Optimization II: Unconstrained Multivariable
|
|
- Gervase Dixon
- 5 years ago
- Views:
Transcription
1 Optimization II: Unconstrained Multivariable CS 205A: Mathematical Methods for Robotics, Vision, and Graphics Justin Solomon CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 1 / 20
2 Unconstrained Multivariable Problems minimize f : R n R CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 2 / 20
3 Recall f( x) Direction of steepest ascent CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 3 / 20
4 Recall f( x) Direction of steepest descent CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 4 / 20
5 Observation If f( x) 0, for sufficiently small α > 0, f( x α f( x)) f( x) CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 5 / 20
6 Gradient Descent Algorithm Iterate until convergence: 1. g k (t) f( x k t f( x k )) 2. Find t 0 minimizing (or decreasing) g k 3. x k+1 x k t f( x k ) CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 6 / 20
7 Stopping Condition f( x k ) 0 Don t forget: Check optimality! CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 7 / 20
8 Line Search g k (t) f( x k t f( x k )) One-dimensional optimization Don t have to minimize completely: Wolfe conditions CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 8 / 20
9 Newton s Method (again!) f( x) f( x k ) + f( x k ) ( x x k ) ( x x k) H f ( x k )( x x k ) CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 9 / 20
10 Newton s Method (again!) f( x) f( x k ) + f( x k ) ( x x k ) ( x x k) H f ( x k )( x x k ) = x k+1 = x k [H f ( x k )] 1 f( x k ) CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 9 / 20
11 Newton s Method (again!) f( x) f( x k ) + f( x k ) ( x x k ) ( x x k) H f ( x k )( x x k ) = x k+1 = x k [H f ( x k )] 1 f( x k ) Consideration: What if H f is not positive (semi-)definite? CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 9 / 20
12 Motivation f might be hard to compute but H f is harder H f might be dense: n 2 CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 10 / 20
13 Quasi-Newton Methods Approximate derivatives to avoid expensive calculations e.g. secant, Broyden,... CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 11 / 20
14 Common Optimization Assumption f known H f unknown or hard to compute CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 12 / 20
15 Quasi-Newton Optimization x k+1 = x k α k B 1 k f( x k) B k H f ( x k ) CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 13 / 20
16 Warning <advanced material> See Nocedal & Wright CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 14 / 20
17 Broyden-Style Update B k+1 ( x k+1 x k ) = f( x k+1 ) f( x k ) CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 15 / 20
18 Additional Considerations B k should be symmetric B k should be positive (semi-)definite CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 16 / 20
19 Davidon-Fletcher-Powell (DFP) min B k+1 B k B k+1 s.t. B k+1 = B k+1 B k+1 ( x k+1 x k ) = f( x k+1 ) f( x k ) CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 17 / 20
20 Observation B k+1 B k small does not mean B 1 is small k+1 B 1 k CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 18 / 20
21 Observation B k+1 B k small does not mean B 1 is small k+1 B 1 k Idea: Try to approximate directly B 1 k CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 18 / 20
22 BFGS Update min H k+1 H k HB k+1 s.t. H k+1 = H k+1 x k+1 x k = H k+1 ( f( x k+1 ) f( x k )) State of the art! CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 19 / 20
23 Lots of Missing Details Choice of Limited-memory alternative Next CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 20 / 20
Optimization II: Unconstrained Multivariable
Optimization II: Unconstrained Multivariable CS 205A: Mathematical Methods for Robotics, Vision, and Graphics Doug James (and Justin Solomon) CS 205A: Mathematical Methods Optimization II: Unconstrained
More information2. Quasi-Newton methods
L. Vandenberghe EE236C (Spring 2016) 2. Quasi-Newton methods variable metric methods quasi-newton methods BFGS update limited-memory quasi-newton methods 2-1 Newton method for unconstrained minimization
More information5 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 informationChapter 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 informationShiqian 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 informationConvex Optimization CMU-10725
Convex Optimization CMU-10725 Quasi Newton Methods Barnabás Póczos & Ryan Tibshirani Quasi Newton Methods 2 Outline Modified Newton Method Rank one correction of the inverse Rank two correction of the
More informationHigher-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 informationMethods that avoid calculating the Hessian. Nonlinear Optimization; Steepest Descent, Quasi-Newton. Steepest Descent
Nonlinear Optimization Steepest Descent and Niclas Börlin Department of Computing Science Umeå University niclas.borlin@cs.umu.se A disadvantage with the Newton method is that the Hessian has to be derived
More informationQuasi-Newton Methods
Newton s Method Pros and Cons Quasi-Newton Methods MA 348 Kurt Bryan Newton s method has some very nice properties: It s extremely fast, at least once it gets near the minimum, and with the simple modifications
More informationQuasi-Newton methods: Symmetric rank 1 (SR1) Broyden Fletcher Goldfarb Shanno February 6, / 25 (BFG. Limited memory BFGS (L-BFGS)
