DOCSLIB.ORG

Iterative Methods for Solving Optimization Problems

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Iterative Methods for Solving Optimization Problems ITERATIVE METHODS FOR SOLVING OPTIMIZATION PROBLEMS Shoham Sabach ITERATIVE METHODS FOR SOLVING OPTIMIZATION PROBLEMS Research Thesis In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Shoham Sabach Submitted to the Senate of the Technion - Israel Institute of Technology Iyar 5772 Haifa May 2012 The research thesis was written under the supervision of Prof. Simeon Reich in the Department of Mathematics Publications: piq Reich, S. and Sabach, S.: Three strong convergence theorems for iterative methods for solving equi- librium problems in reflexive Banach spaces, Optimization Theory and Related Topics, Contemporary Mathematics, vol. 568, Amer. Math. Soc., Providence, RI, 2012, 225{240. piiq Mart´ın-M´arquez,V., Reich, S. and Sabach, S.: Iterative methods for approximating fixed points of Bregman nonexpansive operators, Discrete and Continuous Dynamical Systems, accepted for publi- cation. Impact Factor: 0.986. piiiq Sabach, S.: Products of finitely many resolvents of maximal monotone mappings in reflexive Banach spaces, SIAM Journal on Optimization 21 (2011), 1289{1308. Impact Factor: 2.091. pivq Kassay, G., Reich, S. and Sabach, S.: Iterative methods for solving systems of variational inequalities in reflexive Banach spaces, SIAM J. Optim. 21 (2011), 1319{1344. Impact Factor: 2.091. pvq Censor, Y., Gibali, A., Reich S. and Sabach, S.: The common variational inequality point problem, Set-Valued and Variational Analysis 20 (2012), 229{247. Impact Factor: 0.333. pviq Borwein, J. M., Reich, S. and Sabach, S.: Characterization of Bregman firmly nonexpansive operators using a new type of monotonicity, J. Nonlinear Convex Anal. 12 (2011), 161{183. Impact Factor: 0.738. pviiq Reich, S. and Sabach, S.: A projection method for Bregman firmly nonexpansive operators in reflexive Banach spaces, J. Fixed Point Theory Appl. 9 (2011), 101{116. Impact Factor: 0.545. pviiiq Reich, S. and Sabach, S.: Two strong convergence theorems for Bregman strongly nonexpansive operators in reflexive Banach spaces, Nonlinear Analysis 73 (2010), 122{135. Impact Factor: 1.279. pixq Butnariu, D., Reich, S. and Sabach, S.: A strong convergence theorem for resolvents of monotone operators, J. Convex Anal. 17 (2010), 991{1006. Impact Factor: 0.908. pxq Reich, S. and Sabach, S.: Existence and approximation of fixed points of Bregman firmly nonexpansive mappings in reflexive Banach spaces, Fixed-Point Algorithms for Inverse Problems in Science and Engineering, Springer, New York, 2010, 299{314. pxiq Reich, S. and Sabach, S.: Two strong convergence theorems for a proximal method in reflexive Banach spaces, Numer. Funct. Anal. Optim. 31 (2010), 22{44. Impact Factor: 0.687. pxiiq Reich, S. and Sabach, S.: A strong convergence theorem for a proximal-type algorithm in reflexive Banach spaces, J. Nonlinear Convex Anal. 10 (2009), 471{485. Impact Factor: 0.738. THE GENEROUS FINANCIAL SUPPORT OF THE TECHNION IS GRATEFULLY ACKNOWLEDGED Acknowledgments First and foremost I want to express my sincere gratitude to my advisor Professor Simeon Reich for the continuous support of my Ph.D study and research, for his patience, motivation and immense knowledge. His guidance helped me in all the time of research