Numerical Optimization
|
|
- Eleanore Sharp
- 5 years ago
- Views:
Transcription
1 Numerical Optimization Unit 2: Multivariable optimization problems Che-Rung Lee Scribe: February 28, 2011 (UNIT 2) Numerical Optimization February 28, / 17
2 Partial derivative of a two variable function Given a two variable function f (x 1, x 2 ). The partial derivative of f with respect to x i is f f (x 1 + h, x 2 ) f (x 1, x 2 ) = lim h 0 x 1 h f f (x 1, x 2 + h) f (x 1, x 2 ) = lim h 0 x 2 h The meaning of partial derivative: let F (x 1 ) = f (x 1, v) and G(x 2 ) = f (u, x 2 ), f x 1 (x 1, v) = F (x 1 ). f x 2 (u, x 2 ) = G (x 2 ). (UNIT 2) Numerical Optimization February 28, / 17
3 Gradient Definition The gradient of a function f : R n R is a vector in R n defined as f / x 1 x 1 g = f ( x) =., where x =. f / x n x n (UNIT 2) Numerical Optimization February 28, / 17
4 Directional derivative Definition The directional derivative of a function f : R n R in the direction p is defined as D(f ( x), p) = lim h 0 f ( x + h p) f (x) h. Remark If f : R n R is continuously differentiable in a neighborhood of x, for any vector p. D(f ( x), p) = f (x) T p, (UNIT 2) Numerical Optimization February 28, / 17
5 The descent directions A direction p is called a descent direction of f ( x) at x if D(f ( x 0 ), p) < 0. If f is smooth enough, p is a descent direction if f ( x 0 ) T p < 0. Which direction p makes f ( x 0 + p) decreasing most? Mean Value theorem f ( x 0 + p) = f ( x 0 ) + f ( x 0 + α p) p p = f ( x 0 ) is called the steepest descent direction of f (x) at x 0. f ( x 0 + p) = f ( x 0 ) + f ( x 0 + α p) p f ( x 0 ) f ( x 0 ) T f ( x 0 ) (UNIT 2) Numerical Optimization February 28, / 17
6 The steepest descent algorithm The steepest descent algorithm For k = 1, 2,... until convergence Compute p k = f (x k ) Find α k (0, 1) s,t, F (α k ) = f ( x k + α k p k ) is minimized. x k+1 = x k + α k p k You can use any single variable optimization techniques to compute α k. If F (α k ) = f ( x k + α k p k ) is a quadratic function, α k has a theoretical formula. (will be derived in next slides.) If F (α k ) = f ( x k + α k p k ) is more than a quadratic function, we may approximate it by a quadratic model and use the formula to solve α k. Higher order polynomial approximation will be mentioned in the line search algorithm. (UNIT 2) Numerical Optimization February 28, / 17
7 Quadratic model If f ( x) is a quadratic function, we can write it as f (x, y) = ax 2 + bxy + cy 2 + dx + ey + f (0, 0). If f is smooth, the derivatives of f are Let x = ( x y f x = 2ax + by + d, f y 2 f x 2 = 2a, 2 f y 2 = 2c, ), f ( x) can be expressed as f ( x) = 1 ( 2 x T 2a b b 2c = 2cy + bx + e 2 f x y = ) x + x T ( d e 2 f y x = b. ) + f ( 0). (UNIT 2) Numerical Optimization February 28, / 17
8 Gradient and Hessian The gradient of f, as defined before, is f ( g( x) = f ( x) = x 2a b f = b 2c y ) ( d x + e The second derivative, which is a matrix called Hessian, is f f ( 2 f ( x) = H( x) = x 2 x y 2a b f f = b 2c y x y 2 ) ) Therefore, f ( x) = 1/2 x T H( 0) x + g( 0) T x + f ( 0), f ( x) = H x + g, and 2 f = H In the following lectures, we assume H is symmetric. Thus, H = H T. (UNIT 2) Numerical Optimization February 28, / 17
9 Optimal α k for quadratic model We denote H k = H( x k ), g k = g( x k ), and f k = f ( x k ). Also, H = H( 0), g = g( 0), and f = f ( 0). F (α) = f ( x k + α p k ) = 1 2 ( x k + α p k ) T H( x k + α p k ) + g T ( x k + α p k ) + f ( 0) = 1 2 x k T H x k + g T x k + f ( 0) + α(h x k + g) T p k + α2 2 p k T H p k = f k + α g k T p k + α2 2 p k T k F (α) = g k T p k + α p k T k The optimal solution of α k is at F (α) = 0, which is α k = g k T p k H p k p T k (UNIT 2) Numerical Optimization February 28, / 17
