Topic 3 Iterative Methods for Ax = B
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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. -
Overview of Iterative Linear System Solver Packages
Overview of Iterative Linear System Solver Packages Victor Eijkhout July, 1998 Abstract Description and comparison of several packages for the iterative solu- tion of linear systems of equations. 1 1 Intro duction There are several freely available packages for the iterative solution of linear systems of equations, typically derived from partial di erential equation prob- lems. In this rep ort I will give a brief description of a numberofpackages, and giveaninventory of their features and de ning characteristics. The most imp ortant features of the packages are which iterative metho ds and preconditioners supply; the most relevant de ning characteristics are the interface they present to the user's data structures, and their implementation language. 2 2 Discussion Iterative metho ds are sub ject to several design decisions that a ect ease of use of the software and the resulting p erformance. In this section I will give a global discussion of the issues involved, and how certain p oints are addressed in the packages under review. 2.1 Preconditioners A go o d preconditioner is necessary for the convergence of iterative metho ds as the problem to b e solved b ecomes more dicult. Go o d preconditioners are hard to design, and this esp ecially holds true in the case of parallel pro cessing. Here is a short inventory of the various kinds of preconditioners found in the packages reviewed. 2.1.1 Ab out incomplete factorisation preconditioners Incomplete factorisations are among the most successful preconditioners devel- op ed for single-pro cessor computers. Unfortunately, since they are implicit in nature, they cannot immediately b e used on parallel architectures. -
Jacobi-Davidson, Gauss-Seidel and Successive Over-Relaxation For
Applied Mathematics and Computational Intelligence Volume 6, 2017 [41-52] Jacobi‐Davidson, Gauss‐Seidel and Successive Over‐Relaxation for Solving Systems of Linear Equations Fatini Dalili Mohammed1,a and Mohd Rivaieb 1Department of Computer Sciences and Mathematics, Universiti Teknologi MARA (UiTM) Terengganu, Campus Kuala Terengganu, Malaysia ABSTRACT Linear systems are applied in many applications such as calculating variables, rates, budgets, making a prediction and others. Generally, there are two techniques of solving system of linear equation including direct methods and iterative methods. Some basic solution methods known as direct methods are ineffective in solving many equations in large systems due to slower computation. Due to inability of direct methods, iterative methods are practical to be used in large systems of linear equations as they do not need much storage. In this project, three indirect methods are used to solve large system of linear equations. The methods are Jacobi Davidson, Gauss‐Seidel and Successive Over‐ Relaxation (SOR) which are well known in the field of numerical analysis. The comparative results analysis of the three methods is considered. These three methods are compared based on number of iterations, CPU time and error. The numerical results show that Gauss‐Seidel method and SOR method with ω=1.25 are more efficient than others. This research allows researcher to appreciate the use of iterative techniques for solving systems of linear equations that is widely used in industrial applications. Keywords: system of linear equation, iterative method, Jacobi‐Davidson, Gauss‐Seidel, Successive Over‐Relaxation 1. INTRODUCTION Linear systems are important in our real life. They are applied in various applications such as in calculating variables, rates, budgets, making a prediction and others. -
A Study on Comparison of Jacobi, Gauss-Seidel and Sor Methods For
International Journal of Mathematics Trends and Technology (IJMTT) – Volume 56 Issue 4- April 2018 A Study on Comparison of Jacobi, Gauss- Seidel and Sor Methods for the Solution in System of Linear Equations #1 #2 #3 *4 Dr.S.Karunanithi , N.Gajalakshmi , M.Malarvizhi , M.Saileshwari #Assistant Professor ,* Research Scholar Thiruvalluvar University,Vellore PG & Research Department of Mathematics,Govt.Thirumagal Mills College,Gudiyattam,Vellore Dist,Tamilnadu,India-632602 Abstract — This paper presents three iterative methods for the solution of system of linear equations has been evaluated in this work. The result shows that the Successive Over-Relaxation method is more efficient than the other two iterative methods, number of iterations required to converge to an exact solution. This research will enable analyst to appreciate the use of iterative techniques for understanding the system of linear equations. Keywords — The system of linear equations, Iterative methods, Initial approximation, Jacobi method, Gauss- Seidel method, Successive Over- Relaxation method. 1. INTRODUCTION AND PRELIMINARIES Numerical analysis is the area of mathematics and computer science that creates, analyses, and implements algorithms for solving numerically the problems of continuous mathematics. Such problems originate generally from real-world applications of algebra, geometry and calculus, and they involve variables which vary continuously. These problems occur throughout the natural sciences, social science, engineering, medicine, and business. The solution of system of linear equations can be accomplished by a numerical method which falls in one of two categories: direct or iterative methods. We have so far discussed some direct methods for the solution of system of linear equations and we have seen that these methods yield the solution after an amount of computation that is known advance. -