and writing this thesis. The joy and enthusiasm he has for his research were contagious and motivational for me, even during tough times in my Ph.D. pursuit. Because of his vast knowl- edge and understanding of mathematics, my view of Optimization Theory broadened immensely. It has been a great privilege to learn from him. I finished my master studies at the University of Haifa in 2008 under the supervision of Professor Dan Butnariu who passed away two months after I graduated. Simeon then gave me the option to study for my doctoral degree under his supervision at the Technion. I am grateful to the Department of Mathematics for their help and support. Thanks to Simeon's advice, I had the opportunity to study and work with several amazing mathematicians during my doctoral studies. I am sure that this experience improved my thesis novelty and my research abilities. Among these collaborators, I would like to express my deepest appreciation to Professor Amir Beck form the Technion for his enthusiasm, encourage- ments and able guidance. It was an amazing stimulating environment to study and conduct a fruitful research. My sincere thanks also go to Professors Heinz H. Bauschke, Jonathan M. Borwein, Regina S. Burachik, Yair Censor, Andreas H. Hamel, Gabor Kassay, Elena Resmerita and Kaya C. Yal¸cınfor discussing problems and making observations that improved much the novelty of my thesis. Many many thanks to my friends Dr. Aviv Gibali and Dr. Victoria Mart´ın-M´arquezwho have always been supportive, encouraging and amaz- ing partners for research and fun. Most of all, I wish to thank my parents Eyal and Yael, my wife Orli and my son Jonathan for sharing these unforgettable years with me. Table of Contents List of Tables iii List of Figures iv List of Algorithms v Abstract 1 List of Symbols 2 Introduction 4 1 Preliminaries 12 1.1 Functions . 13 1.1.1 Lower Semicontinuity, Continuity and Lipschitz Continuity . 13 1.1.2 Subdifferentiability . 15 1.1.3 G^ateauxand Fr´echet Differentiability . 17 1.1.4 The Fenchel Conjugate . 19 1.1.5 Geometry of Banach Spaces . 21 1.2 Bregman Distances . 23 1.2.1 Legendre Functions . 25 1.2.2 Totally Convex Functions . 26 1.2.3 The Bregman Projection . 37 1.2.4 Properties of Bregman Distances . 44 1.3 Operators . 54 1.3.1 Bregman Nonexpansive Operators . 55 1.4 Monotone Mappings . 59 1.4.1 Bregman Inverse Strongly Monotone Mappings . 64 2 Fixed Point Properties of Bregman Nonexpansive Operators 66 2.1 Properties of Bregman Nonexpansive Operators . 66 2.1.1 Characterization of BFNE Operators . 81 2.1.2 Examples of BFNE Operators in Euclidean Spaces . 84 3 Iterative Methods for Approximating Fixed Points 88 3.1 Picard's Iteration for Bregman Nonexpansive Operators . 89 3.2 Mann's Iteration for Bregman Nonexpansive Operators . 91 3.3 Haugazeau's Iteration for Bregman Nonexpansive Operators . 94 3.3.1 Convergence Analysis . 96 i Table of Contents - Continued 3.4 An Implicit Method for Approximating Fixed Points . 103 4 Iterative Methods for Approximating Zeroes 107 4.1 Properties of f-Resolvents . 109 4.2 Examples of f-Resolvents . 113 4.2.1 Examples of BS-Resolvents . 113 4.2.2 Examples of FD-Resolvents . 115 4.2.3 Examples of f-Resolvents in Hilbert Spaces . 116 4.3 Iterative Methods Based on Haugazeau's Algorithm . 117 4.3.1 The Solodov-Svaiter Iterative Method . 117 4.4 Iterative Methods Based on the Bauschke-Combettes Algorithm . 