10 Optimal condition Theorem (Necessary and sufficient condition of optimality) Let f : R n R be continuously differentiable in D. If x D is a local minimizer, f ( x ) = 0 and 2 f ( x) is positive semidefinite. If f ( x ) = 0 and 2 f ( x) is positive definite, then x is a local minimizer. Definition A matrix H is called positive definite if for any nonzero vector v R n, v H v > 0. H is called positive semidefinite if v H v 0 for all v R n. H is negative definite or negative semidefinite if H is positive definite or positive semidefinite. H is indefinite if it is neither positive semidefinite nor negative semidefinite. (UNIT 2) Numerical Optimization February 28, / 17
11 Convergence of the steepest descent method Theorem (Convergence theorem of the steepest descent method) If the steepest descent method converges to a local minimizer x, where 2 f ( x) is positive definite, and e max and e min are the largest and the smallest eigenvalue of 2 f ( x), then Definition x k+1 x lim k x k x ( ) emax e min e max + e min For a scalar λ and an unit vector v, (λ, v) is an eigenpair of of a matrix H if Hv = λv. The scalar λ is called an eigenvalue of H, and v is called an eigenvector. (UNIT 2) Numerical Optimization February 28, / 17
12 Newton s method We use the quadratic model to find the step length α k. Can we use the quadratic model to find the search direction p k? Yes, we can. Recall the quadratic model (now p is the variable.) f ( x k + p) 1 2 p T H k p + p T g k + f k Compute the gradient p f ( x k + p) = H k p + g k The solution of p f ( x k + p) = 0 is p k = H 1 k g k. Newton s method uses p k as the search direction Newton s method 1 Given an initial guess x 0 2 For k = 0, 1, 2,... until converge x k+1 = x k H 1 k g k. (UNIT 2) Numerical Optimization February 28, / 17
13 Descent direction The direction p k = H 1 k g k is called Newton s direction Is p k a descent direction? (what s the definition of descent directions?) We only need to check if g T k p k < 0. g T k p k = g T k H 1 k g k. Thus, p k is a descent direction if H 1 is positive definite. For a symmetric matrix H, the following conditions are equivalent H is positive definite. H 1 is positive definite. All the eigenvalues of H are positive. (UNIT 2) Numerical Optimization February 28, / 17
14 Some properties of eigenvalues/eigenvectors A symmetric matrix H, of order n has n real eigenvalues and n real and linearly independent (orthogonal) eigenvectors Hv 1 = λ 1 v 1, Hv 2 = λ 2 v 2,..., Hv n = λ n v n λ 1 λ 2 Let V = [v 1 v 2... v n ], Λ =, HV = V Λ.... λn If λ 1, λ 2,..., λ n are nonzero, since H = V ΛV 1, 1/λ 1 H 1 = V Λ 1 V 1, Λ 1 1/λ 2 =... 1/λn The eigenvalues of H 1 are 1 λ 1, 1 λ 2,..., 1 λ n. (UNIT 2) Numerical Optimization February 28, / 17
15 How to solve H p = g? For a symmetric positive definite matrix H, H p = g can be solved by Cholesky decomposition, which is similar to LU decomposition, but is only half computational cost of LU decomposition. h 11 h 12 h 13 Let H = h 21 h 22 h 23, where h 12 = h 21, h 13 = h 31, h 23 = h 32. h 31 h 32 h 33 Cholesky decomposition makes H = LL T, where L is a lower l 11 triangular matrix, L = l 21 l 22 l 31 l 32 l 33 Using Cholesky decomposition, H p = g can be solved by 1 Compute H = LL T 2 p = (L T ) 1 L 1 g In Matlab, use p = H \ g. Don t use inv(h). (UNIT 2) Numerical Optimization February 28, / 17
16 The Cholesky decomposition For i = 1, 2,..., n l ii = h ii For j = i + 1, i + 2,..., n l ji = h ji l ii For k = i + 1, i + 2,..., j h jk = h jk l ji l ki h 11 h 12 h 13 h 21 h 22 h 23 h 31 h 32 h 33 l 11 = h 11 l 21 = h 21 /l 11 l 31 = h 31 /l 11 =LL T = l 2 11 l 11 l 21 l 11 l 31 l 11 l 21 l l2 22 l 21 l 31 + l 22 l 32 l 11 l 33 l 21 l 31 + l 22 l 32 l l l2 33 h (2) 22 = h 22 l 21 l 21 h (2) 32 = h 32 l 21 l 31 h (2) 33 = h 33 l 31 l 31 l 22 = h (2) 22 l 32 = h (2) 32 /l 22 l 33 = h (2) 33 l 32l 32 (UNIT 2) Numerical Optimization February 28, / 17