Quasi-Newton methods: Symmetric rank 1 (SR1) Broyden Fletcher Goldfarb Shanno (BFGS) Limited memory BFGS (L-BFGS) February 6, 2014 Quasi-Newton methods: Symmetric rank 1 (SR1) Broyden Fletcher Goldfarb
More informationComparative study of Optimization methods for Unconstrained Multivariable Nonlinear Programming Problems
International Journal of Scientific and Research Publications, Volume 3, Issue 10, October 013 1 ISSN 50-3153 Comparative study of Optimization methods for Unconstrained Multivariable Nonlinear Programming
More informationConjugate Gradients I: Setup
Conjugate Gradients I: Setup CS 205A: Mathematical Methods for Robotics, Vision, and Graphics Justin Solomon CS 205A: Mathematical Methods Conjugate Gradients I: Setup 1 / 22 Time for Gaussian Elimination
More informationQuasi-Newton Methods. Zico Kolter (notes by Ryan Tibshirani, Javier Peña, Zico Kolter) Convex Optimization
Quasi-Newton Methods Zico Kolter (notes by Ryan Tibshirani, Javier Peña, Zico Kolter) Convex Optimization 10-725 Last time: primal-dual interior-point methods Given the problem min x f(x) subject to h(x)
More informationAlgorithms for Constrained Optimization
1 / 42 Algorithms for Constrained Optimization ME598/494 Lecture Max Yi Ren Department of Mechanical Engineering, Arizona State University April 19, 2015 2 / 42 Outline 1. Convergence 2. Sequential quadratic
More informationQuasi-Newton Methods. Javier Peña Convex Optimization /36-725
Quasi-Newton Methods Javier Peña Convex Optimization 10-725/36-725 Last time: primal-dual interior-point methods Consider the problem min x subject to f(x) Ax = b h(x) 0 Assume f, h 1,..., h m are convex
More informationLecture 18: November Review on Primal-dual interior-poit methods
10-725/36-725: Convex Optimization Fall 2016 Lecturer: Lecturer: Javier Pena Lecture 18: November 2 Scribes: Scribes: Yizhu Lin, Pan Liu Note: LaTeX template courtesy of UC Berkeley EECS dept. Disclaimer:
More informationNewton 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 informationECS550NFB Introduction to Numerical Methods using Matlab Day 2
ECS550NFB Introduction to Numerical Methods using Matlab Day 2 Lukas Laffers lukas.laffers@umb.sk Department of Mathematics, University of Matej Bel June 9, 2015 Today Root-finding: find x that solves
More informationProgramming, 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 informationMATH 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 informationLecture V. Numerical Optimization
Lecture V Numerical Optimization Gianluca Violante New York University Quantitative Macroeconomics G. Violante, Numerical Optimization p. 1 /19 Isomorphism I We describe minimization problems: to maximize
More informationStatistics 580 Optimization Methods
Statistics 580 Optimization Methods Introduction Let fx be a given real-valued function on R p. The general optimization problem is to find an x ɛ R p at which fx attain a maximum or a minimum. It is of
More informationLecture 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 informationImproving L-BFGS Initialization for Trust-Region Methods in Deep Learning
Improving L-BFGS Initialization for Trust-Region Methods in Deep Learning Jacob Rafati http://rafati.net jrafatiheravi@ucmerced.edu Ph.D. Candidate, Electrical Engineering and Computer Science University
More informationOptimization: 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 informationNonlinear 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 information1 Numerical optimization
Contents 1 Numerical optimization 5 1.1 Optimization of single-variable functions............ 5 1.1.1 Golden Section Search................... 6 1.1. Fibonacci Search...................... 8 1. Algorithms
More informationNumerical solutions of nonlinear systems of equations
Numerical solutions of nonlinear systems of equations Tsung-Ming Huang Department of Mathematics National Taiwan Normal University, Taiwan E-mail: min@math.ntnu.edu.tw August 28, 2011 Outline 1 Fixed points
More informationNumerical Optimization Professor Horst Cerjak, Horst Bischof, Thomas Pock Mat Vis-Gra SS09
Numerical Optimization 1 Working Horse in Computer Vision Variational Methods Shape Analysis Machine Learning Markov Random Fields Geometry Common denominator: optimization problems 2 Overview of Methods
More informationNonlinear 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 informationOPER 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 informationPart 4: IIR Filters Optimization Approach. Tutorial ISCAS 2007
Part 4: IIR Filters Optimization Approach Tutorial ISCAS 2007 Copyright 2007 Andreas Antoniou Victoria, BC, Canada Email: aantoniou@ieee.org July 24, 2007 Frame # 1 Slide # 1 A. Antoniou Part4: IIR Filters
More informationEAD 115. Numerical Solution of Engineering and Scientific Problems. David M. Rocke Department of Applied Science
EAD 115 Numerical Solution of Engineering and Scientific Problems David M. Rocke Department of Applied Science Multidimensional Unconstrained Optimization Suppose we have a function f() of more than one
More information1 Numerical optimization
Contents Numerical optimization 5. Optimization of single-variable functions.............................. 5.. Golden Section Search..................................... 6.. Fibonacci Search........................................
