DOCSLIB.ORG

Optimization and Approximation

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Optimization and Approximation Optimization and Approximation Elisa Riccietti12 1Thanks to Stefania Bellavia, University of Florence. 2Reference book: Numerical Optimization, Nocedal and Wright, Springer 2 Contents I I part: nonlinear optimization 5 1 Prerequisites 7 1.1 Necessary and sufficient conditions . .9 1.2 Convex functions . 10 1.3 Quadratic functions . 11 2 Iterative methods 13 2.1 Directions for line-search methods . 14 2.1.1 Direction of steepest descent . 14 2.1.2 Newton's direction . 15 2.1.3 Quasi-Newton directions . 16 2.2 Rates of convergence . 16 2.3 Steepest descent method for quadratic functions . 17 2.4 Convergence of Newton's method . 19 3 Line-search methods 23 3.1 Armijo and Wolfe conditions . 23 3.2 Convergence of line-search methods . 27 3.3 Backtracking . 30 3.4 Newton's method . 32 4 Quasi-Newton method 33 4.1 BFGS method . 34 4.2 Global convergence of the BFGS method . 38 5 Nonlinear least-squares problems 41 5.1 Background: modelling, regression . 41 5.2 General concepts . 41 5.3 Linear least-squares problems . 43 5.4 Algorithms for nonlinear least-squares problems . 44 5.4.1 Gauss-Newton method . 44 5.5 Levenberg-Marquardt method . 45 6 Constrained optimization 47 6.1 One equality constraint . 48 6.2 One inequality constraint . 50 6.3 First order optimality conditions . 52 3 4 CONTENTS 6.4 Second order optimality conditions . 58 7 Optimization methods for Machine Learning 61 II Linear and integer programming 63 8 Linear programming 65 8.1 How to rewrite an LP in standard form . 66 8.2 Primal and dual problems . 69 8.3 Convex and strictly convex problems . 71 8.4 Geometry of Ω . 72 8.5 Simplex method . 77 8.5.1 How to choose a starting vertex of Ω . 84 8.5.2 Generalization of the algorithm to the degenerate case . 85 8.5.3 Advantages and disadvantages of the simplex method . 87 9 Flow networks problems 89 9.1 Minimum cost flow problem . 89 9.2 Maximum flow problem . 91 9.2.1 How to find a starting point of Ω . 99 Part I I part: nonlinear optimization 5 Chapter 1 Prerequisites n Let A be an open set of R and let n f : A ⊆ R −! R T x = (x1; : : : ; xn) 7−! f(x) An unconstrained optimization problem is a problem of the form min f(x); x where f is called the objective function. In the following we will assume f to be a nonlinear function. 1. If f is differentiable in x (i.e. if there exist all the partial derivatives of f in x), the n gradient of f in x is rf(x) 2 R : 0@f(x)1 B @x1 C B . C rf(x) = B . C : B . C @@f(x)A @xn n×n 2. If f is two times differentiable in x, the Hessian matrix of f in x is H(x) 2 R 0 T 1 0 @2f(x) @2f(x) 1 @f(x) ··· r B @x @x @x @x C B @x1 C B 1 1 1 n C B C B . C B . C H(x) = Hf (x) = B . C = B . C : B C B C @ @2f(x) @2f(x) A B @f(x)T C ··· @ r A @xn@x1 @xn@xn @xn If f 2 C2(x) then H(x) is a symmetric matrix. 3. Let us remind the first-order Taylor formula with Lagrange form of the remainder. Let f 2 C1(A). Let x; x + h 2 A with h 6= 0 such that the segment fx + th j t 2 [0; 1]g whose endpoints are x and x + h is contained in A. Then it exists t 2 (0; 1), depending on x and h, such that f(x + h) = f(x) + rf(x + th)T h: (1.1) 7 8 CHAPTER 1. PREREQUISITES 4. Let us remind the second-order Taylor formula with Lagrange form of the remainder. Let f 2 C2(A). Let x; x + h 2 A with h 6= 0 such that the segment fx + th j t 2 [0; 1]g whose endpoints are x and x + h is contained in A. Then it exists t 2 (0; 1), depending on x and h, such that 1 f(x + h) = f(x) + rf(x)T h + hT H(x + th)h: (1.2) 2 5. f is convex in A if 8x; y 2 A f(tx + (1 − t)y) ≤ tf(x) + (1 − t)f(y); 8t 2 [0; 1]: 6. f is strictly convex in A if 8x; y 2 A f(tx + (1 − t)y) < tf(x) + (1 − t)f(y); 8t 2 (0; 1): Definition 1.0.1. Let x∗ 2 A. • x∗ is a local minimizer or a local minimum point for f if it exists a neighbourhood Ω of x∗ such that f(x∗) ≤ f(x) 8x 2 Ω: f(x∗) is a local minimum of f. • x∗ is a global minimizer or a global minimum point for f if f(x∗) ≤ f(x) 8x 2 A: f(x∗) is a global minimum of f. • x∗ is a stationary point for f if rf(x∗) = 0: Definition 1.0.2. Directional derivatives n Let p 2 R and f differentiable in a neighbourhood of x. The directional derivative of f in x with respect to the direction p is defined as @f f(x + hp) − f(x) (x) = lim : (1.3) @p h!0 h It holds (see TD1) @f (x) = rf(x)T p: @p Definition 1.0.3. Descent directions n A direction p 2 R is a descent direction for f in x if @f (x) = rf(x)T p < 0; @p π i i.e., if the angle # between p and rf(x) is such that # 2 ; π . 2 If rf(x) 6= 0, we can always find a descent direction: that of the antigradient −∇f(x). From (1.3) it follows that if p is a descent direction, then it exists h¯ > 0 such that f(x + hp) − f(x) < 0 8h 2 (0; h¯): 1.1. NECESSARY AND SUFFICIENT CONDITIONS 9 1.1 Necessary and sufficient conditions In this section we give necessary and sufficient conditions for a point to be a minimum point. Theorem 1.1.1. First order necessary condition Let f 2 C1(Ω) in a neighbourhood Ω of x∗. x∗ is a minimizer for f (in Ω) =) rf(x∗) = 0, i.e. x∗ is a stationary point for f: Proof. We prove it by contradiction. Let us assume that rf(x∗) 6= 0. Let p = −∇f(x∗) the antigradient of f in x∗, clearly p 6= 0. The function g(x) = rf(x)T p is such that g(x∗) = rf(x∗)T p = −∇f(x∗)T rf(x∗) = −k∇f(x∗)k2 < 0: Then, as f 2 C1(Ω) and so g is continuous in Ω, it will remain negative in a neighbourhood of ∗ x , i.e., it exists T 2 R; T > 0 such that 8t 2 [0;T ] 0 > g(x∗ + tp) = rf(x∗ + tp)T p: (1.4) From (1.1), 8τ 2 (0;T ), it exists t 2 (0; 1) such that f(x∗+τp) = f(x∗)+rf(x∗+ tτ p)T τp = f(x∗)+τ rf(x∗ + t0p)T p < f(x∗): |{z}0 | {z } = t 2(0;T ) < 0 from (1.4) Then x∗ cannot be a minimum point for f, which leads us to a contradiction. This is just a necessary conditions, all minimizers are stationary points but not all stationary points are minimizers, they may be maximizers or saddle points. Theorem 1.1.2. Second order necessary condition Let f 2 C2(Ω) for a neighbourhood Ω of x∗. x∗ is a minimum point for f (in Ω) =) H(x∗) is positive semidefinite: Proof. We do the proof by contradiction. Let us assume that H(x∗) is not positive semidefinite, n T ∗ i.e. that it exists p 2 R ; p 6= 0 such that p H(x )p < 0: Let us define g(x) := pT H(x)p; ∗ 2 it holds g(x ) < 0. Then, as f 2 C (Ω) and so g is continuous in Ω, it exists T 2 R; T > 0 such that 8t 2 [0;T ] 0 > g(x∗ + tp) = pT H(x∗ + tp)p: (1.5) From (1.2), 8τ 2 (0;T ), it exists t 2 (0; 1) such that 1 f(x∗ + τp) = f(x∗) + rf(x∗)T τp + (τp)T H(x∗ + tτ p)τp = 2 | {z } |{z}0 = 0 from Theorem 1.1.1 = t 2(0;T ) 1 = f(x∗) + τ 2 pT H(x∗ + t0p)p < f(x∗): 2 | {z } < 0 from (1.5) Then x∗ cannot be a minimum point for f, which leads us to a contradiction. 10 CHAPTER 1. PREREQUISITES In the following theorem we show a sufficient condition: if this condition is satisfied, we are sure to have a minimum point. This is a second order condition: first order derivatives are not enough to establish a sufficient condition. Theorem 1.1.3. Sufficient second-order condition Let f 2 C2(Ω) for a neighbourhood Ω of x∗. rf(x∗) = 0 =) x∗ is a minimum point for f: H(x∗) is positive definite ∗ ∗ ∗ Proof. As H(x ) is positive definite, it exists a neighbourhood B = BT (x ) of x such that n 8x 2 B the matrix H(x) remains positive definite. Then for every p 2 R , 8τ 2 (0;T ) it exists t 2 (0; 1) such that 1 f(x∗ + τp) = f(x∗) + rf(x∗)T τp + (τp)T H(x∗ + tτ p)τp = 2 | {z } |{z}0 = 0 = t 2(0;T ) 1 = f(x∗) + τ 2 pT H(x∗ + t0p)p > f(x∗); 2 | {z } 2B | {z } > 0 i.e.
