Optimization Methods. Lecture 19: Line Searches and Newton s Method

Size: px
Start display at page:

Download "Optimization Methods. Lecture 19: Line Searches and Newton s Method"

Transcription

1 15.93 Optimization Methods Lecture 19: Line Searches and Newton s Method

2 1 Last Lecture Necessary Conditions for Optimality (identifies candidates) x local min f(x ) =, f(x ) PSD Slide 1 Sufficient Conditions for Optimality f(x ) = f(x) psd x B ǫ (x ) x loc. min. Characterizations of Convexity a. f convex f(y) f(x) + f(x) (y x) x, y b. f convex f(x) PSD x f convex then global min f(x ) = Steepest descent.1 The algorithm Step Given x, set :=. Step 1 Step d := f(x ). If d ǫ, then stop. Solve min λ h(λ) := f(x + λd ) for the step-length λ, perhaps chosen by an exact or inexact line-search. Step 3 Set x +1 x + λ d, + 1. Go to Step 1. 3 Outline 1. Bisection Method - Armijo s Rule Slide Slide 3. Motivation for Newton s method 3. Newton s method 4. Quadratic rate of convergence 5. Modification for global convergence 4 Choices of step sizes Min λ f(x + λd ) Slide 4 Limited Minimization: Min λ [,s] f(x + λd ) Constant stepsize λ = s constant 1

3 Diminishing stepsize: λ, Armijo Rule λ = 4.1 Bisection Line- Search Algorithm Convex functions λ := arg min h(λ) := argmin f(x + λd ) λ λ Slide 5 If f(x) is convex, h(λ) is convex. Find λ for which h (λ) = h (λ) = f(x + λd ) d 4.1. Algorithm Step. Set =. Set λ L := and λ U := λˆ. Slide 6 Step. Set λ = If h λ U + λ L and compute h (λ ). (λ ) >, re-set λ U := λ. Set + 1. If h (λ ) <, re-set λ L := λ. Set + 1. If h (λ ) =, stop Analysis At the th iteration of the bisection algorithm, the current interval [λ L, λ U ] must contain a point λ : h (λ ) = Slide 7 At the th iteration of the bisection algorithm, the length of the current interval [λ L, λ U ] is ( ) length = 1 (λˆ). A value of λ such that λ λ ǫ can be found in at most ( ) λˆ log ǫ steps of the bisection algorithm.

4 4.1.4 Example Slide 8 h(λ) = (x 1 + λd 1 ).6(x + λd ) + 4(x 3 + λd 3 ) (x 4 + λd 4 ) log(x i + λd i ) i=1 ( ) 4 4 log (5 x i λ d i i=1 i=1 where (x 1, x, x 3, x 4 ) = (1, 1, 1, 1) and (d 1, d, d 3, d 4 ) = ( 1,.6, 4,.5). Slide 9 and h(λ) = λ log(1 λ) log(1 +.6λ) log(1 4λ) log(1.5λ) log( λ) 1.6 h (λ) = λ 1 +.6λ 1 4λ λ λ Slide Slide λl λ u h (λ ) 1. NaN.5 NaN Slide 1 3

5 l λ λ u h (λ ) Armijo Rule Slide 13 Bisection is accurate but may be expensive in practice Need cheap method guaranteeing sufficient accuracy Inexact line search method. Requires two parameters: ǫ (, 1), σ > 1. h (λ) = h() + λǫh () λ acceptable by Armijo s rule if: h(λ ) h (λ ) h(σλ ) h (σλ ) (prevents the step size be small) (i.e. f(x + λd ) f(x ) + λǫ f(x ) d ) Slide 14 We get a range of acceptable stepsizes. Step : Set =, λ = λ > Step : If h(λ ) h (λ ), choose λ stop. If h(λ ) > h (λ ) let λ +1 1 = σλ (for example, σ = ) 5 Newton s method 5.1 History Steepest Descent is simple but slow Newton s method complex but fast Origins not clear Slide 15 Raphson became member of the Royal Society in 1691 for his boo Analysis Aequationum Universalis with Newton method. Raphson published it 5 years before Newton. 4

6 6 Newton s method 6.1 Motivation Consider min f(x) Slide 16 Taylor series expansion around x f(x) g(x) = f(x ) + f(x ) (x x )+ 1 (x x ) f( x)(x x ) Instead of minf(x), solve min g(x), i.e., g(x) = f(x ) + f(x )(x x ) = x x = ( f(x )) 1 f(x ) The direction d = ( f(x )) 1 f(x ) is the Newton direction 6. The algorithm Step Given x, set :=. Step 1 d := ( f(x )) 1 f(x ). If d ǫ, then stop. Step Set x +1 x + d, + 1. Go to Step Comments The method assumes that f(x ) is nonsingular at each iteration There is no guaranteee that f(x +1 ) f(x ) We can augment the algorithm with a line-search: Slide 17 Slide 18 min f(x + λd ) λ 6.4 Properties Theorem If H = f(x ) is PD, then d is a descent direction: f(x ) d < Proof: f(x ) d = f(x ) H 1 f(x ) < if H 1 is PD. But, Slide 19 < v (H 1 ) HH 1 v = v H 1 v H 1 is PD 5

7 6.5 Example 1 f(x) = 7x log x, 1 x = = f (x) = 7 1, f (x) = x ( d = (f (x)) 1 1 f (x) = x 1 x ) 1 ( 7 1 x ( x+1 = x + x 7(x ) ) = x 7(x ) ) = x 7x Slide Slide x E11-11E x x x Example Slide f(x 1, x ) = log(1 x 1 x ) log x 1 log x 1 1 f(x 1, x ) = 1 x 1 x x x 1 x x ( ) ( ) ( ) x 1 x x 1 1 x 1 x f(x 1, x ) = ( ) ( ) ( ) x 1 x 1 x 1 x x 1 Slide 3 ( ) 1 1, x ) = 3, 1 3 (x, f(x 1, x ) =