17 Convergence of Newton s method Theorem Suppose f is twice differentiable. 2 f is continuous in a neighborhood of x and 2 f ( x ) is positive definite, and if x 0 is sufficiently close to x, the sequence converges to x quadratically. Three problems of Newton s method 1 H may not be positive definite Modified Newton s method + Line search. 2 H is expensive to compute Quasi-Newton. 3 H 1 is expensive to compute Conjugate gradient. (UNIT 2) Numerical Optimization February 28, / 17
MATH 5720: Unconstrained Optimization Hung Phan, UMass Lowell September 13, 2018
MATH 57: Unconstrained Optimization Hung Phan, UMass Lowell September 13, 18 1 Global and Local Optima Let a function f : S R be defined on a set S R n Definition 1 (minimizers and maximizers) (i) x S
More informationLine 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 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 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 Taylor s Theorem Can often approximate a function by a polynomial The error in the approximation
More informationVasil Khalidov & Miles Hansard. C.M. Bishop s PRML: Chapter 5; Neural Networks
C.M. Bishop s PRML: Chapter 5; Neural Networks Introduction The aim is, as before, to find useful decompositions of the target variable; t(x) = y(x, w) + ɛ(x) (3.7) t(x n ) and x n are the observations,
More informationConjugate Gradient (CG) Method
Conjugate Gradient (CG) Method by K. Ozawa 1 Introduction In the series of this lecture, I will introduce the conjugate gradient method, which solves efficiently large scale sparse linear simultaneous
More informationScientific 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 informationChapter 3 Transformations
Chapter 3 Transformations An Introduction to Optimization Spring, 2014 Wei-Ta Chu 1 Linear Transformations A function is called a linear transformation if 1. for every and 2. for every If we fix the bases
More informationLinear Algebra in Actuarial Science: Slides to the lecture
Linear Algebra in Actuarial Science: Slides to the lecture Fall Semester 2010/2011 Linear Algebra is a Tool-Box Linear Equation Systems Discretization of differential equations: solving linear equations
More informationAM 205: lecture 18. Last time: optimization methods Today: conditions for optimality
AM 205: lecture 18 Last time: optimization methods Today: conditions for optimality Existence of Global Minimum For example: f (x, y) = x 2 + y 2 is coercive on R 2 (global min. at (0, 0)) f (x) = x 3
More informationPaul Schrimpf. October 18, UBC Economics 526. Unconstrained optimization. Paul Schrimpf. Notation and definitions. First order conditions
Unconstrained UBC Economics 526 October 18, 2013 .1.2.3.4.5 Section 1 Unconstrained problem x U R n F : U R. max F (x) x U Definition F = max x U F (x) is the maximum of F on U if F (x) F for all x U and
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 information, b = 0. (2) 1 2 The eigenvectors of A corresponding to the eigenvalues λ 1 = 1, λ 2 = 3 are
Quadratic forms We consider the quadratic function f : R 2 R defined by f(x) = 2 xt Ax b T x with x = (x, x 2 ) T, () where A R 2 2 is symmetric and b R 2. We will see that, depending on the eigenvalues
More information14. 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 informationECE580 Exam 1 October 4, Please do not write on the back of the exam pages. Extra paper is available from the instructor.