128 5 Applications - Equilibrium, Variational and Convex Feasibility Problems136 5.1 Equilibrium Problems . 137 5.1.1 Properties of Resolvents of Bifunctions . 137 5.1.2 Iterative Methods for Solving Equilibrium Problems . 144 5.2 Variational Inequalities . 146 5.2.1 Properties of Anti-Resolvents . 148 5.2.2 Properties of Generalized Resolvents . 151 5.2.3 Iterative Methods for Solving Variational Inequalities . 152 5.3 Convex Feasibility Problems . 156 5.3.1 A Numerical Example . 156 6 Minimal Norm-Like Solutions of Convex Optimizations Problems 161 6.1 Problem Formulation . 162 6.1.1 Stage by Stage Solution . 163 6.2 Mathematical Toolbox . 164 6.2.1 The Gradient Mapping . 165 6.2.2 Cutting Planes . 165 6.3 The Minimal Norm Gradient Algorithm . 170 6.4 Convergence Analysis . 172 6.5 A Numerical Example - A Portfolio Optimization Problem . 177 Index 182 References 182 List of Tables 2.1 Examples of BS-BFNE operators ......................... 85 2.2 Examples of FD-BFNE operators ......................... 87 2.3 Conditions for T to be a BFNE operator ..................... 87 4.1 Examples of f-Resolvents ............................. 116 5.1 The first 10 iterations of Algorithm (5.3.1) with x0 p¡1{2; 3q . 158 5.2 The first 10 iterations of Algorithm (5.3.1) with x0 p3; 3q . 160 iii List of Figures 1.1 The Boltzmann-Shannon entropy . 31 1.2 The Fermi-Dirac entropy . 32 1.3 Implications between types of Bregman nonexpansivity . 57 4.1 The Lambert W function . 113 5.1 Geometric illustration of Algorithm (5.3.1) with x0 p¡1{2; 3q . 158 5.2 Plot of the first 10 iterations of Algorithm (5.3.1) with x0 p¡1{2; 3q . 159 5.3 Geometric illustration of Algorithm (5.3.1) with x0 p3; 3q . 159 iv List of Algorithms Picard Iterative Method . 88 Krasnoselski Iterative Method . 88 Mann Iterative Method . 89 Haugazeau Iterative Method . 89 f-Mann Iterative Method . 91 f-Haugazeau Iterative Method . 94 Minimal Norm-Like Picard Iterative Method . 95 Bauschke-Combettes Iterative Method . 95 f-Bauschke-Combettes Iterative Method . 95 Minimal Norm-Like Bauschke-Combettes Iterative Method . 96 Proximal Point Algorithm . 117 Solodov-Svaiter Proximal Point Algorithm . 117 Kamimura-Takahashi Proximal Point Algorithm . 118 Bauschke-Combettes Proximal Point Algorithm I . 118 Minimal Norm-Like Proximal Point Algorithm I . 119 Minimal Norm-Like Proximal Point Algorithm II . 124 Minimal Norm-Like Proximal Point Algorithm III . 125 Bauschke-Combettes Proximal Point Algorithm II . 128 Wei-Zhou Proximal Point Algorithm . 129 f-Bauschke-Combettes Proximal Point Algorithm I . 129 f-Bauschke-Combettes Proximal Point Algorithm II . 130 Sabach Proximal Point Algorithm I . 131 Sabach Proximal Point Algorithm II . 131 Picard Iterative Method for Solving Equilibrium Problems . 145 Sabach Iterative Method for Solving Equilibrium Problems . 145 f-Mann Iterative Method for Solving Variational Inequalities . 153 Minimal Norm-Like Iterative Method for Solving Variational Inequalities I . 153 Minimal Norm-Like Iterative Method for Solving Variational Inequalities II . 155 The Minimal Norm Gradient Method (Known Lipschitz Constant) . 170 The Minimal Norm Gradient Method (Unknown Lipschitz Constant) . 171 v Abstract This dissertation concerns iterative methods for solving diverse optimiza- tion problems in infinite-dimensional and Euclidean spaces.