Load more

Recommended publications
  • Hybrid Whale Optimization Algorithm with Modified Conjugate Gradient Method to Solve Global Optimization Problems
    Open Access Library Journal 2020, Volume 7, e6459 ISSN Online: 2333-9721 ISSN Print: 2333-9705 Hybrid Whale Optimization Algorithm with Modified Conjugate Gradient Method to Solve Global Optimization Problems Layth Riyadh Khaleel, Ban Ahmed Mitras Department of Mathematics, College of Computer Sciences & Mathematics, Mosul University, Mosul, Iraq How to cite this paper: Khaleel, L.R. and Abstract Mitras, B.A. (2020) Hybrid Whale Optimi- zation Algorithm with Modified Conjugate Whale Optimization Algorithm (WOA) is a meta-heuristic algorithm. It is a Gradient Method to Solve Global Opti- new algorithm, it simulates the behavior of Humpback Whales in their search mization Problems. Open Access Library for food and migration. In this paper, a modified conjugate gradient algo- Journal, 7: e6459. https://doi.org/10.4236/oalib.1106459 rithm is proposed by deriving new conjugate coefficient. The sufficient des- cent and the global convergence properties for the proposed algorithm proved. Received: May 25, 2020 Novel hybrid algorithm of the Whale Optimization Algorithm (WOA) pro- Accepted: June 27, 2020 posed with modified conjugate gradient Algorithm develops the elementary Published: June 30, 2020 society that randomly generated as the primary society for the Whales opti- Copyright © 2020 by author(s) and Open mization algorithm using the characteristics of the modified conjugate gra- Access Library Inc. dient algorithm. The efficiency of the hybrid algorithm measured by applying This work is licensed under the Creative it to (10) of the optimization functions of high measurement with different Commons Attribution International License (CC BY 4.0). dimensions and the results of the hybrid algorithm were very good in com- http://creativecommons.org/licenses/by/4.0/ parison with the original algorithm.
    [Show full text]
  • Lecture Notes on Numerical Optimization
    Lecture Notes on Numerical Optimization Miguel A.´ Carreira-Perpi˜n´an EECS, University of California, Merced December 30, 2020 These are notes for a one-semester graduate course on numerical optimisation given by Prof. Miguel A.´ Carreira-Perpi˜n´an at the University of California, Merced. The notes are largely based on the book “Numerical Optimization” by Jorge Nocedal and Stephen J. Wright (Springer, 2nd ed., 2006), with some additions. These notes may be used for educational, non-commercial purposes. c 2005–2020 Miguel A.´ Carreira-Perpi˜n´an 1 Introduction Goal: describe the basic concepts & main state-of-the-art algorithms for continuous optimiza- • tion. The optimization problem: • c (x)=0, i equality constraints (scalar) min f(x) s.t. i ∈E x Rn ci(x) 0, i inequality constraints (scalar) ∈ ≥ ∈ I x: variables (vector); f(x): objective function (scalar). Feasible region: set of points satisfying all constraints. max f min f. ≡ − − 2 2 2 x1 x2 0 Ex. (fig. 1.1): minx1,x2 (x1 2) +(x2 1) s.t. − ≤ • − − x1 + x2 2. ≤ Ex.: transportation problem (LP) • x a i (capacity of factory i) j ij ≤ i ∀ min cijxij s.t. i xij bj i (demand of shop j) xij P ≥ ∀ { } i,j xij 0 i, j (nonnegative production) X P ≥ ∀ cij: shipping cost; xij: amount of product shipped from factory i to shop j. Ex.: LSQ problem: fit a parametric model (e.g. line, polynomial, neural net...) to a data set. Ex. 2.1 • Optimization algorithms are iterative: build sequence of points that converges to the solution. • Needs good initial point (often by prior knowledge).