ECE580 Exam 1 October 4, 2012 1 Name: Solution Score: /100 You must show ALL of your work for full credit. This exam is closed-book. Calculators may NOT be used. Please leave fractions as fractions, etc.
More informationREVIEW OF DIFFERENTIAL CALCULUS
REVIEW OF DIFFERENTIAL CALCULUS DONU ARAPURA 1. Limits and continuity To simplify the statements, we will often stick to two variables, but everything holds with any number of variables. Let f(x, y) be
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 informationConstrained optimization. Unconstrained optimization. One-dimensional. Multi-dimensional. Newton with equality constraints. Active-set method.
Optimization Unconstrained optimization One-dimensional Multi-dimensional Newton s method Basic Newton Gauss- Newton Quasi- Newton Descent methods Gradient descent Conjugate gradient Constrained optimization
More informationLinear Algebra II Lecture 22
Linear Algebra II Lecture 22 Xi Chen University of Alberta March 4, 24 Outline Characteristic Polynomial, Eigenvalue, Eigenvector and Eigenvalue, Eigenvector and Let T : V V be a linear endomorphism. We
More informationFundamentals of Unconstrained Optimization
dalmau@cimat.mx Centro de Investigación en Matemáticas CIMAT A.C. Mexico Enero 2016 Outline Introduction 1 Introduction 2 3 4 Optimization Problem min f (x) x Ω where f (x) is a real-valued function The
More informationB553 Lecture 5: Matrix Algebra Review
B553 Lecture 5: Matrix Algebra Review Kris Hauser January 19, 2012 We have seen in prior lectures how vectors represent points in R n and gradients of functions. Matrices represent linear transformations
More informationConjugate Gradient Method
Conjugate Gradient Method Hung M Phan UMass Lowell April 13, 2017 Throughout, A R n n is symmetric and positive definite, and b R n 1 Steepest Descent Method We present the steepest descent method for
More information1 Last time: least-squares problems
MATH Linear algebra (Fall 07) Lecture Last time: least-squares problems Definition. If A is an m n matrix and b R m, then a least-squares solution to the linear system Ax = b is a vector x R n such that
More informationLINEAR ALGEBRA QUESTION BANK
LINEAR ALGEBRA QUESTION BANK () ( points total) Circle True or False: TRUE / FALSE: If A is any n n matrix, and I n is the n n identity matrix, then I n A = AI n = A. TRUE / FALSE: If A, B are n n matrices,
More informationNumerical Methods - Numerical Linear Algebra
Numerical Methods - Numerical Linear Algebra Y. K. Goh Universiti Tunku Abdul Rahman 2013 Y. K. Goh (UTAR) Numerical Methods - Numerical Linear Algebra I 2013 1 / 62 Outline 1 Motivation 2 Solving Linear
More informationChapter 2: Unconstrained Extrema
Chapter 2: Unconstrained Extrema Math 368 c Copyright 2012, 2013 R Clark Robinson May 22, 2013 Chapter 2: Unconstrained Extrema 1 Types of Sets Definition For p R n and r > 0, the open ball about p of
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 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 informationforms Christopher Engström November 14, 2014 MAA704: Matrix factorization and canonical forms Matrix properties Matrix factorization Canonical forms
Christopher Engström November 14, 2014 Hermitian LU QR echelon Contents of todays lecture Some interesting / useful / important of matrices Hermitian LU QR echelon Rewriting a as a product of several matrices.