Load more

Recommended publications
  • A Learning-Based Iterative Method for Solving Vehicle
    Published as a conference paper at ICLR 2020 ALEARNING-BASED ITERATIVE METHOD FOR SOLVING VEHICLE ROUTING PROBLEMS Hao Lu ∗ Princeton University, Princeton, NJ 08540 [email protected] Xingwen Zhang ∗ & Shuang Yang Ant Financial Services Group, San Mateo, CA 94402 fxingwen.zhang,[email protected] ABSTRACT This paper is concerned with solving combinatorial optimization problems, in par- ticular, the capacitated vehicle routing problems (CVRP). Classical Operations Research (OR) algorithms such as LKH3 (Helsgaun, 2017) are inefficient and dif- ficult to scale to larger-size problems. Machine learning based approaches have recently shown to be promising, partly because of their efficiency (once trained, they can perform solving within minutes or even seconds). However, there is still a considerable gap between the quality of a machine learned solution and what OR methods can offer (e.g., on CVRP-100, the best result of learned solutions is between 16.10-16.80, significantly worse than LKH3’s 15.65). In this paper, we present “Learn to Improve” (L2I), the first learning based approach for CVRP that is efficient in solving speed and at the same time outperforms OR methods. Start- ing with a random initial solution, L2I learns to iteratively refine the solution with an improvement operator, selected by a reinforcement learning based controller. The improvement operator is selected from a pool of powerful operators that are customized for routing problems. By combining the strengths of the two worlds, our approach achieves the new state-of-the-art results on CVRP, e.g., an average cost of 15.57 on CVRP-100. 1 INTRODUCTION In this paper, we focus on an important class of combinatorial optimization, vehicle routing problems (VRP), which have a wide range of applications in logistics.
    [Show full text]
  • The Generalized Simplex Method for Minimizing a Linear Form Under Linear Inequality Restraints
    Pacific Journal of Mathematics THE GENERALIZED SIMPLEX METHOD FOR MINIMIZING A LINEAR FORM UNDER LINEAR INEQUALITY RESTRAINTS GEORGE BERNARD DANTZIG,ALEXANDER ORDEN AND PHILIP WOLFE Vol. 5, No. 2 October 1955 THE GENERALIZED SIMPLEX METHOD FOR MINIMIZING A LINEAR FORM UNDER LINEAR INEQUALITY RESTRAINTS GEORGE B. DANTZIG, ALEX ORDEN, PHILIP WOLFE 1. Background and summary. The determination of "optimum" solutions of systems of linear inequalities is assuming increasing importance as a tool for mathematical analysis of certain problems in economics, logistics, and the theory of games [l;5] The solution of large systems is becoming more feasible with the advent of high-speed digital computers; however, as in the related problem of inversion of large matrices, there are difficulties which remain to be resolved connected with rank. This paper develops a theory for avoiding as- sumptions regarding rank of underlying matrices which has import in applica- tions where little or nothing is known about the rank of the linear inequality system under consideration. The simplex procedure is a finite iterative method which deals with problems involving linear inequalities in a manner closely analogous to the solution of linear equations or matrix inversion by Gaussian elimination. Like the latter it is useful in proving fundamental theorems on linear algebraic systems. For example, one form of the fundamental duality theorem associated with linear inequalities is easily shown as a direct consequence of solving the main prob- lem. Other forms can be obtained by trivial manipulations (for a fuller discus- sion of these interrelations, see [13]); in particular, the duality theorem [8; 10; 11; 12] leads directly to the Minmax theorem for zero-sum two-person games [id] and to a computational method (pointed out informally by Herman Rubin and demonstrated by Robert Dorfman [la]) which simultaneously yields optimal strategies for both players and also the value of the game.