8 1 x x x x E E E-16 7 Quadratic convergence Recall z n+1 z z = lim z n, limsup = δ < n n z n z Slide 4 Example: z n = a n, < a < 1, z = z n+1 z a n+1 z n z = = 1 a n Intuitive statement Theorem Suppose f(x) C 3 (thrice cont. diff/ble), x : f(x ) = and f(x ) is nonsingular. If Newton s method is started sufficiently close to x, the sequence of iterates converges quadratically to x. Lemma: Suppose f(x) C 3 on R n and x a given pt. Then ǫ >, β > : x y ǫ Slide 5 Slide 6 f(x) f(y) H(y)(x y) β x y 7. Formal statement Theorem Suppose f twice cont. diff/ble. Let g(x) = f(x), H(x) = f(x), and x : g(x ) =. Suppose f(x) is twice differentiable and its Hessian satisfies: H(x ) h Slide 7 H(x) H(x ) L x x, x : x x β (Reminder: If A is a square matrix: A = max Ax = max{ λ 1,..., λ n }) x: x =1 7

9 Suppose ( ) Slide 8 x x h min β, = γ. 3L Let x be the next iterate in the Newton method. Then, ( ) x x x x h and 3L x x < x x < γ. 7.3 Proof x x Slide 9 = x H(x) 1 g(x) x ( ) = x x + H(x) 1 g(x ) g(x) = x x + H(x) 1 1 H(x + t(x x))(x x)dt (Lemma 1) x x Now = H(x) 1 (H(x + t(x x)) H(x)) (x x)dt 1 H(x) 1 1 H(x + t(x x)) H(x) x x dt 1 H(x) 1 x x L x x t dt = 1 L H(x) 1 x x h H(x ) = H(x) + H(x ) H(x) H(x) + H(x ) H(x) H(x) + L x x H(x) h L x x H(x) 1 1 H (x) 1 h L x x Slide 3 Slide 31 Slide 3 8

10 x x x x L (h L x x ) x x L (h 3 h) = 3L x x h 7.4 Lemma 1 Fix w. Let φ(t) = g(x + t(x x)) w Slide 33 φ() = g(x) w, φ(1) = g(x ) w φ (t) = w H(x + t(x x))(x x) 1 φ(1) = φ() + φ (t)dt w 1 w : w (g(x ) g(x)) = H(x + t(x x))(x x)dt Slide 34 1 g(x ) g(x) = H(x + t(x x))(x x)dt 7.5 Critical comments The iterates from Newton s method are equally attracted to local min and local max. Slide 35 We do not now β, h, L in general. Note, however, that they are only used in the proof, not in the algorithm. We do not assume convexity, only that H(x ) is nonsingular and not badly behaved near x. Slide Properties of Convergence Proposition: Let r = x x and C = 3L. If x x < γ then h r 1 (Cr ) C Slide 37 9

11 Proposition: If x x > ǫ > x x < ǫ 1 log( Cǫ ) log( log( 1 ) Cr log 8 Solving systems of equations g(x) = g(x t+1 ) g(x t ) + g(x t )(x t+1 x t ) x t+1 = x t ( g(x t )) 1 g(x t ) Application in optimization: g(x) = f(x) 9 Modifications for global convergence Perform line search Slide 38 Slide 39 When Hessian is singular or near singular, use: ( ) 1 d = f(x ) + D f(x ) D is a diagonal matrix: f(x ) + D is PD 1 Summary 1. Line search methods: Bisection Method. Armijo s Rule. Slide 4. The Newton s method 3. Quadratic rate of convergence 4. Modification for global convergence 1

12 MIT OpenCourseWare J / 6.55J Optimization Methods Fall 9 For information about citing these materials or our Terms of Use, visit:

1 Newton s Method. Suppose we want to solve: x R. At x = x, f (x) can be approximated by:

1 Newton s Method. Suppose we want to solve: x R. At x = x, f (x) can be approximated by: Newton s Method Suppose we want to solve: (P:) min f (x) At x = x, f (x) can be approximated by: n x R. f (x) h(x) := f ( x)+ f ( x) T (x x)+ (x x) t H ( x)(x x), 2 which is the quadratic Taylor expansion

More information

The Steepest Descent Algorithm for Unconstrained Optimization

The Steepest Descent Algorithm for Unconstrained Optimization The Steepest Descent Algorithm for Unconstrained Optimization Robert M. Freund February, 2014 c 2014 Massachusetts Institute of Technology. All rights reserved. 1 1 Steepest Descent Algorithm The problem

More information

Line Search Methods for Unconstrained Optimisation

Line Search Methods for Unconstrained Optimisation Line Search Methods for Unconstrained Optimisation Lecture 8, Numerical Linear Algebra and Optimisation Oxford University Computing Laboratory, MT 2007 Dr Raphael Hauser (hauser@comlab.ox.ac.uk) The Generic

More information

Optimization Methods. Lecture 18: Optimality Conditions and. Gradient Methods. for Unconstrained Optimization

Optimization Methods. Lecture 18: Optimality Conditions and. Gradient Methods. for Unconstrained Optimization 5.93 Optimization Methods Lecture 8: Optimality Conditions and Gradient Methods for Unconstrained Optimization Outline. Necessary and sucient optimality conditions Slide. Gradient m e t h o d s 3. The

More information

Optimization Tutorial 1. Basic Gradient Descent

Optimization Tutorial 1. Basic Gradient Descent E0 270 Machine Learning Jan 16, 2015 Optimization Tutorial 1 Basic Gradient Descent Lecture by Harikrishna Narasimhan Note: This tutorial shall assume background in elementary calculus and linear algebra.