More information(One Dimension) Problem: for a function f(x), find x 0 such that f(x 0 ) = 0. f(x)
Solving Nonlinear Equations & Optimization One Dimension Problem: or a unction, ind 0 such that 0 = 0. 0 One Root: The Bisection Method This one s guaranteed to converge at least to a singularity, i not
More informationOptimization Methods
Optimization Methods Decision making Examples: determining which ingredients and in what quantities to add to a mixture being made so that it will meet specifications on its composition allocating available
More informationIntroduction. New Nonsmooth Trust Region Method for Unconstraint Locally Lipschitz Optimization Problems
New Nonsmooth Trust Region Method for Unconstraint Locally Lipschitz Optimization Problems Z. Akbari 1, R. Yousefpour 2, M. R. Peyghami 3 1 Department of Mathematics, K.N. Toosi University of Technology,
More informationOptimization Methods for Machine Learning
Optimization Methods for Machine Learning Sathiya Keerthi Microsoft Talks given at UC Santa Cruz February 21-23, 2017 The slides for the talks will be made available at: http://www.keerthis.com/ Introduction
More informationIntroduction. A Modified Steepest Descent Method Based on BFGS Method for Locally Lipschitz Functions. R. Yousefpour 1
A Modified Steepest Descent Method Based on BFGS Method for Locally Lipschitz Functions R. Yousefpour 1 1 Department Mathematical Sciences, University of Mazandaran, Babolsar, Iran; yousefpour@umz.ac.ir
More informationAM 205: lecture 19. Last time: Conditions for optimality Today: Newton s method for optimization, survey of optimization methods
AM 205: lecture 19 Last time: Conditions for optimality Today: Newton s method for optimization, survey of optimization methods Optimality Conditions: Equality Constrained Case As another example of equality
More informationUnconstrained 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 informationMaria Cameron. f(x) = 1 n
Maria Cameron 1. Local algorithms for solving nonlinear equations Here we discuss local methods for nonlinear equations r(x) =. These methods are Newton, inexact Newton and quasi-newton. We will show that
More informationORIE 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 informationMultivariate Newton Minimanization
Multivariate Newton Minimanization Optymalizacja syntezy biosurfaktantu Rhamnolipid Rhamnolipids are naturally occuring glycolipid produced commercially by the Pseudomonas aeruginosa species of bacteria.