    [Show full text]
  • An Efficient Three-Term Iterative Method for Estimating Linear
    mathematics Article An Efficient Three-Term Iterative Method for Estimating Linear Approximation Models in Regression Analysis Siti Farhana Husin 1,*, Mustafa Mamat 1, Mohd Asrul Hery Ibrahim 2 and Mohd Rivaie 3 1 Faculty of Informatics and Computing, Universiti Sultan Zainal Abidin, Terengganu 21300, Malaysia; [email protected] 2 Faculty of Entrepreneurship and Business, Universiti Malaysia Kelantan, Kelantan 16100, Malaysia; [email protected] 3 Department of Computer Sciences and Mathematics, Universiti Teknologi Mara, Terengganu 54000, Malaysia; [email protected] * Correspondence: [email protected] Received: 15 April 2020; Accepted: 20 May 2020; Published: 15 June 2020 Abstract: This study employs exact line search iterative algorithms for solving large scale unconstrained optimization problems in which the direction is a three-term modification of iterative method with two different scaled parameters. The objective of this research is to identify the effectiveness of the new directions both theoretically and numerically. Sufficient descent property and global convergence analysis of the suggested methods are established. For numerical experiment purposes, the methods are compared with the previous well-known three-term iterative method and each method is evaluated over the same set of test problems with different initial points. Numerical results show that the performances of the proposed three-term methods are more efficient and superior to the existing method. These methods could also produce an approximate linear regression equation to solve the regression model. The findings of this study can help better understanding of the applicability of numerical algorithms that can be used in estimating the regression model. Keywords: steepest descent method; large-scale unconstrained optimization; regression model 1.
    [Show full text]
  • Revised Simplex Algorithm - 1
    Lecture 9: Linear Programming: Revised Simplex and Interior Point Methods (a nice application of what we have learned so far) Prof. Krishna R. Pattipati Dept. of Electrical and Computer Engineering University of Connecticut Contact: [email protected] (860) 486-2890 ECE 6435 Fall 2008 Adv Numerical Methods in Sci Comp October 22, 2008 Copyright ©2004 by K. Pattipati Lecture Outline What is Linear Programming (LP)? Why do we need to solve Linear-Programming problems? • L1 and L∞ curve fitting (i.e., parameter estimation using 1-and ∞-norms) • Sample LP applications • Transportation Problems, Shortest Path Problems, Optimal Control, Diet Problem Methods for solving LP problems • Revised Simplex method • Ellipsoid method….not practical • Karmarkar’s projective scaling (interior point method) Implementation issues of the Least-Squares subproblem of Karmarkar’s method ….. More in Linear Programming and Network Flows course Comparison of Simplex and projective methods References 1. Dimitris Bertsimas and John N. Tsisiklis, Introduction to Linear Optimization, Athena Scientific, Belmont, MA, 1997. 2. I. Adler, M. G. C. Resende, G. Vega, and N. Karmarkar, “An Implementation of Karmarkar’s Algorithm for Linear Programming,” Mathematical Programming, Vol. 44, 1989, pp. 297-335. 3. I. Adler, N. Karmarkar, M. G. C. Resende, and G. Vega, “Data Structures and Programming Techniques for the Implementation of Karmarkar’s Algorithm,” ORSA Journal on Computing, Vol. 1, No. 2, 1989. 2 Copyright ©2004 by K. Pattipati What is Linear Programming? One of the most celebrated problems since 1951 • Major breakthroughs: • Dantzig: Simplex method (1947-1949) • Khachian: Ellipsoid method (1979) - Polynomial complexity, but not competitive with the Simplex → not practical.
    [Show full text]
  • A Non-Iterative Method for Optimizing Linear Functions in Convex Linearly Constrained Spaces
    A non-iterative method for optimizing linear functions in convex linearly constrained spaces GERARDO FEBRES*,** * Departamento de Procesos y Sistemas, Universidad Simón Bolívar, Venezuela. ** Laboratorio de Evolución, Universidad Simón Bolívar, Venezuela. Received… This document introduces a new strategy to solve linear optimization problems. The strategy is based on the bounding condition each constraint produces on each one of the problem’s dimension. The solution of a linear optimization problem is located at the intersection of the constraints defining the extreme vertex. By identifying the constraints that limit the growth of the objective function value, we formulate linear equations system leading to the optimization problem’s solution. The most complex operation of the algorithm is the inversion of a matrix sized by the number of dimensions of the problem. Therefore, the algorithm’s complexity is comparable to the corresponding to the classical Simplex method and the more recently developed Linear Programming algorithms. However, the algorithm offers the advantage of being non-iterative. Key Words: linear optimization; optimization algorithm; constraint redundancy; extreme points. 1. INTRODUCTION Since the apparition of Dantzig’s Simplex Algorithm in 1947, linear optimization has become a widely used method to model multidimensional decision problems. Appearing even before the establishment of digital computers, the first version of the Simplex algorithm has remained for long time as the most effective way to solve large scale linear problems; an interesting fact perhaps explained by the Dantzig [1] in a report written for Stanford University in 1987, where he indicated that even though most large scale problems are formulated with sparse matrices, the inverse of these matrixes are generally dense [1], thus producing an important burden to record and track these matrices as the algorithm advances toward the problem solution.