More information

Numerisches Rechnen. (für Informatiker) M. Grepl P. Esser & G. Welper & L. Zhang. Institut für Geometrie und Praktische Mathematik RWTH Aachen

Numerisches Rechnen. (für Informatiker) M. Grepl P. Esser & G. Welper & L. Zhang. Institut für Geometrie und Praktische Mathematik RWTH Aachen Numerisches Rechnen (für Informatiker) M. Grepl P. Esser & G. Welper & L. Zhang Institut für Geometrie und Praktische Mathematik RWTH Aachen Wintersemester 2011/12 IGPM, RWTH Aachen Numerisches Rechnen

More information

5 Overview of algorithms for unconstrained optimization

5 Overview of algorithms for unconstrained optimization IOE 59: NLP, Winter 22 c Marina A. Epelman 9 5 Overview of algorithms for unconstrained optimization 5. General optimization algorithm Recall: we are attempting to solve the problem (P) min f(x) s.t. x

More information

E5295/5B5749 Convex optimization with engineering applications. Lecture 8. Smooth convex unconstrained and equality-constrained minimization

E5295/5B5749 Convex optimization with engineering applications. Lecture 8. Smooth convex unconstrained and equality-constrained minimization E5295/5B5749 Convex optimization with engineering applications Lecture 8 Smooth convex unconstrained and equality-constrained minimization A. Forsgren, KTH 1 Lecture 8 Convex optimization 2006/2007 Unconstrained

More information

Optimality conditions for unconstrained optimization. Outline

Optimality conditions for unconstrained optimization. Outline Optimality conditions for unconstrained optimization Daniel P. Robinson Department of Applied Mathematics and Statistics Johns Hopkins University September 13, 2018 Outline 1 The problem and definitions

More information

Lecture 4 - The Gradient Method Objective: find an optimal solution of the problem

Lecture 4 - The Gradient Method Objective: find an optimal solution of the problem Lecture 4 - The Gradient Method Objective: find an optimal solution of the problem min{f (x) : x R n }. The iterative algorithms that we will consider are of the form x k+1 = x k + t k d k, k = 0, 1,...

More information

Lecture 4 - The Gradient Method Objective: find an optimal solution of the problem

Lecture 4 - The Gradient Method Objective: find an optimal solution of the problem Lecture 4 - The Gradient Method Objective: find an optimal solution of the problem min{f (x) : x R n }. The iterative algorithms that we will consider are of the form x k+1 = x k + t k d k, k = 0, 1,...

More information

Optimization. Yuh-Jye Lee. March 21, Data Science and Machine Intelligence Lab National Chiao Tung University 1 / 29

Optimization. Yuh-Jye Lee. March 21, Data Science and Machine Intelligence Lab National Chiao Tung University 1 / 29 Optimization Yuh-Jye Lee Data Science and Machine Intelligence Lab National Chiao Tung University March 21, 2017 1 / 29 You Have Learned (Unconstrained) Optimization in Your High School Let f (x) = ax

More information

1 Overview. 2 A Characterization of Convex Functions. 2.1 First-order Taylor approximation. AM 221: Advanced Optimization Spring 2016

1 Overview. 2 A Characterization of Convex Functions. 2.1 First-order Taylor approximation. AM 221: Advanced Optimization Spring 2016 AM 221: Advanced Optimization Spring 2016 Prof. Yaron Singer Lecture 8 February 22nd 1 Overview In the previous lecture we saw characterizations of optimality in linear optimization, and we reviewed the

More information

Unconstrained minimization of smooth functions

Unconstrained minimization of smooth functions Unconstrained minimization of smooth functions We want to solve min x R N f(x), where f is convex. In this section, we will assume that f is differentiable (so its gradient exists at every point), and

More information

Lecture 3: Linesearch methods (continued). Steepest descent methods

Lecture 3: Linesearch methods (continued). Steepest descent methods Lecture 3: Linesearch methods (continued). Steepest descent methods Coralia Cartis, Mathematical Institute, University of Oxford C6.2/B2: Continuous Optimization Lecture 3: Linesearch methods (continued).

More information

Algorithms for constrained local optimization

Algorithms for constrained local optimization Algorithms for constrained local optimization Fabio Schoen 2008 http://gol.dsi.unifi.it/users/schoen Algorithms for constrained local optimization p. Feasible direction methods Algorithms for constrained

More information

4 Newton Method. Unconstrained Convex Optimization 21. H(x)p = f(x). Newton direction. Why? Recall second-order staylor series expansion:

4 Newton Method. Unconstrained Convex Optimization 21. H(x)p = f(x). Newton direction. Why? Recall second-order staylor series expansion: Unconstrained Convex Optimization 21 4 Newton Method H(x)p = f(x). Newton direction. Why? Recall second-order staylor series expansion: f(x + p) f(x)+p T f(x)+ 1 2 pt H(x)p ˆf(p) In general, ˆf(p) won

More information

Newton s Method. Ryan Tibshirani Convex Optimization /36-725

Newton s Method. Ryan Tibshirani Convex Optimization /36-725 Newton s Method Ryan Tibshirani Convex Optimization 10-725/36-725 1 Last time: dual correspondences Given a function f : R n R, we define its conjugate f : R n R, Properties and examples: f (y) = max x

More information

Newton s Method. Javier Peña Convex Optimization /36-725

Newton s Method. Javier Peña Convex Optimization /36-725 Newton s Method Javier Peña Convex Optimization 10-725/36-725 1 Last time: dual correspondences Given a function f : R n R, we define its conjugate f : R n R, f ( (y) = max y T x f(x) ) x Properties and

More information

You should be able to...