More informationIntroduction to gradient descent
6-1: Introduction to gradient descent Prof. J.C. Kao, UCLA Introduction to gradient descent Derivation and intuitions Hessian 6-2: Introduction to gradient descent Prof. J.C. Kao, UCLA Introduction Our
More informationMidterm for Introduction to Numerical Analysis I, AMSC/CMSC 466, on 10/29/2015
Midterm for Introduction to Numerical Analysis I, AMSC/CMSC 466, on 10/29/2015 The test lasts 1 hour and 15 minutes. No documents are allowed. The use of a calculator, cell phone or other equivalent electronic
More informationUnconstrained 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 informationCS 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 informationChap 3. Linear Algebra
Chap 3. Linear Algebra Outlines 1. Introduction 2. Basis, Representation, and Orthonormalization 3. Linear Algebraic Equations 4. Similarity Transformation 5. Diagonal Form and Jordan Form 6. Functions
More information1 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 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 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 informationNumerical methods for eigenvalue problems
Numerical methods for eigenvalue problems D. Löchel Supervisors: M. Hochbruck und M. Tokar Mathematisches Institut Heinrich-Heine-Universität Düsseldorf GRK 1203 seminar february 2008 Outline Introduction
More informationNeural Network Training
Neural Network Training Sargur Srihari Topics in Network Training 0. Neural network parameters Probabilistic problem formulation Specifying the activation and error functions for Regression Binary classification
More informationMatrices and Linear transformations
Matrices and Linear transformations We have been thinking of matrices in connection with solutions to linear systems of equations like Ax = b. It is time to broaden our horizons a bit and start thinking
More informationSymmetric matrices and dot products
Symmetric matrices and dot products Proposition An n n matrix A is symmetric iff, for all x, y in R n, (Ax) y = x (Ay). Proof. If A is symmetric, then (Ax) y = x T A T y = x T Ay = x (Ay). If equality
More informationReview problems for MA 54, Fall 2004.
Review problems for MA 54, Fall 2004. Below are the review problems for the final. They are mostly homework problems, or very similar. If you are comfortable doing these problems, you should be fine on
More informationArc Search Algorithms
Arc Search Algorithms Nick Henderson and Walter Murray Stanford University Institute for Computational and Mathematical Engineering November 10, 2011 Unconstrained Optimization minimize x D F (x) where
More informationLecture 6 Positive Definite Matrices
Linear Algebra Lecture 6 Positive Definite Matrices Prof. Chun-Hung Liu Dept. of Electrical and Computer Engineering National Chiao Tung University Spring 2017 2017/6/8 Lecture 6: Positive Definite Matrices
More informationMotivation: We have already seen an example of a system of nonlinear equations when we studied Gaussian integration (p.8 of integration notes)
AMSC/CMSC 460 Computational Methods, Fall 2007 UNIT 5: Nonlinear Equations Dianne P. O Leary c 2001, 2002, 2007 Solving Nonlinear Equations and Optimization Problems Read Chapter 8. Skip Section 8.1.1.
More informationthe method of steepest descent
MATH 3511 Spring 2018 the method of steepest descent http://www.phys.uconn.edu/ rozman/courses/m3511_18s/ Last modified: February 6, 2018 Abstract The Steepest Descent is an iterative method for solving
More informationLagrange multipliers. Portfolio optimization. The Lagrange multipliers method for finding constrained extrema of multivariable functions.