    [Show full text]
  • Download Thesisadobe
    Exploiting the Intrinsic Structures of Simultaneous Localization and Mapping by Kasra Khosoussi Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the Centre for Autonomous Systems Faculty of Engineering and Information Technology University of Technology Sydney March 2017 Declaration I certify that the work in this thesis has not previously been submitted for a degree nor has it been submitted as part of requirements for a degree except as fully acknowledged within the text. I also certify that the thesis has been written by me. Any help that I have received in my research work and the preparation of the thesis itself has been acknowledged. In addition, I certify that all information sources and literature used are indicated in the thesis. Signature: Date: i Abstract Imagine a robot which is assigned a complex task that requires it to navigate in an unknown environment. This situation frequently arises in a wide spectrum of applications; e.g., sur- gical robots used for diagnosis and treatment operate inside the body, domestic robots have to operate in people’s houses, and Mars rovers explore Mars. To navigate safely in such environments, the robot needs to constantly estimate its location and, simultane- ously, create a consistent representation of the environment by, e.g., building a map. This problem is known as simultaneous localization and mapping (SLAM). Over the past three decades, SLAM has always been a central topic in mobile robotics. Tremendous progress has been made during these years in efficiently solving SLAM using a variety of sensors in large-scale environments.
    [Show full text]
  • Arxiv:1806.10197V1 [Math.NA] 26 Jun 2018 Tive
    LINEARLY CONVERGENT NONLINEAR CONJUGATE GRADIENT METHODS FOR A PARAMETER IDENTIFICATION PROBLEMS MOHAMED KAMEL RIAHI?;† AND ISSAM AL QATTAN‡ ABSTRACT. This paper presents a general description of a parameter estimation inverse prob- lem for systems governed by nonlinear differential equations. The inverse problem is presented using optimal control tools with state constraints, where the minimization process is based on a first-order optimization technique such as adaptive monotony-backtracking steepest descent technique and nonlinear conjugate gradient methods satisfying strong Wolfe conditions. Global convergence theory of both methods is rigorously established where new linear convergence rates have been reported. Indeed, for the nonlinear non-convex optimization we show that under the Lipschitz-continuous condition of the gradient of the objective function we have a linear conver- gence rate toward a stationary point. Furthermore, nonlinear conjugate gradient method has also been shown to be linearly convergent toward stationary points where the second derivative of the objective function is bounded. The convergence analysis in this work has been established in a general nonlinear non-convex optimization under constraints framework where the considered time-dependent model could whether be a system of coupled ordinary differential equations or partial differential equations. Numerical evidence on a selection of popular nonlinear models is presented to support the theoretical results. Nonlinear Conjugate gradient methods, Nonlinear Optimal control and Convergence analysis and Dynamical systems and Parameter estimation and Inverse problem 1. INTRODUCTION Linear and nonlinear dynamical systems are popular approaches to model the behavior of complex systems such as neural network, biological systems and physical phenomena. These models often need to be tailored to experimental data through optimization of parameters in- volved in the mathematical formulations.