You should be able to... Lecture Outline Gradient Projection Algorithm Constant Step Length, Varying Step Length, Diminishing Step Length Complexity Issues Gradient Projection With Exploration Projection Solving QPs: active set

More information

Nonlinear Programming

Nonlinear Programming Nonlinear Programming Kees Roos e-mail: C.Roos@ewi.tudelft.nl URL: http://www.isa.ewi.tudelft.nl/ roos LNMB Course De Uithof, Utrecht February 6 - May 8, A.D. 2006 Optimization Group 1 Outline for week

More information

Part 2: Linesearch methods for unconstrained optimization. Nick Gould (RAL)

Part 2: Linesearch methods for unconstrained optimization. Nick Gould (RAL) Part 2: Linesearch methods for unconstrained optimization Nick Gould (RAL) minimize x IR n f(x) MSc course on nonlinear optimization UNCONSTRAINED MINIMIZATION minimize x IR n f(x) where the objective

More information

8 Numerical methods for unconstrained problems

8 Numerical methods for unconstrained problems 8 Numerical methods for unconstrained problems Optimization is one of the important fields in numerical computation, beside solving differential equations and linear systems. We can see that these fields

More information

2.098/6.255/ Optimization Methods Practice True/False Questions

2.098/6.255/ Optimization Methods Practice True/False Questions 2.098/6.255/15.093 Optimization Methods Practice True/False Questions December 11, 2009 Part I For each one of the statements below, state whether it is true or false. Include a 1-3 line supporting sentence

More information

Introduction to Nonlinear Optimization Paul J. Atzberger

Introduction to Nonlinear Optimization Paul J. Atzberger Introduction to Nonlinear Optimization Paul J. Atzberger Comments should be sent to: atzberg@math.ucsb.edu Introduction We shall discuss in these notes a brief introduction to nonlinear optimization concepts,

More information

Unconstrained optimization

Unconstrained optimization Chapter 4 Unconstrained optimization An unconstrained optimization problem takes the form min x Rnf(x) (4.1) for a target functional (also called objective function) f : R n R. In this chapter and throughout

More information

Nonlinear Optimization

Nonlinear Optimization Nonlinear Optimization (Com S 477/577 Notes) Yan-Bin Jia Nov 7, 2017 1 Introduction Given a single function f that depends on one or more independent variable, we want to find the values of those variables

More information

CS 435, 2018 Lecture 6, Date: 12 April 2018 Instructor: Nisheeth Vishnoi. Newton s Method

CS 435, 2018 Lecture 6, Date: 12 April 2018 Instructor: Nisheeth Vishnoi. Newton s Method CS 435, 28 Lecture 6, Date: 2 April 28 Instructor: Nisheeth Vishnoi Newton s Method In this lecture we derive and analyze Newton s method which is an example of a second order optimization method. We argue

More information

Lecture 14: October 17

Lecture 14: October 17 1-725/36-725: Convex Optimization Fall 218 Lecture 14: October 17 Lecturer: Lecturer: Ryan Tibshirani Scribes: Pengsheng Guo, Xian Zhou Note: LaTeX template courtesy of UC Berkeley EECS dept. Disclaimer:

More information

1. Nonlinear Equations. This lecture note excerpted parts from Michael Heath and Max Gunzburger. f(x) = 0

1. Nonlinear Equations. This lecture note excerpted parts from Michael Heath and Max Gunzburger. f(x) = 0 Numerical Analysis 1 1. Nonlinear Equations This lecture note excerpted parts from Michael Heath and Max Gunzburger. Given function f, we seek value x for which where f : D R n R n is nonlinear. f(x) =

More information

Nonlinear Optimization for Optimal Control

Nonlinear Optimization for Optimal Control Nonlinear Optimization for Optimal Control Pieter Abbeel UC Berkeley EECS Many slides and figures adapted from Stephen Boyd [optional] Boyd and Vandenberghe, Convex Optimization, Chapters 9 11 [optional]

More information

10.34: Numerical Methods Applied to Chemical Engineering. Lecture 7: Solutions of nonlinear equations Newton-Raphson method

10.34: Numerical Methods Applied to Chemical Engineering. Lecture 7: Solutions of nonlinear equations Newton-Raphson method 10.34: Numerical Methods Applied to Chemical Engineering Lecture 7: Solutions of nonlinear equations Newton-Raphson method 1 Recap Singular value decomposition Iterative solutions to linear equations 2

More information

1. Method 1: bisection. The bisection methods starts from two points a 0 and b 0 such that

1. Method 1: bisection. The bisection methods starts from two points a 0 and b 0 such that Chapter 4 Nonlinear equations 4.1 Root finding Consider the problem of solving any nonlinear relation g(x) = h(x) in the real variable x. We rephrase this problem as one of finding the zero (root) of a

More information

Unconstrained optimization I Gradient-type methods

Unconstrained optimization I Gradient-type methods Unconstrained optimization I Gradient-type methods Antonio Frangioni Department of Computer Science University of Pisa www.di.unipi.it/~frangio frangio@di.unipi.it Computational Mathematics for Learning

More information

Optimization: Nonlinear Optimization without Constraints. Nonlinear Optimization without Constraints 1 / 23

Optimization: Nonlinear Optimization without Constraints. Nonlinear Optimization without Constraints 1 / 23 Optimization: Nonlinear Optimization without Constraints Nonlinear Optimization without Constraints 1 / 23 Nonlinear optimization without constraints Unconstrained minimization min x f(x) where f(x) is