Chapter 9 Lagrange multipliers Portfolio optimization The Lagrange multipliers method for finding constrained extrema of multivariable functions 91 Lagrange multipliers Optimization problems often require
More informationMath 5630: Conjugate Gradient Method Hung M. Phan, UMass Lowell March 29, 2019
Math 563: Conjugate Gradient Method Hung M. Phan, UMass Lowell March 29, 219 hroughout, A R n n is symmetric and positive definite, and b R n. 1 Steepest Descent Method We present the steepest descent
More informationAM 205: lecture 19. Last time: Conditions for optimality, Newton s method for optimization Today: survey of optimization methods
AM 205: lecture 19 Last time: Conditions for optimality, Newton s method for optimization Today: survey of optimization methods Quasi-Newton Methods General form of quasi-newton methods: x k+1 = x k α
More informationMATHEMATICS FOR COMPUTER VISION WEEK 8 OPTIMISATION PART 2. Dr Fabio Cuzzolin MSc in Computer Vision Oxford Brookes University Year
MATHEMATICS FOR COMPUTER VISION WEEK 8 OPTIMISATION PART 2 1 Dr Fabio Cuzzolin MSc in Computer Vision Oxford Brookes University Year 2013-14 OUTLINE OF WEEK 8 topics: quadratic optimisation, least squares,
More informationLecture # 11 The Power Method for Eigenvalues Part II. The power method find the largest (in magnitude) eigenvalue of. A R n n.
Lecture # 11 The Power Method for Eigenvalues Part II The power method find the largest (in magnitude) eigenvalue of It makes two assumptions. 1. A is diagonalizable. That is, A R n n. A = XΛX 1 for some
More informationMATH 304 Linear Algebra Lecture 33: Bases of eigenvectors. Diagonalization.
MATH 304 Linear Algebra Lecture 33: Bases of eigenvectors. Diagonalization. Eigenvalues and eigenvectors of an operator Definition. Let V be a vector space and L : V V be a linear operator. A number λ
More informationIterative methods for Linear System of Equations. Joint Advanced Student School (JASS-2009)
Iterative methods for Linear System of Equations Joint Advanced Student School (JASS-2009) Course #2: Numerical Simulation - from Models to Software Introduction In numerical simulation, Partial Differential
More informationPositive Definite Matrix
1/29 Chia-Ping Chen Professor Department of Computer Science and Engineering National Sun Yat-sen University Linear Algebra Positive Definite, Negative Definite, Indefinite 2/29 Pure Quadratic Function
More informationISM206 Lecture Optimization of Nonlinear Objective with Linear Constraints
ISM206 Lecture Optimization of Nonlinear Objective with Linear Constraints Instructor: Prof. Kevin Ross Scribe: Nitish John October 18, 2011 1 The Basic Goal The main idea is to transform a given constrained
More informationOptimization. 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 informationLecture Notes: Geometric Considerations in Unconstrained Optimization
Lecture Notes: Geometric Considerations in Unconstrained Optimization James T. Allison February 15, 2006 The primary objectives of this lecture on unconstrained optimization are to: Establish connections
More information22.4. Numerical Determination of Eigenvalues and Eigenvectors. Introduction. Prerequisites. Learning Outcomes
Numerical Determination of Eigenvalues and Eigenvectors 22.4 Introduction In Section 22. it was shown how to obtain eigenvalues and eigenvectors for low order matrices, 2 2 and. This involved firstly solving
More informationE5295/5B5749 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 informationLecture 10 - Eigenvalues problem
Lecture 10 - Eigenvalues problem Department of Computer Science University of Houston February 28, 2008 1 Lecture 10 - Eigenvalues problem Introduction Eigenvalue problems form an important class of problems
More informationIterative methods for Linear System
Iterative methods for Linear System JASS 2009 Student: Rishi Patil Advisor: Prof. Thomas Huckle Outline Basics: Matrices and their properties Eigenvalues, Condition Number Iterative Methods Direct and
More informationCheat Sheet for MATH461
Cheat Sheet for MATH46 Here is the stuff you really need to remember for the exams Linear systems Ax = b Problem: We consider a linear system of m equations for n unknowns x,,x n : For a given matrix A
More informationIterative Methods for Solving A x = b
Iterative Methods for Solving A x = b A good (free) online source for iterative methods for solving A x = b is given in the description of a set of iterative solvers called templates found at netlib: http
More informationHomework 2 Foundations of Computational Math 2 Spring 2019
Homework 2 Foundations of Computational Math 2 Spring 2019 Problem 2.1 (2.1.a) Suppose (v 1,λ 1 )and(v 2,λ 2 ) are eigenpairs for a matrix A C n n. Show that if λ 1 λ 2 then v 1 and v 2 are linearly independent.