    [Show full text]
  • Lecture Notes on Iterative Optimization Algorithms
    Charles L. Byrne Department of Mathematical Sciences University of Massachusetts Lowell December 8, 2014 Lecture Notes on Iterative Optimization Algorithms Contents Preface vii 1 Overview and Examples 1 1.1 Overview . 1 1.2 Auxiliary-Function Methods . 2 1.2.1 Barrier-Function Methods: An Example . 3 1.2.2 Barrier-Function Methods: Another Example . 3 1.2.3 Barrier-Function Methods for the Basic Problem 4 1.2.4 The SUMMA Class . 5 1.2.5 Cross-Entropy Methods . 5 1.2.6 Alternating Minimization . 6 1.2.7 Penalty-Function Methods . 6 1.3 Fixed-Point Methods . 7 1.3.1 Gradient Descent Algorithms . 7 1.3.2 Projected Gradient Descent . 8 1.3.3 Solving Ax = b ................... 9 1.3.4 Projected Landweber Algorithm . 9 1.3.5 The Split Feasibility Problem . 10 1.3.6 Firmly Nonexpansive Operators . 10 1.3.7 Averaged Operators . 11 1.3.8 Useful Properties of Operators on H . 11 1.3.9 Subdifferentials and Subgradients . 12 1.3.10 Monotone Operators . 13 1.3.11 The Baillon{Haddad Theorem . 14 2 Auxiliary-Function Methods and Examples 17 2.1 Auxiliary-Function Methods . 17 2.2 Majorization Minimization . 17 2.3 The Method of Auslander and Teboulle . 18 2.4 The EM Algorithm . 19 iii iv Contents 3 The SUMMA Class 21 3.1 The SUMMA Class of Algorithms . 21 3.2 Proximal Minimization . 21 3.2.1 The PMA . 22 3.2.2 Difficulties with the PMA . 22 3.2.3 All PMA are in the SUMMA Class . 23 3.2.4 Convergence of the PMA .
    [Show full text]
  • Topic 3 Iterative Methods for Ax = B
    Topic 3 Iterative methods for Ax = b 3.1 Introduction Earlier in the course, we saw how to reduce the linear system Ax = b to echelon form using elementary row operations. Solution methods that rely on this strategy (e.g. LU factorization) are robust and efficient, and are fundamental tools for solving the systems of linear equations that arise in practice. These are known as direct methods, since the solution x is obtained following a single pass through the relevant algorithm. We are now going to look at some alternative approaches that fall into the category of iterative methods . These techniques can only be applied to square linear systems ( n equations in n unknowns), but this is of course a common and important case. 45 Iterative methods for Ax = b begin with an approximation to the solution, x0, then seek to provide a series of improved approximations x1, x2, … that converge to the exact solution. For the engineer, this approach is appealing because it can be stopped as soon as the approximations xi have converged to an acceptable precision, which might be something as crude as 10 −3. With a direct method, bailing out early is not an option; the process of elimination and back-substitution has to be carried right through to completion, or else abandoned altogether. By far the main attraction of iterative methods, however, is that for certain problems (particularly those where the matrix A is large and sparse ) they are much faster than direct methods. On the other hand, iterative methods can be unreliable; for some problems they may exhibit very slow convergence, or they may not converge at all.