    [Show full text]
  • Unconstrained Optimization
    Chapter 4 Unconstrained optimization An unconstrained optimization problem takes the form min f(x) (4.1) x∈Rn for a target functional (also called objective function) f : Rn → R. In this chapter and throughout most of our discussion on optimization, we will assume that f is sufficiently smooth, that is, at least continuously differentiable. In most applications of optimization problem, one is usually interested in a global minimizer x∗, which satisfies f(x∗) ≤ f(x) for all x in Rn (or at least for all x in the domain of interest). Unless f is particularly nice, optimization algorithms are often not guaranteed to yield global minima but only yield local minima. A point x∗ is called a local minimizer if there is a neighborhood N such that f(x∗) ≤ f(x) for all x ∈ N . Similarly, x∗ is called a strict local minimizer f(x∗) < f(x) for all x ∈N with x 6= x∗. 4.1 Fundamentals Sufficient and necessary conditions for local minimizers can be developed from the Taylor expansion of f. Let us recall Example 2.3: If f is two times continuously differentiable then 1 T f(x + h)= f(x)+ ∇f(x)T h + h H(x)h + O(khk3), (4.2) 2 2 m where ∇f is the gradient and H = ∂ f is the Hessian [matrice Hessienne] ∂xi∂xj i,j=1 of f. Ä ä Theorem 4.1 (First-order necessary condition) If x∗ is a local mini- mizer and f is continuously differentiable in an open neighborhood of x∗ then ∇f(x∗) = 0.
    [Show full text]
  • Numerical Optimization
    This is page iii Printer: Opaque this Jorge Nocedal Stephen J. Wright Numerical Optimization Second Edition This is pag Printer: O Jorge Nocedal Stephen J. Wright EECS Department Computer Sciences Department Northwestern University University of Wisconsin Evanston, IL 60208-3118 1210 West Dayton Street USA Madison, WI 53706–1613 [email protected] USA [email protected] Series Editors: Thomas V. Mikosch Stephen M. Robinson University of Copenhagen Department of Industrial and Systems Laboratory of Actuarial Mathematics Engineering DK-1017 Copenhagen University of Wisconsin Denmark 1513 University Avenue [email protected] Madison, WI 53706–1539 USA Sidney I. Resnick [email protected] Cornell University School of Operations Research and Industrial Engineering Ithaca, NY 14853 USA [email protected] Mathematics Subject Classification (2000): 90B30, 90C11, 90-01, 90-02 Library of Congress Control Number: 2006923897 ISBN-10: 0-387-30303-0 ISBN-13: 978-0387-30303-1 Printed on acid-free paper. C 2006 Springer Science+Business Media, LLC. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.
    [Show full text]
  • Line-Search Methods for Smooth Unconstrained Optimization
    Outline 1 Generic Linesearch Framework 2 Computing a descent direction pk (search direction) Line-Search Methods for Smooth Unconstrained Optimization Steepest descent direction Modified Newton direction Quasi-Newton directions for medium scale problems Limited-memory quasi-Newton directions for large-scale problems Daniel P. Robinson Linear CG method for large-scale problems Department of Applied Mathematics and Statistics Johns Hopkins University 3 Choosing the step length αk (linesearch) Backtracking-Armijo linesearch Wolfe conditions Strong Wolfe conditions September 17, 2020 4 Complete Algorithms Steepest descent backtracking Armijo linesearch method Modified Newton backtracking-Armijo linesearch method Modified Newton linesearch method based on the Wolfe conditions Quasi-Newton linesearch method based on the Wolfe conditions 1 / 106 2 / 106 The basic idea The problem minimize f(x) n consider descent methods, i.e., f(xk+1) < f(xk) x2R n calculate a search direction pk from xk where the objective function f : R ! R ensure that this direction is a descent direction, i.e., 1 2 assume that f 2 C (sometimes C ) and is Lipschitz continuous T gk pk < 0 if gk 6= 0 in practice this assumption may be violated, but the algorithms we develop may still work so that, for small steps along pk, the objective function f will be reduced in practice it is very rare to be able to provide an explicit minimizer calculate a suitable steplength αk > 0 so that we consider iterative methods: given starting guess x0, generate sequence f(xk + αkpk) < fk fxkg for
    [Show full text]
  • Nonsmooth Optimization Via Bfgs