More informationLinear Algebra Massoud Malek
CSUEB Linear Algebra Massoud Malek Inner Product and Normed Space In all that follows, the n n identity matrix is denoted by I n, the n n zero matrix by Z n, and the zero vector by θ n An inner product
More informationSection 7.3: SYMMETRIC MATRICES AND ORTHOGONAL DIAGONALIZATION
Section 7.3: SYMMETRIC MATRICES AND ORTHOGONAL DIAGONALIZATION When you are done with your homework you should be able to Recognize, and apply properties of, symmetric matrices Recognize, and apply properties
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 informationMATH 1553-C MIDTERM EXAMINATION 3
MATH 553-C MIDTERM EXAMINATION 3 Name GT Email @gatech.edu Please read all instructions carefully before beginning. Please leave your GT ID card on your desk until your TA scans your exam. Each problem
More information1. 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 informationLecture 2 - Unconstrained Optimization Definition[Global Minimum and Maximum]Let f : S R be defined on a set S R n. Then
Lecture 2 - Unconstrained Optimization Definition[Global Minimum and Maximum]Let f : S R be defined on a set S R n. Then 1. x S is a global minimum point of f over S if f (x) f (x ) for any x S. 2. x S
More informationMath 411 Preliminaries
Math 411 Preliminaries Provide a list of preliminary vocabulary and concepts Preliminary Basic Netwon s method, Taylor series expansion (for single and multiple variables), Eigenvalue, Eigenvector, Vector
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 informationmin f(x). (2.1) Objectives consisting of a smooth convex term plus a nonconvex regularization term;
Chapter 2 Gradient Methods The gradient method forms the foundation of all of the schemes studied in this book. We will provide several complementary perspectives on this algorithm that highlight the many
More informationLinear Algebra Practice Problems
Linear Algebra Practice Problems Math 24 Calculus III Summer 25, Session II. Determine whether the given set is a vector space. If not, give at least one axiom that is not satisfied. Unless otherwise stated,
More informationMultidisciplinary System Design Optimization (MSDO)
Multidisciplinary System Design Optimization (MSDO) Numerical Optimization II Lecture 8 Karen Willcox 1 Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Today s Topics Sequential
More informationECE580 Fall 2015 Solution to Midterm Exam 1 October 23, Please leave fractions as fractions, but simplify them, etc.
ECE580 Fall 2015 Solution to Midterm Exam 1 October 23, 2015 1 Name: Solution Score: /100 This exam is closed-book. You must show ALL of your work for full credit. Please read the questions carefully.