More information

Optimization. Escuela de Ingeniería Informática de Oviedo. (Dpto. de Matemáticas-UniOvi) Numerical Computation Optimization 1 / 30

Optimization. Escuela de Ingeniería Informática de Oviedo. (Dpto. de Matemáticas-UniOvi) Numerical Computation Optimization 1 / 30 Optimization Escuela de Ingeniería Informática de Oviedo (Dpto. de Matemáticas-UniOvi) Numerical Computation Optimization 1 / 30 Unconstrained optimization Outline 1 Unconstrained optimization 2 Constrained

More information

Lecture 15 Newton Method and Self-Concordance. October 23, 2008

Lecture 15 Newton Method and Self-Concordance. October 23, 2008 Newton Method and Self-Concordance October 23, 2008 Outline Lecture 15 Self-concordance Notion Self-concordant Functions Operations Preserving Self-concordance Properties of Self-concordant Functions Implications

More information

Conditional Gradient (Frank-Wolfe) Method

Conditional Gradient (Frank-Wolfe) Method Conditional Gradient (Frank-Wolfe) Method Lecturer: Aarti Singh Co-instructor: Pradeep Ravikumar Convex Optimization 10-725/36-725 1 Outline Today: Conditional gradient method Convergence analysis Properties

More information

Unconstrained Optimization

Unconstrained Optimization 1 / 36 Unconstrained Optimization ME598/494 Lecture Max Yi Ren Department of Mechanical Engineering, Arizona State University February 2, 2015 2 / 36 3 / 36 4 / 36 5 / 36 1. preliminaries 1.1 local approximation

More information

FIXED POINT ITERATION

FIXED POINT ITERATION FIXED POINT ITERATION The idea of the fixed point iteration methods is to first reformulate a equation to an equivalent fixed point problem: f (x) = 0 x = g(x) and then to use the iteration: with an initial

More information

Methods for Unconstrained Optimization Numerical Optimization Lectures 1-2

Methods for Unconstrained Optimization Numerical Optimization Lectures 1-2 Methods for Unconstrained Optimization Numerical Optimization Lectures 1-2 Coralia Cartis, University of Oxford INFOMM CDT: Modelling, Analysis and Computation of Continuous Real-World Problems Methods

More information

Nonlinear equations. Norms for R n. Convergence orders for iterative methods

Nonlinear equations. Norms for R n. Convergence orders for iterative methods Nonlinear equations Norms for R n Assume that X is a vector space. A norm is a mapping X R with x such that for all x, y X, α R x = = x = αx = α x x + y x + y We define the following norms on the vector

More information

10. Unconstrained minimization

10. Unconstrained minimization Convex Optimization Boyd & Vandenberghe 10. Unconstrained minimization terminology and assumptions gradient descent method steepest descent method Newton s method self-concordant functions implementation

More information

MATH 4211/6211 Optimization Quasi-Newton Method

MATH 4211/6211 Optimization Quasi-Newton Method MATH 4211/6211 Optimization Quasi-Newton Method Xiaojing Ye Department of Mathematics & Statistics Georgia State University Xiaojing Ye, Math & Stat, Georgia State University 0 Quasi-Newton Method Motivation:

More information

17 Solution of Nonlinear Systems

17 Solution of Nonlinear Systems 17 Solution of Nonlinear Systems We now discuss the solution of systems of nonlinear equations. An important ingredient will be the multivariate Taylor theorem. Theorem 17.1 Let D = {x 1, x 2,..., x m

More information

The Conjugate Gradient Method

The Conjugate Gradient Method The Conjugate Gradient Method Lecture 5, Continuous Optimisation Oxford University Computing Laboratory, HT 2006 Notes by Dr Raphael Hauser (hauser@comlab.ox.ac.uk) The notion of complexity (per iteration)

More information

Unconstrained minimization

Unconstrained minimization CSCI5254: Convex Optimization & Its Applications Unconstrained minimization terminology and assumptions gradient descent method steepest descent method Newton s method self-concordant functions 1 Unconstrained

More information

Selected Topics in Optimization. Some slides borrowed from

Selected Topics in Optimization. Some slides borrowed from Selected Topics in Optimization Some slides borrowed from http://www.stat.cmu.edu/~ryantibs/convexopt/ Overview Optimization problems are almost everywhere in statistics and machine learning. Input Model

More information

Math 164: Optimization Barzilai-Borwein Method

Math 164: Optimization Barzilai-Borwein Method Math 164: Optimization Barzilai-Borwein Method Instructor: Wotao Yin Department of Mathematics, UCLA Spring 2015 online discussions on piazza.com Main features of the Barzilai-Borwein (BB) method The BB

More information

Lecture 7 Unconstrained nonlinear programming

Lecture 7 Unconstrained nonlinear programming Lecture 7 Unconstrained nonlinear programming Weinan E 1,2 and Tiejun Li 2 1 Department of Mathematics, Princeton University, weinan@princeton.edu 2 School of Mathematical Sciences, Peking University,

More information

Constrained optimization: direct methods (cont.)