    [Show full text]
  • On Some Iterative Methods for Solving Systems of Linear Equations
    Computational and Applied Mathematics Journal 2015; 1(2): 21-28 Published online February 20, 2015 (http://www.aascit.org/journal/camj) On Some Iterative Methods for Solving Systems of Linear Equations Fadugba Sunday Emmanuel Department of Mathematical Sciences, Ekiti State University, Ado Ekiti, Nigeria Email address [email protected], [email protected] Citation Fadugba Sunday Emmanuel. On Some Iterative Methods for Solving Systems of Linear Equations. Computational and Applied Mathematics Journal. Vol. 1, No. 2, 2015, pp. 21-28. Keywords Abstract Iterative Method, This paper presents some iterative methods for solving system of linear equations Jacobi Method, namely the Jacobi method and the modified Jacobi method. The Jacobi method is an Modified Jacobi Method algorithm for solving system of linear equations with largest absolute values in each row and column dominated by the diagonal elements. The modified Jacobi method also known as the Gauss Seidel method or the method of successive displacement is useful for the solution of system of linear equations. The comparative results analysis of the Received: January 17, 2015 two methods was considered. We also discussed the rate of convergence of the Jacobi Revised: January 27, 2015 method and the modified Jacobi method. Finally, the results showed that the modified Accepted: January 28, 2015 Jacobi method is more efficient, accurate and converges faster than its counterpart “the Jacobi Method”. 1. Introduction In computational mathematics, an iterative method attempts to solve a system of linear equations by finding successive approximations to the solution starting from an initial guess. This approach is in contrast to direct methods, which attempt to solve the problem by a finite sequence of some operations and in the absence of rounding errors.
    [Show full text]
  • Matrices and Matroids for Systems Analysis (Algorithms And
    Algorithms and Combinatorics Volume 20 Editorial Board R.L. Graham, La Jolla B. Korte, Bonn L. Lovasz,´ Budapest A. Wigderson, Princeton G.M. Ziegler, Berlin Kazuo Murota Matrices and Matroids for Systems Analysis 123 Kazuo Murota Department of Mathematical Informatics Graduate School of Information Science and Technology University of Tokyo Tokyo, 113-8656 Japan [email protected] ISBN 978-3-642-03993-5 e-ISBN 978-3-642-03994-2 DOI 10.1007/978-3-642-03994-2 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2009937412 c Springer-Verlag Berlin Heidelberg 2000, first corrected softcover printing 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: deblik, Berlin Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Preface Interplay between matrix theory and matroid theory is the main theme of this book, which offers a matroid-theoretic approach to linear algebra and, reciprocally, a linear-algebraic approach to matroid theory.
    [Show full text]
  • Lecture 20 1 Linear Equations
    U.C. Berkeley — CS270: Algorithms Lecture 20 Professor Vazirani and Professor Rao Scribe: Anupam Last revised Lecture 20 1 Linear Equations Gaussian elimination solves the linear system Ax = b in time O(n3). If A is a symmetric positive semidefinite matrix then there are iterative methods for solving linear systems in A. Each iteration in an iterative method amounts to matrix vector multiplications, the number of iterations for convergence to a vector at distance from the true solution for these methods is bounded in terms of the condition number κ = λmax , λmin (i) Steepest descent: O(κ log 1/) √ (ii) Conjugate gradient: O( κ log 1/). Preconditioning tries to improve the performance of iterative methods by multiplying by a matrix B−1 such that the system B−1Ax = B−1b is ‘easy’ to solve. Trivially A is the best preconditioner for the system, but we are looking for a matrix that is ‘easy’ to invert. Intuitively finding a preconditioner is like finding a matrix that is like A but easy to invert. We will review the iterative methods, then use a low stretch spanning tree as a precon- ditioners for solving Laplacian linear systems, and finally see that a preconditioner obtained by adding some edges to the tree preconditioner achieves a nearly linear time solution. 1.1 Steepest Descent The steepest descent method is the analog of finding the minima of a function by following n the derivative in higher dimensions. The solution to Ax = b is the point in R where the 1 T quadratic form f(x) = 2 x Ax − bx + c achieves the minimum value.
