithm is described from the definition of the formula to discretize, passing by the definition of the elements, like getting the stiffness matrix and the mass matrix to finally get the displacement, speed and acceleration for each node in each time step

Schnelle SVM Regularisierungspfadberechnung: Theorie und Implementierung

Im Fokus dieser Diplomarbeit steht die Implementierung und theoretische Analyse eines neuen Algorithmus zur Berechnung des gesamten Lösungspfades von allgemeinen Support Vector Machines bezüglich ihres Regularisierungsparameters. Dessen optimaler Wert wird im Ergebnis durch einen solchen Pfad deutlich leichter auffindbar. Erreicht wird dieses Ziel mit Hilfe der nicht-approximativen Criss-Cross Methode aus dem Bereich der linearen Komplementaritätsprobleme. Neben dem geometrischen Verhalten dieser Methode wird insbesondere auf deren effi ziente Initialisierung zu Beginn eines Lösungspfades eingegangen. Darüber hinaus zeigt diese Arbeit auf, dass auch Probleme der Conjoint Analyse in Support Vector Machines überführt und entsprechend gelöst werden können. Abschließend werden die theoretischen Resultate anhand von Conjoint-Analyse-Datensätzen und solchen für Support Vector Machines veranschaulich

Two dimensional search algorithms for linear programming

Linear programming is one of the most important classes of optimization problems. These mathematical models have been used by academics and practitioners to solve numerous real world applications. Quickly solving linear programs impacts decision makers from both the public and private sectors. Substantial research has been performed to solve this class of problems faster, and the vast majority of the solution techniques can be categorized as one dimensional search algorithms. That is, these methods successively move from one solution to another solution by solving a one dimensional subspace linear program at each iteration. This dissertation proposes novel algorithms that move between solutions by repeatedly solving a two dimensional subspace linear program. Computational experiments demonstrate the potential of these newly developed algorithms and show an average improvement of nearly 25% in solution time when compared to the corresponding one dimensional search version. This dissertation\u27s research creates the core concept of these two dimensional search algorithms, which is a fast technique to determine an optimal basis and an optimal solution to linear programs with only two variables. This method, called the slope algorithm, compares the slope formed by the objective function with the slope formed by each constraint to determine a pair of constraints that intersect at an optimal basis and an optimal solution. The slope algorithm is implemented within a simplex framework to perform two dimensional searches. This results in the double pivot simplex method. Differently than the well-known simplex method, the double pivot simplex method simultaneously pivots up to two basic variables with two nonbasic variables at each iteration. The theoretical computational complexity of the double pivot simplex method is identical to the simplex method. Computational results show that this new algorithm reduces the number of pivots to solve benchmark instances by approximately 40% when compared to the classical implementation of the simplex method, and 20% when compared to the primal simplex implementation of CPLEX, a high performance mathematical programming solver. Solution times of some random linear programs are also improved by nearly 25% on average. This dissertation also presents a novel technique, called the ratio algorithm, to find an optimal basis and an optimal solution to linear programs with only two constraints. When the ratio algorithm is implemented within a simplex framework to perform two dimensional searches, it results in the double pivot dual simplex method. In this case, the double pivot dual simplex method behaves similarly to the dual simplex method, but two variables are exchanged at every step. Two dimensional searches are also implemented within an interior point framework. This dissertation creates a set of four two dimensional search interior point algorithms derived from primal and dual affine scaling and logarithmic barrier search directions. Each iteration of these techniques quickly solves a two dimensional subspace linear program formed by the intersection of two search directions and the feasible region of the linear program. Search directions are derived by orthogonally partitioning the objective function vector, which allows these novel methods to improve the objective function value at each step by at least as much as the corresponding one dimensional search version. Computational experiments performed on benchmark linear programs demonstrate that these two dimensional search interior point algorithms improve the average solution time by approximately 12% and the average number of iterations by 15%. In conclusion, this dissertation provides a change of paradigm in linear programming optimization algorithms. Implementing two dimensional searches within both a simplex and interior point framework typically reduces the computational time and number of iterations to solve linear programs. Furthermore, this dissertation sets the stage for future research topics in multidimensional search algorithms to solve not only linear programs but also other critical classes of optimization methods. Consequently, this dissertation\u27s research can become one of the first steps to change how commercial and open source mathematical programming software will solve optimization problems

Développement de méthodes parallèles pour des problèmes de grande taille

Portée des travaux -- Choix des outils informatiques -- Analyse du potentiel parallèle -- Résolution de la relaxation linéaire en parallèle -- Plus court chemin avec contraintes de ressources en parallèle