Constrained optimization: direct methods (cont.) Constrained optimization: direct methods (cont.) Jussi Hakanen Post-doctoral researcher jussi.hakanen@jyu.fi Direct methods Also known as methods of feasible directions Idea in a point x h, generate a

More information

Introduction to unconstrained optimization - direct search methods

Introduction to unconstrained optimization - direct search methods Introduction to unconstrained optimization - direct search methods Jussi Hakanen Post-doctoral researcher jussi.hakanen@jyu.fi Structure of optimization methods Typically Constraint handling converts the

More information

The Randomized Newton Method for Convex Optimization

The Randomized Newton Method for Convex Optimization The Randomized Newton Method for Convex Optimization Vaden Masrani UBC MLRG April 3rd, 2018 Introduction We have some unconstrained, twice-differentiable convex function f : R d R that we want to minimize:

More information

Convex Optimization. Problem set 2. Due Monday April 26th

Convex Optimization. Problem set 2. Due Monday April 26th Convex Optimization Problem set 2 Due Monday April 26th 1 Gradient Decent without Line-search In this problem we will consider gradient descent with predetermined step sizes. That is, instead of determining

More information

Proximal Newton Method. Zico Kolter (notes by Ryan Tibshirani) Convex Optimization

Proximal Newton Method. Zico Kolter (notes by Ryan Tibshirani) Convex Optimization Proximal Newton Method Zico Kolter (notes by Ryan Tibshirani) Convex Optimization 10-725 Consider the problem Last time: quasi-newton methods min x f(x) with f convex, twice differentiable, dom(f) = R

More information

Computational Optimization. Mathematical Programming Fundamentals 1/25 (revised)

Computational Optimization. Mathematical Programming Fundamentals 1/25 (revised) Computational Optimization Mathematical Programming Fundamentals 1/5 (revised) If you don t know where you are going, you probably won t get there. -from some book I read in eight grade If you do get there,

More information

Shiqian Ma, MAT-258A: Numerical Optimization 1. Chapter 3. Gradient Method

Shiqian Ma, MAT-258A: Numerical Optimization 1. Chapter 3. Gradient Method Shiqian Ma, MAT-258A: Numerical Optimization 1 Chapter 3 Gradient Method Shiqian Ma, MAT-258A: Numerical Optimization 2 3.1. Gradient method Classical gradient method: to minimize a differentiable convex

More information

Examination paper for TMA4180 Optimization I

Examination paper for TMA4180 Optimization I Department of Mathematical Sciences Examination paper for TMA4180 Optimization I Academic contact during examination: Phone: Examination date: 26th May 2016 Examination time (from to): 09:00 13:00 Permitted

More information

Lecture 17: October 27

Lecture 17: October 27 0-725/36-725: Convex Optimiation Fall 205 Lecturer: Ryan Tibshirani Lecture 7: October 27 Scribes: Brandon Amos, Gines Hidalgo Note: LaTeX template courtesy of UC Berkeley EECS dept. Disclaimer: These

More information

Higher-Order Methods

Higher-Order Methods Higher-Order Methods Stephen J. Wright 1 2 Computer Sciences Department, University of Wisconsin-Madison. PCMI, July 2016 Stephen Wright (UW-Madison) Higher-Order Methods PCMI, July 2016 1 / 25 Smooth

More information

Lecture 5: Gradient Descent. 5.1 Unconstrained minimization problems and Gradient descent

Lecture 5: Gradient Descent. 5.1 Unconstrained minimization problems and Gradient descent 10-725/36-725: Convex Optimization Spring 2015 Lecturer: Ryan Tibshirani Lecture 5: Gradient Descent Scribes: Loc Do,2,3 Disclaimer: These notes have not been subjected to the usual scrutiny reserved for

More information

Complexity analysis of second-order algorithms based on line search for smooth nonconvex optimization

Complexity analysis of second-order algorithms based on line search for smooth nonconvex optimization Complexity analysis of second-order algorithms based on line search for smooth nonconvex optimization Clément Royer - University of Wisconsin-Madison Joint work with Stephen J. Wright MOPTA, Bethlehem,

More information

MATH 4211/6211 Optimization Basics of Optimization Problems

MATH 4211/6211 Optimization Basics of Optimization Problems MATH 4211/6211 Optimization Basics of Optimization Problems Xiaojing Ye Department of Mathematics & Statistics Georgia State University Xiaojing Ye, Math & Stat, Georgia State University 0 A standard minimization

More information

Numerical Optimization

Numerical Optimization Numerical Optimization Unit 2: Multivariable optimization problems Che-Rung Lee Scribe: February 28, 2011 (UNIT 2) Numerical Optimization February 28, 2011 1 / 17 Partial derivative of a two variable function

More information

Sub-Sampled Newton Methods

Sub-Sampled Newton Methods Sub-Sampled Newton Methods F. Roosta-Khorasani and M. W. Mahoney ICSI and Dept of Statistics, UC Berkeley February 2016 F. Roosta-Khorasani and M. W. Mahoney (UCB) Sub-Sampled Newton Methods Feb 2016 1

More information

Chapter 3 Numerical Methods

Chapter 3 Numerical Methods Chapter 3 Numerical Methods Part 2 3.2 Systems of Equations 3.3 Nonlinear and Constrained Optimization 1 Outline 3.2 Systems of Equations 3.3 Nonlinear and Constrained Optimization Summary 2 Outline 3.2

More information

Lecture 13 Newton-type Methods A Newton Method for VIs. October 20, 2008

Lecture 13 Newton-type Methods A Newton Method for VIs. October 20, 2008 Lecture 13 Newton-type Methods A Newton Method for VIs October 20, 2008 Outline Quick recap of Newton methods for composite functions Josephy-Newton methods for VIs A special case: mixed complementarity

More information

Programming, numerics and optimization

Programming, numerics and optimization Programming, numerics and optimization Lecture C-3: Unconstrained optimization II Łukasz Jankowski ljank@ippt.pan.pl Institute of Fundamental Technological Research Room 4.32, Phone +22.8261281 ext. 428

More information

Proximal Newton Method. Ryan Tibshirani Convex Optimization /36-725

Proximal Newton Method. Ryan Tibshirani Convex Optimization /36-725 Proximal Newton Method Ryan Tibshirani Convex Optimization 10-725/36-725 1 Last time: primal-dual interior-point method Given the problem min x subject to f(x) h i (x) 0, i = 1,... m Ax = b where f, h

More information

CS 323: Numerical Analysis and Computing

CS 323: Numerical Analysis and Computing CS 323: Numerical Analysis and Computing MIDTERM #2 Instructions: This is an open notes exam, i.e., you are allowed to consult any textbook, your class notes, homeworks, or any of the handouts from us.