More informationMaximum Likelihood Estimation
Maximum Likelihood Estimation Prof. C. F. Jeff Wu ISyE 8813 Section 1 Motivation What is parameter estimation? A modeler proposes a model M(θ) for explaining some observed phenomenon θ are the parameters
More informationOptimization 2. CS5240 Theoretical Foundations in Multimedia. Leow Wee Kheng
Optimization 2 CS5240 Theoretical Foundations in Multimedia Leow Wee Kheng Department of Computer Science School of Computing National University of Singapore Leow Wee Kheng (NUS) Optimization 2 1 / 38
More informationNonlinearOptimization
1/35 NonlinearOptimization Pavel Kordík Department of Computer Systems Faculty of Information Technology Czech Technical University in Prague Jiří Kašpar, Pavel Tvrdík, 2011 Unconstrained nonlinear optimization,
More informationConvex 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 informationLecture 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 informationGeometry optimization
Geometry optimization Trygve Helgaker Centre for Theoretical and Computational Chemistry Department of Chemistry, University of Oslo, Norway European Summer School in Quantum Chemistry (ESQC) 211 Torre
More informationQuasi-Newton methods for minimization
Quasi-Newton methods for minimization Lectures for PHD course on Numerical optimization Enrico Bertolazzi DIMS Universitá di Trento November 21 December 14, 2011 Quasi-Newton methods for minimization 1
More informationNonlinearity Root-finding Bisection Fixed Point Iteration Newton s Method Secant Method Conclusion. Nonlinear Systems
Nonlinear Systems CS 205A: Mathematical Methods for Robotics, Vision, and Graphics Justin Solomon CS 205A: Mathematical Methods Nonlinear Systems 1 / 24 Part III: Nonlinear Problems Not all numerical problems
More informationMethods 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 informationOptimization and Root Finding. Kurt Hornik
Optimization and Root Finding Kurt Hornik Basics Root finding and unconstrained smooth optimization are closely related: Solving ƒ () = 0 can be accomplished via minimizing ƒ () 2 Slide 2 Basics Root finding
More informationON THE CONNECTION BETWEEN THE CONJUGATE GRADIENT METHOD AND QUASI-NEWTON METHODS ON QUADRATIC PROBLEMS
ON THE CONNECTION BETWEEN THE CONJUGATE GRADIENT METHOD AND QUASI-NEWTON METHODS ON QUADRATIC PROBLEMS Anders FORSGREN Tove ODLAND Technical Report TRITA-MAT-203-OS-03 Department of Mathematics KTH Royal
More informationImproved Damped Quasi-Newton Methods for Unconstrained Optimization
Improved Damped Quasi-Newton Methods for Unconstrained Optimization Mehiddin Al-Baali and Lucio Grandinetti August 2015 Abstract Recently, Al-Baali (2014) has extended the damped-technique in the modified
More informationENSIEEHT-IRIT, 2, rue Camichel, Toulouse (France) LMS SAMTECH, A Siemens Business,15-16, Lower Park Row, BS1 5BN Bristol (UK)
Quasi-Newton updates with weighted secant equations by. Gratton, V. Malmedy and Ph. L. oint Report NAXY-09-203 6 October 203 0.5 0 0.5 0.5 0 0.5 ENIEEH-IRI, 2, rue Camichel, 3000 oulouse France LM AMECH,
More informationGradient-Based Optimization
Multidisciplinary Design Optimization 48 Chapter 3 Gradient-Based Optimization 3. Introduction In Chapter we described methods to minimize (or at least decrease) a function of one variable. While problems
More informationGlobal Convergence of Perry-Shanno Memoryless Quasi-Newton-type Method. 1 Introduction
ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.11(2011) No.2,pp.153-158 Global Convergence of Perry-Shanno Memoryless Quasi-Newton-type Method Yigui Ou, Jun Zhang
More informationProximal 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 informationStep lengths in BFGS method for monotone gradients
Noname manuscript No. (will be inserted by the editor) Step lengths in BFGS method for monotone gradients Yunda Dong Received: date / Accepted: date Abstract In this paper, we consider how to directly
More informationUnconstrained Multivariate Optimization
Unconstrained Multivariate Optimization Multivariate optimization means optimization of a scalar function of a several variables: and has the general form: y = () min ( ) where () is a nonlinear scalar-valued
More informationMath 408A: Non-Linear Optimization
February 12 Broyden Updates Given g : R n R n solve g(x) = 0. Algorithm: Broyden s Method Initialization: x 0 R n, B 0 R n n Having (x k, B k ) compute (x k+1, B x+1 ) as follows: Solve B k s k = g(x
More informationSTAT Advanced Bayesian Inference