    NONSMOOTH OPTIMIZATION VIA BFGS ADRIAN S. LEWIS∗ AND MICHAEL L. OVERTON† Abstract. We investigate the BFGS algorithm with an inexact line search when applied to non- smooth functions, not necessarily convex. We define a suitable line search and show that it generates a sequence of nested intervals containing points satisfying the Armijo and weak Wolfe conditions, as- suming only absolute continuity. We also prove that the line search terminates for all semi-algebraic functions. The analysis of the convergence of BFGS using this line search seems very challenging; our theoretical results are limited to the univariate case. However, we systematically investigate the numerical behavior of BFGS with the inexact line search on various classes of examples. The method consistently converges to local minimizers on all but the most difficult class of examples, and even in that case, the method converges to points that are apparently Clarke stationary. Furthermore, the convergence rate is observed to be linear with respect to the number of function evaluations, with a rate of convergence that varies in an unexpectedly consistent way with the problem parameters. When the problem is sufficiently difficult, convergence may not be observed, but this seems to be due to rounding error caused by ill-conditioning. We try to give insight into why BFGS works as well as it does, and we conclude with a bold conjecture. Key words. BFGS, quasi-Newton, nonsmooth, nonconvex, line search, R-linear convergence AMS subject classifications. 90C30, 65K05 1. Introduction. This paper is concerned with the behavior of the BFGS (Broy- den-Fletcher-Goldfarb-Shanno) variable metric (quasi-Newton) method with an inex- act line search applied to nonsmooth functions, convex or nonconvex.
    [Show full text]
  • Gauss–Newton Optimization for Phase Recovery from the Bispectrum James L
    1 Gauss–Newton Optimization for Phase Recovery from the Bispectrum James L. Herring, James Nagy, and Lars Ruthotto Abstract—Phase recovery from the bispectrum is a central This work focuses on the problem of phase recovery from problem in speckle interferometry which can be posed as an an object’s collected bispectrum. The broader problem of optimization problem minimizing a weighted nonlinear least- phase retrieval occurs in numerous engineering fields and squares objective function. We look at two different formu- lations of the phase recovery problem from the literature, the sciences including astronomy, electronmicroscopy, crys- both of which can be minimized with respect to either the tallography, and optical imaging. For a recent overview of recovered phase or the recovered image. Previously, strategies applications and challenges, we recommend [4]. The prob- for solving these formulations have been limited to gradient lem of phase recovery from the bispectrum has its origins descent or quasi-Newton methods. This paper explores Gauss– in astronomical imaging for low-light images in the visible Newton optimization schemes for the problem of phase recovery from the bispectrum. We implement efficient Gauss–Newton spectrum [2], [3], [5], [6], [7], [8]. More recently, the problem optimization schemes for all the formulations. For the two of has been of interest in multiple applications including long- these formulations which optimize with respect to the recovered range horizontal and slant path imaging [9], [10], [11] and image, we also extend to projected Gauss–Newton to enforce phase recovery in underwater imaging [12], [13]. The general element-wise lower and upper bounds on the pixel intensities problem of phase retrieval from Fourier measurements also of the recovered image.
    [Show full text]
  • The Wolfe Epsilon Steepest Descent Algorithm
    Applied Mathematical Sciences, Vol. 8, 2014, no. 55, 2731 - 2741 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.43175 The Wolfe Epsilon Steepest Descent Algorithm Hakima Degaichia Department of Mathematics, University of Tebessa 12000 Tebessa, Algeria Rachid Benzine LANOS, Badji Mokhtar University, Annaba, Algeria Copyright c 2014 Hakima Degaichia and Rachid Benzine. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this article, we study the unconstrained minimization problem n (P ) min f(x): x R : f 2 g where f : Rn R is a continously di¤erentiable function. We introduce a new ! algorithm which accelerates the convergence of the steepest descent method. We study the global convergence of the new algorithm, named the Wolfe epsilon steep- est descent algorithm, by using Wolfe inexact line search ([35],[36]). In [16], [33], Benzine, Djeghaba and Rahali studied the same problem by using exact and Armijo inexact line serachs. Numerical tests show that Wolfe Epsilon steepest descent Al- gorithm accelerates the convergence of the gradient method and is more successful than Armijo Epsilon Steepest descent, Exact Epsilon steepest descent algorithm and Steepest descent algorithm. Mathematics Subject Classi…cation: 90C26; 65H10 Keywords: Unconstrained optimization. Gradient Method. Epsilon steep- est descent Algorithm. Wolfe line search. Armijo line search. Exact line search. Global convergence 2732 Hakima Degaichia and Rachid Benzine 1. INTRODUCTION Our problem is to minimize a function of n variables n (1.1) min f(x); x R ; f 2 g where f : Rn R is smooth and its gradient g is available.