More information

ORIE 6326: Convex Optimization. Quasi-Newton Methods

ORIE 6326: Convex Optimization. Quasi-Newton Methods ORIE 6326: Convex Optimization Quasi-Newton Methods Professor Udell Operations Research and Information Engineering Cornell April 10, 2017 Slides on steepest descent and analysis of Newton s method adapted

More information

14. Nonlinear equations

14. Nonlinear equations L. Vandenberghe ECE133A (Winter 2018) 14. Nonlinear equations Newton method for nonlinear equations damped Newton method for unconstrained minimization Newton method for nonlinear least squares 14-1 Set

More information

Self-Concordant Barrier Functions for Convex Optimization

Self-Concordant Barrier Functions for Convex Optimization Appendix F Self-Concordant Barrier Functions for Convex Optimization F.1 Introduction In this Appendix we present a framework for developing polynomial-time algorithms for the solution of convex optimization

More information

Convex Optimization. 9. Unconstrained minimization. Prof. Ying Cui. Department of Electrical Engineering Shanghai Jiao Tong University

Convex Optimization. 9. Unconstrained minimization. Prof. Ying Cui. Department of Electrical Engineering Shanghai Jiao Tong University Convex Optimization 9. Unconstrained minimization Prof. Ying Cui Department of Electrical Engineering Shanghai Jiao Tong University 2017 Autumn Semester SJTU Ying Cui 1 / 40 Outline Unconstrained minimization

More information

Lecture 5: September 15

Lecture 5: September 15 10-725/36-725: Convex Optimization Fall 2015 Lecture 5: September 15 Lecturer: Lecturer: Ryan Tibshirani Scribes: Scribes: Di Jin, Mengdi Wang, Bin Deng Note: LaTeX template courtesy of UC Berkeley EECS

More information

OPER 627: Nonlinear Optimization Lecture 14: Mid-term Review

OPER 627: Nonlinear Optimization Lecture 14: Mid-term Review OPER 627: Nonlinear Optimization Lecture 14: Mid-term Review Department of Statistical Sciences and Operations Research Virginia Commonwealth University Oct 16, 2013 (Lecture 14) Nonlinear Optimization

More information

18.657: Mathematics of Machine Learning

18.657: Mathematics of Machine Learning 8.657: Mathematics of Machine Learning Lecturer: Philippe Rigollet Lecture Scribe: Kevin Li Oct. 4, 05. CONVEX OPTIMIZATION FOR MACHINE LEARNING In this lecture, we will cover the basics of convex optimization

More information

UNDERGROUND LECTURE NOTES 1: Optimality Conditions for Constrained Optimization Problems

UNDERGROUND LECTURE NOTES 1: Optimality Conditions for Constrained Optimization Problems UNDERGROUND LECTURE NOTES 1: Optimality Conditions for Constrained Optimization Problems Robert M. Freund February 2016 c 2016 Massachusetts Institute of Technology. All rights reserved. 1 1 Introduction

More information

Scientific Computing: Optimization

Scientific Computing: Optimization Scientific Computing: Optimization Aleksandar Donev Courant Institute, NYU 1 donev@courant.nyu.edu 1 Course MATH-GA.2043 or CSCI-GA.2112, Spring 2012 March 8th, 2011 A. Donev (Courant Institute) Lecture

More information

NUMERICAL METHODS. x n+1 = 2x n x 2 n. In particular: which of them gives faster convergence, and why? [Work to four decimal places.

NUMERICAL METHODS. x n+1 = 2x n x 2 n. In particular: which of them gives faster convergence, and why? [Work to four decimal places. NUMERICAL METHODS 1. Rearranging the equation x 3 =.5 gives the iterative formula x n+1 = g(x n ), where g(x) = (2x 2 ) 1. (a) Starting with x = 1, compute the x n up to n = 6, and describe what is happening.

More information

Primal/Dual Decomposition Methods

Primal/Dual Decomposition Methods Primal/Dual Decomposition Methods Daniel P. Palomar Hong Kong University of Science and Technology (HKUST) ELEC5470 - Convex Optimization Fall 2018-19, HKUST, Hong Kong Outline of Lecture Subgradients

More information

Root Finding (and Optimisation)

Root Finding (and Optimisation) Root Finding (and Optimisation) M.Sc. in Mathematical Modelling & Scientific Computing, Practical Numerical Analysis Michaelmas Term 2018, Lecture 4 Root Finding The idea of root finding is simple we want

More information

GRADIENT = STEEPEST DESCENT

GRADIENT = STEEPEST DESCENT GRADIENT METHODS GRADIENT = STEEPEST DESCENT Convex Function Iso-contours gradient 0.5 0.4 4 2 0 8 0.3 0.2 0. 0 0. negative gradient 6 0.2 4 0.3 2.5 0.5 0 0.5 0.5 0 0.5 0.4 0.5.5 0.5 0 0.5 GRADIENT DESCENT

More information

IFT Lecture 2 Basics of convex analysis and gradient descent

IFT Lecture 2 Basics of convex analysis and gradient descent IF 6085 - Lecture Basics of convex analysis and gradient descent his version of the notes has not yet been thoroughly checked. Please report any bugs to the scribes or instructor. Scribes: Assya rofimov,

More information

An example of slow convergence for Newton s method on a function with globally Lipschitz continuous Hessian