1 / 8 STAT 625 - Advanced Bayesian Inference Meng Li Department of Statistics March 5, 2018 Distributional approximations 2 / 8 Distributional approximations are useful for quick inferences, as starting
More information1 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 informationNonlinearity Root-finding Bisection Fixed Point Iteration Newton s Method Secant Method Conclusion. Nonlinear Systems
Nonlinear Systems CS 205A: Mathematical Methods for Robotics, Vision, and Graphics Doug James (and Justin Solomon) CS 205A: Mathematical Methods Nonlinear Systems 1 / 27 Part III: Nonlinear Problems Not
More information17 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 informationNONSMOOTH VARIANTS OF POWELL S BFGS CONVERGENCE THEOREM
NONSMOOTH VARIANTS OF POWELL S BFGS CONVERGENCE THEOREM JIAYI GUO AND A.S. LEWIS Abstract. The popular BFGS quasi-newton minimization algorithm under reasonable conditions converges globally on smooth
More informationFALL 2018 MATH 4211/6211 Optimization Homework 4
FALL 2018 MATH 4211/6211 Optimization Homework 4 This homework assignment is open to textbook, reference books, slides, and online resources, excluding any direct solution to the problem (such as solution
More informationMATH 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 informationEECS260 Optimization Lecture notes
EECS260 Optimization Lecture notes Based on Numerical Optimization (Nocedal & Wright, Springer, 2nd ed., 2006) Miguel Á. Carreira-Perpiñán EECS, University of California, Merced May 2, 2010 1 Introduction
More informationIntroduction 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 informationPOLI 8501 Introduction to Maximum Likelihood Estimation
POLI 8501 Introduction to Maximum Likelihood Estimation Maximum Likelihood Intuition Consider a model that looks like this: Y i N(µ, σ 2 ) So: E(Y ) = µ V ar(y ) = σ 2 Suppose you have some data on Y,
More informationHomework 3 Conjugate Gradient Descent, Accelerated Gradient Descent Newton, Quasi Newton and Projected Gradient Descent
Homework 3 Conjugate Gradient Descent, Accelerated Gradient Descent Newton, Quasi Newton and Projected Gradient Descent CMU 10-725/36-725: Convex Optimization (Fall 2017) OUT: Sep 29 DUE: Oct 13, 5:00
More informationOptimization Algorithms on Riemannian Manifolds with Applications
Optimization Algorithms on Riemannian Manifolds with Applications Wen Huang Coadvisor: Kyle A. Gallivan Coadvisor: Pierre-Antoine Absil Florida State University Catholic University of Louvain November
More informationData Mining (Mineria de Dades)
Data Mining (Mineria de Dades) Lluís A. Belanche belanche@lsi.upc.edu Soft Computing Research Group Dept. de Llenguatges i Sistemes Informàtics (Software department) Universitat Politècnica de Catalunya
More informationNumerical Analysis of Electromagnetic Fields
Pei-bai Zhou Numerical Analysis of Electromagnetic Fields With 157 Figures Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest Contents Part 1 Universal Concepts
More informationUniversity of Maryland at College Park. limited amount of computer memory, thereby allowing problems with a very large number
Limited-Memory Matrix Methods with Applications 1 Tamara Gibson Kolda 2 Applied Mathematics Program University of Maryland at College Park Abstract. The focus of this dissertation is on matrix decompositions
More informationOrdinary Differential Equations II
Ordinary Differential Equations II CS 205A: Mathematical Methods for Robotics, Vision, and Graphics Justin Solomon CS 205A: Mathematical Methods Ordinary Differential Equations II 1 / 33 Almost Done! Last
More informationChapter 2. Optimization. Gradients, convexity, and ALS
Chapter 2 Optimization Gradients, convexity, and ALS Contents Background Gradient descent Stochastic gradient descent Newton s method Alternating least squares KKT conditions 2 Motivation We can solve
More informationA Primer on Multidimensional Optimization
A Primer on Multidimensional Optimization Prof. Dr. Florian Rupp German University of Technology in Oman (GUtech) Introduction to Numerical Methods for ENG & CS (Mathematics IV) Spring Term 2016 Eercise
More informationCoE 3SK3 Computer Aided Engineering Tutorial: Unconstrained Optimization
CoE 3SK3 Computer Aided Engineering Tutorial: Unconstrained Optimization Jie Cao caoj23@grads.ece.mcmaster.ca Department of Electrical and Computer Engineering McMaster University Feb. 2, 2010 Outline
More informationEigenproblems II: Computation