    [Show full text]
  • A NONLINEAR RESPONSE MODEL for SINGLE NUCLEOTIDE POLYMORPHISM DETECTION ASSAYS by Drew Philip Kouri Submitted in Partial Fulfill
    A NONLINEAR RESPONSE MODEL FOR SINGLE NUCLEOTIDE POLYMORPHISM DETECTION ASSAYS by Drew Philip Kouri Submitted in partial fulfillment of the requirements For the degree of Masters of Science Thesis Adviser: Dr. Peter Thomas Department of Mathematics CASE WESTERN RESERVE UNIVERSITY August, 2008 CASE WESTERN RESERVE UNIVERSITY SCHOOL OF GRADUATE STUDIES We hereby approve the thesis/dissertation of Drew P. Kouri _____________________________________________________ Masters of Science candidate for the ______________________degree *. Peter J. Thomas (signed)_______________________________________________ (chair of the committee) Daniela Calvetti ________________________________________________ Peter A. Zimmerman ________________________________________________ James Alexander ________________________________________________ ________________________________________________ ________________________________________________ March 31, 2008 (date) _______________________ *We also certify that written approval has been obtained for any proprietary material contained therein. Contents List of Tables iv List of Figures v List of Abbreviations vi Abstract vii 1 Introduction 1 2 Laboratory Methods 4 3 The Model 6 3.1 Polmerase Chain Reaction (PCR) . 6 3.2 Ligase Detection Reaction (LDR) . 8 3.3 Fluorescent Microsphere Assay (FMA) . 9 4 Simulated Data Generation 10 5 Parameter Estimation 12 5.1 Nonlinear Least Squares . 13 5.1.1 The Levenberg-Marquardt Method . 14 5.1.2 Stopping Criterion . 15 5.1.3 Scaling . 16 5.2 Global Convergence . 18 5.2.1 Finite Differences Jacobian Approximation . 18 i 5.2.2 The Line Search . 18 5.2.3 The Hook Step . 21 5.2.4 The Double Dogleg . 23 5.3 Maximum Likelihood . 24 6 Analysis and Results 25 6.1 Nonlinear Least Squares Parameter Estimates . 28 6.2 Conditional Probability Distribution Analysis . 28 7 Discussion 29 8 Conclusions 30 9 Appendix 31 9.1 Histogram SNP Analysis Algorithm .
    [Show full text]
  • A Survey of Optimization Methods from a Machine Learning Perspective
    1 A Survey of Optimization Methods from a Machine Learning Perspective Shiliang Sun, Zehui Cao, Han Zhu, and Jing Zhao Abstract—Machine learning develops rapidly, which has made high-order optimization methods, in which Newton’s method many theoretical breakthroughs and is widely applied in various is a typical example; and heuristic derivative-free optimization fields. Optimization, as an important part of machine learning, methods, in which the coordinate descent method is a has attracted much attention of researchers. With the exponential growth of data amount and the increase of model complexity, representative. optimization methods in machine learning face more and more As the representative of first-order optimization methods, challenges. A lot of work on solving optimization problems or the stochastic gradient descent method [1], [2], as well as improving optimization methods in machine learning has been its variants, has been widely used in recent years and is proposed successively. The systematic retrospect and summary evolving at a high speed. However, many users pay little of the optimization methods from the perspective of machine learning are of great significance, which can offer guidance attention to the characteristics or application scope of these for both developments of optimization and machine learning methods. They often adopt them as black box optimizers, research. In this paper, we first describe the optimization which may limit the functionality of the optimization methods. problems in machine learning. Then, we introduce the principles In this paper, we comprehensively introduce the fundamental and progresses of commonly used optimization methods. Next, optimization methods. Particularly, we systematically explain we summarize the applications and developments of optimization methods in some popular machine learning fields.
    [Show full text]