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. -
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. -
Solving Linear Systems: Iterative Methods and Sparse Systems
Solving Linear Systems: Iterative Methods and Sparse Systems COS 323 Last time • Linear system: Ax = b • Singular and ill-conditioned systems • Gaussian Elimination: A general purpose method – Naïve Gauss (no pivoting) – Gauss with partial and full pivoting – Asymptotic analysis: O(n3) • Triangular systems and LU decomposition • Special matrices and algorithms: – Symmetric positive definite: Cholesky decomposition – Tridiagonal matrices • Singularity detection and condition numbers Today: Methods for large and sparse systems • Rank-one updating with Sherman-Morrison • Iterative refinement • Fixed-point and stationary methods – Introduction – Iterative refinement as a stationary method – Gauss-Seidel and Jacobi methods – Successive over-relaxation (SOR) • Solving a system as an optimization problem • Representing sparse systems Problems with large systems • Gaussian elimination, LU decomposition (factoring step) take O(n3) • Expensive for big systems! • Can get by more easily with special matrices – Cholesky decomposition: for symmetric positive definite A; still O(n3) but halves storage and operations – Band-diagonal: O(n) storage and operations • What if A is big? (And not diagonal?) Special Example: Cyclic Tridiagonal • Interesting extension: cyclic tridiagonal • Could derive yet another special case algorithm, but there’s a better way Updating Inverse • Suppose we have some fast way of finding A-1 for some matrix A • Now A changes in a special way: A* = A + uvT for some n×1 vectors u and v • Goal: find a fast way of computing (A*)-1 -
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. -
Numerical Mathematics
4 Iterative Methods for Solving Linear Systems Iterative methods formally yield the solution x of a linear system after an infinite number of steps. At each step they require the computation of the residual of the system. In the case of a full matrix, their computational cost is therefore of the order of n2 operations for each iteration, to be 2 3 compared with an overall cost of the order of 3 n operations needed by direct methods. Iterative methods can therefore become competitive with direct methods provided the number of iterations that are required to con- verge (within a prescribed tolerance) is either independent of n or scales sublinearly with respect to n. In the case of large sparse matrices, as discussed in Section 3.9, direct methods may be unconvenient due to the dramatic fill-in, although ex- tremely efficient direct solvers can be devised on sparse matrices featuring special structures like, for example, those encountered in the approximation of partial differential equations (see Chapters 12 and 13). Finally, we notice that, when A is ill-conditioned, a combined use of direct and iterative methods is made possible by preconditioning techniques that will be addressed in Section 4.3.2. 4.1 On the Convergence of Iterative Methods The basic idea of iterative methods is to construct a sequence of vectors x(k) that enjoy the property of convergence x = lim x(k), (4.1) k →∞ 124 4. Iterative Methods for Solving Linear Systems where x is the solution to (3.2). In practice, the iterative process is stopped at the minimum value of n such that x(n) x <ε, where ε is a fixed tolerance and is any convenient vector∥ norm.− ∥ However, since the exact solution is obviously∥·∥ not available, it is necessary to introduce suitable stopping criteria to monitor the convergence of the iteration (see Section 4.6). -
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. -
Linear Iterative Methods
Chapter 6 Linear Iterative Methods 6.1 Motivation In Chapter 3 we learned that, in general, solving the linear system of equations n n n 3 Ax = b with A C ⇥ and b C requires (n ) operations. This is too 2 2 O expensive in practice. The high cost begs the following questions: Are there lower cost options? Is an approximation of x good enough? How would such an approximation be generated? Often times we can find schemes that have a much lower cost of computing an approximate solution to x. As an alternative to the direct methods that we studied in the previous chapters, in the present chapter we will describe so called iteration methods for n 1 constructing sequences, x k 1 C , with the desire that x k x := A− b, { }k=1 ⇢ ! as k . The idea is that, given some ">0, we look for a k N such that !1 2 x x " k − k k with respect to some norm. In this context, " is called the stopping tolerance. In other words, we want to make certain the error is small in norm. But a word of caution. Usually, we do not have a direct way of approximating the error. The residual is more readily available. Suppose that x k is an ap- proximation of x = A 1b. The error is e = x x and the residual is − k − k r = b Ax = Ae . Recall that, k − k k e r k k k (A)k k k. x b k k k k r k Thus, when (A) is large, k b k , which is easily computable, may not be a good k k ek indicator of the size of the relative error k x k , which is not directly computable. -
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 .