Eigenproblems II: Computation CS 205A: Mathematical Methods for Robotics, Vision, and Graphics Doug James (and Justin Solomon) CS 205A: Mathematical Methods Eigenproblems II: Computation 1 / 31 Setup A
More informationMA/OR/ST 706: Nonlinear Programming Midterm Exam Instructor: Dr. Kartik Sivaramakrishnan INSTRUCTIONS
MA/OR/ST 706: Nonlinear Programming Midterm Exam Instructor: Dr. Kartik Sivaramakrishnan INSTRUCTIONS 1. Please write your name and student number clearly on the front page of the exam. 2. The exam is
More informationStochastic Optimization Algorithms Beyond SG
Stochastic Optimization Algorithms Beyond SG Frank E. Curtis 1, Lehigh University involving joint work with Léon Bottou, Facebook AI Research Jorge Nocedal, Northwestern University Optimization Methods
More informationImage restoration. An example in astronomy
Image restoration Convex approaches: penalties and constraints An example in astronomy Jean-François Giovannelli Groupe Signal Image Laboratoire de l Intégration du Matériau au Système Univ. Bordeaux CNRS
More informationPreconditioned conjugate gradient algorithms with column scaling
Proceedings of the 47th IEEE Conference on Decision and Control Cancun, Mexico, Dec. 9-11, 28 Preconditioned conjugate gradient algorithms with column scaling R. Pytla Institute of Automatic Control and
More informationQuasi-Newton Methods
Quasi-Newton Methods Werner C. Rheinboldt These are excerpts of material relating to the boos [OR00 and [Rhe98 and of write-ups prepared for courses held at the University of Pittsburgh. Some further references
More informationStatic unconstrained optimization
Static unconstrained optimization 2 In unconstrained optimization an objective function is minimized without any additional restriction on the decision variables, i.e. min f(x) x X ad (2.) with X ad R
More informationLine Search Methods. Shefali Kulkarni-Thaker
1 BISECTION METHOD Line Search Methods Shefali Kulkarni-Thaker Consider the following unconstrained optimization problem min f(x) x R Any optimization algorithm starts by an initial point x 0 and performs
More informationABSTRACT 1. INTRODUCTION
A DIAGONAL-AUGMENTED QUASI-NEWTON METHOD WITH APPLICATION TO FACTORIZATION MACHINES Aryan Mohtari and Amir Ingber Department of Electrical and Systems Engineering, University of Pennsylvania, PA, USA Big-data
More informationGRADIENT-TYPE METHODS FOR UNCONSTRAINED OPTIMIZATION
GRADIENT-TYPE METHODS FOR UNCONSTRAINED OPTIMIZATION By GUAN HUI SHAN A project report submitted in partial fulfilment of the requirements for the award of Bachelor of Science (Hons.) Applied Mathematics
More informationMachine Learning CS 4900/5900. Lecture 03. Razvan C. Bunescu School of Electrical Engineering and Computer Science
Machine Learning CS 4900/5900 Razvan C. Bunescu School of Electrical Engineering and Computer Science bunescu@ohio.edu Machine Learning is Optimization Parametric ML involves minimizing an objective function
More informationP1: JYD /... CB495-08Drv CB495/Train KEY BOARDED March 24, :7 Char Count= 0 Part II Estimation 183
Part II Estimation 8 Numerical Maximization 8.1 Motivation Most estimation involves maximization of some function, such as the likelihood function, the simulated likelihood function, or squared moment
More informationSTUDYING THE BASIN OF CONVERGENCE OF METHODS FOR COMPUTING PERIODIC ORBITS
Tutorials and Reviews International Journal of Bifurcation and Chaos, Vol. 21, No. 8 (2011) 2079 2106 c World Scientific Publishing Company DOI: 10.1142/S0218127411029653 STUDYING THE BASIN OF CONVERGENCE
More informationMath 409/509 (Spring 2011)
Math 409/509 (Spring 2011) Instructor: Emre Mengi Study Guide for Homework 2 This homework concerns the root-finding problem and line-search algorithms for unconstrained optimization. Please don t hesitate
More informationBindel, Fall 2011 Intro to Scientific Computing (CS 3220) Week 6: Monday, Mar 7. e k+1 = 1 f (ξ k ) 2 f (x k ) e2 k.
Problem du jour Week 6: Monday, Mar 7 Show that for any initial guess x 0 > 0, Newton iteration on f(x) = x 2 a produces a decreasing sequence x 1 x 2... x n a. What is the rate of convergence if a = 0?
More informationNumerical 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 informationLecture 14 Ellipsoid method
S. Boyd EE364 Lecture 14 Ellipsoid method idea of localization methods bisection on R center of gravity algorithm ellipsoid method 14 1 Localization f : R n R convex (and for now, differentiable) problem:
More informationIntroduction 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