    [Show full text]
  • 6. Iterative Methods for Linear Systems
    TU M ¨unchen 6. Iterative Methods for Linear Systems The stepwise approach to the solution... by minimization Miriam Mehl: 6. Iterative Methods for Linear Systems The stepwise approach to the solution... by minimization, January 14, 2013 1 TU M ¨unchen 6.3. Large Sparse Systems of Linear Equations II – Minimization Methods Formulation as a Problem of Minimization • One of the best known solution methods, the method of conjugate gradients (cg), is based on a different principle than relaxation or smoothing. To see that, we will tackle the problem of solving systems of linear equations using a detour. ; • In the following, let A 2 Rn n be a symmetric and positive definite matrix, i.e. A = AT and xT Ax > 0 8x 6= 0. In this case, solving the linear system Ax = b is equivalent to minimizing the quadratic function 1 f (x) := xT Ax − bT x + c 2 for an arbitrary scalar constant c 2 R. • As A is positive definite, the hyperplane given by z := f (x) defines a paraboloid in + Rn 1 with n-dimensional ellipsoids as isosurfaces f (x) = const:, and f has a global minimum in x. The equivalence of the problems is obvious: 1 1 f 0(x) = AT x + Ax − b = Ax − b = −r(x) = 0 , Ax = b : 2 2 Miriam Mehl: 6. Iterative Methods for Linear Systems The stepwise approach to the solution... by minimization, January 14, 2013 2 TU M ¨unchen A Simple Two-Dimensional Example Compute the intersection point of two lines given by 2x = 0 and y = 0. This corresponds to the linear system y 2 0 x 0 y=0 x = : 0 1 y 0 2x=0 This is equivalento to minimizing the quadratic function 1 f (x; y) = −x2+ y 2: 2 graph of f (x) isolines f (x) = c Miriam Mehl: 6.
    [Show full text]
  • Chapter 5 Iterative Methods for Solving Linear Systems
    Chapter 5 Iterative Methods for Solving Linear Systems 5.1 Convergence of Sequences of Vectors and Matrices In Chapter 2 we have discussed some of the main methods for solving systems of linear equations. These methods are direct methods,inthesensethattheyyieldexact solutions (assuming infinite precision!). Another class of methods for solving linear systems con- sists in approximating solutions using iterative methods. 387 388 CHAPTER 5. ITERATIVE METHODS FOR SOLVING LINEAR SYSTEMS The basic idea is this: Given a linear system Ax = b (with A asquareinvertiblematrix),findanothermatrix B and a vector c,suchthat 1. The matrix I B is invertible − 2. The unique solution x of the system Ax = b is iden- tical to the unique solution u of the system e u = Bue+ c, and then, starting from any vector u0,computethese- quence (uk)givenby uk+1 = Buk + c, k N. 2 Under certain conditions (to be clarified soon), the se- quence (uk)convergestoalimitu which is the unique solution of u = Bu + c,andthusofAx = b. e 5.1. CONVERGENCE OF SEQUENCES OF VECTORS AND MATRICES 389 Let (E, )beanormedvectorspace.Recallthata kk sequence (uk)ofvectorsuk E converges to a limit u E,ifforevery✏>0, there2 some natural number N such2 that u u ✏, for all k N. k k − k ≥ We write u =limuk. k 7!1 If E is a finite-dimensional vector space and dim(E)= n,weknowfromTheorem4.3thatanytwonormsare equivalent, and if we choose the norm ,weseethat kk1 the convergence of the sequence of vectors uk is equivalent to the convergence of the n sequences of scalars formed by the components of these vectors (over any basis).
    [Show full text]