An example of slow convergence for Newton s method on a function with globally Lipschitz continuous Hessian An example of slow convergence for Newton s method on a function with globally Lipschitz continuous Hessian C. Cartis, N. I. M. Gould and Ph. L. Toint 3 May 23 Abstract An example is presented where Newton

More information

Design and Analysis of Algorithms Lecture Notes on Convex Optimization CS 6820, Fall Nov 2 Dec 2016

Design and Analysis of Algorithms Lecture Notes on Convex Optimization CS 6820, Fall Nov 2 Dec 2016 Design and Analysis of Algorithms Lecture Notes on Convex Optimization CS 6820, Fall 206 2 Nov 2 Dec 206 Let D be a convex subset of R n. A function f : D R is convex if it satisfies f(tx + ( t)y) tf(x)

More information

Gradient Descent. Ryan Tibshirani Convex Optimization /36-725

Gradient Descent. Ryan Tibshirani Convex Optimization /36-725 Gradient Descent Ryan Tibshirani Convex Optimization 10-725/36-725 Last time: canonical convex programs Linear program (LP): takes the form min x subject to c T x Gx h Ax = b Quadratic program (QP): like

More information

Computational Optimization. Convexity and Unconstrained Optimization 1/29/08 and 2/1(revised)

Computational Optimization. Convexity and Unconstrained Optimization 1/29/08 and 2/1(revised) Computational Optimization Convexity and Unconstrained Optimization 1/9/08 and /1(revised) Convex Sets A set S is convex if the line segment joining any two points in the set is also in the set, i.e.,

More information

CS 542G: Robustifying Newton, Constraints, Nonlinear Least Squares

CS 542G: Robustifying Newton, Constraints, Nonlinear Least Squares CS 542G: Robustifying Newton, Constraints, Nonlinear Least Squares Robert Bridson October 29, 2008 1 Hessian Problems in Newton Last time we fixed one of plain Newton s problems by introducing line search

More information

6. Proximal gradient method

6. Proximal gradient method L. Vandenberghe EE236C (Spring 2013-14) 6. Proximal gradient method motivation proximal mapping proximal gradient method with fixed step size proximal gradient method with line search 6-1 Proximal mapping

More information

Nonlinear Optimization: What s important?

Nonlinear Optimization: What s important? Nonlinear Optimization: What s important? Julian Hall 10th May 2012 Convexity: convex problems A local minimizer is a global minimizer A solution of f (x) = 0 (stationary point) is a minimizer A global

More information

Solution Methods. Richard Lusby. Department of Management Engineering Technical University of Denmark

Solution Methods. Richard Lusby. Department of Management Engineering Technical University of Denmark Solution Methods Richard Lusby Department of Management Engineering Technical University of Denmark Lecture Overview (jg Unconstrained Several Variables Quadratic Programming Separable Programming SUMT

More information

Proximal Gradient Descent and Acceleration. Ryan Tibshirani Convex Optimization /36-725

Proximal Gradient Descent and Acceleration. Ryan Tibshirani Convex Optimization /36-725 Proximal Gradient Descent and Acceleration Ryan Tibshirani Convex Optimization 10-725/36-725 Last time: subgradient method Consider the problem min f(x) with f convex, and dom(f) = R n. Subgradient method:

More information

5 Quasi-Newton Methods

5 Quasi-Newton Methods Unconstrained Convex Optimization 26 5 Quasi-Newton Methods If the Hessian is unavailable... Notation: H = Hessian matrix. B is the approximation of H. C is the approximation of H 1. Problem: Solve min

More information

ECE580 Partial Solution to Problem Set 3

ECE580 Partial Solution to Problem Set 3 ECE580 Fall 2015 Solution to Problem Set 3 October 23, 2015 1 ECE580 Partial Solution to Problem Set 3 These problems are from the textbook by Chong and Zak, 4th edition, which is the textbook for the

More information

Chapter 4. Unconstrained optimization

Chapter 4. Unconstrained optimization Chapter 4. Unconstrained optimization Version: 28-10-2012 Material: (for details see) Chapter 11 in [FKS] (pp.251-276) A reference e.g. L.11.2 refers to the corresponding Lemma in the book [FKS] PDF-file

More information

IE 5531: Engineering Optimization I

IE 5531: Engineering Optimization I IE 5531: Engineering Optimization I Lecture 15: Nonlinear optimization Prof. John Gunnar Carlsson November 1, 2010 Prof. John Gunnar Carlsson IE 5531: Engineering Optimization I November 1, 2010 1 / 24

More information

Math 473: Practice Problems for Test 1, Fall 2011, SOLUTIONS

Math 473: Practice Problems for Test 1, Fall 2011, SOLUTIONS Math 473: Practice Problems for Test 1, Fall 011, SOLUTIONS Show your work: 1. (a) Compute the Taylor polynomials P n (x) for f(x) = sin x and x 0 = 0. Solution: Compute f(x) = sin x, f (x) = cos x, f

More information

D(f/g)(P ) = D(f)(P )g(p ) f(p )D(g)(P ). g 2 (P )

D(f/g)(P ) = D(f)(P )g(p ) f(p )D(g)(P ). g 2 (P ) We first record a very useful: 11. Higher derivatives Theorem 11.1. Let A R n be an open subset. Let f : A R m and g : A R m be two functions and suppose that P A. Let λ A be a scalar. If f and g are differentiable

More information

Lecture 10: September 26

Lecture 10: September 26 0-725: Optimization Fall 202 Lecture 0: September 26 Lecturer: Barnabas Poczos/Ryan Tibshirani Scribes: Yipei Wang, Zhiguang Huo Note: LaTeX template courtesy of UC Berkeley EECS dept. Disclaimer: These

More information