More informationHW3 - Due 02/06. Each answer must be mathematically justified. Don t forget your name. 1 2, A = 2 2
HW3 - Due 02/06 Each answer must be mathematically justified Don t forget your name Problem 1 Find a 2 2 matrix B such that B 3 = A, where A = 2 2 If A was diagonal, it would be easy: we would just take
More information8 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 informationElements of linear algebra
Elements of linear algebra Elements of linear algebra A vector space S is a set (numbers, vectors, functions) which has addition and scalar multiplication defined, so that the linear combination c 1 v
More informationTHE EIGENVALUE PROBLEM
THE EIGENVALUE PROBLEM Let A be an n n square matrix. If there is a number λ and a column vector v 0 for which Av = λv then we say λ is an eigenvalue of A and v is an associated eigenvector. Note that
More informationCS137 Introduction to Scientific Computing Winter Quarter 2004 Solutions to Homework #3
CS137 Introduction to Scientific Computing Winter Quarter 2004 Solutions to Homework #3 Felix Kwok February 27, 2004 Written Problems 1. (Heath E3.10) Let B be an n n matrix, and assume that B is both
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 informationComputational Methods CMSC/AMSC/MAPL 460. Eigenvalues and Eigenvectors. Ramani Duraiswami, Dept. of Computer Science
Computational Methods CMSC/AMSC/MAPL 460 Eigenvalues and Eigenvectors Ramani Duraiswami, Dept. of Computer Science Eigen Values of a Matrix Recap: A N N matrix A has an eigenvector x (non-zero) with corresponding
More informationGradient Descent. Dr. Xiaowei Huang
Gradient Descent Dr. Xiaowei Huang https://cgi.csc.liv.ac.uk/~xiaowei/ Up to now, Three machine learning algorithms: decision tree learning k-nn linear regression only optimization objectives are discussed,
More informationUnconstrained 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 informationReview for Exam 2 Ben Wang and Mark Styczynski
Review for Exam Ben Wang and Mark Styczynski This is a rough approximation of what we went over in the review session. This is actually more detailed in portions than what we went over. Also, please note
More informationCPSC 540: Machine Learning
CPSC 540: Machine Learning Gradient Descent, Newton-like Methods Mark Schmidt University of British Columbia Winter 2017 Admin Auditting/registration forms: Submit them in class/help-session/tutorial this
More informationLINEAR SYSTEMS (11) Intensive Computation
LINEAR SYSTEMS () Intensive Computation 27-8 prof. Annalisa Massini Viviana Arrigoni EXACT METHODS:. GAUSSIAN ELIMINATION. 2. CHOLESKY DECOMPOSITION. ITERATIVE METHODS:. JACOBI. 2. GAUSS-SEIDEL 2 CHOLESKY
More informationLinear Algebra II Lecture 13
Linear Algebra II Lecture 13 Xi Chen 1 1 University of Alberta November 14, 2014 Outline 1 2 If v is an eigenvector of T : V V corresponding to λ, then v is an eigenvector of T m corresponding to λ m since
More informationLecture 15 Review of Matrix Theory III. Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore
Lecture 15 Review of Matrix Theory III Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore Matrix An m n matrix is a rectangular or square array of
More informationThere are six more problems on the next two pages
Math 435 bg & bu: Topics in linear algebra Summer 25 Final exam Wed., 8/3/5. Justify all your work to receive full credit. Name:. Let A 3 2 5 Find a permutation matrix P, a lower triangular matrix L with
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 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 informationAlgebra II. Paulius Drungilas and Jonas Jankauskas
Algebra II Paulius Drungilas and Jonas Jankauskas Contents 1. Quadratic forms 3 What is quadratic form? 3 Change of variables. 3 Equivalence of quadratic forms. 4 Canonical form. 4 Normal form. 7 Positive
More informationFunctions of Several Variables
Functions of Several Variables The Unconstrained Minimization Problem where In n dimensions the unconstrained problem is stated as f() x variables. minimize f()x x, is a scalar objective function of vector
More informationHere each term has degree 2 (the sum of exponents is 2 for all summands). A quadratic form of three variables looks as
Reading [SB], Ch. 16.1-16.3, p. 375-393 1 Quadratic Forms A quadratic function f : R R has the form f(x) = a x. Generalization of this notion to two variables is the quadratic form Q(x 1, x ) = a 11 x
More informationIntroduction to Unconstrained Optimization: Part 2
Introduction to Unconstrained Optimization: Part 2 James Allison ME 555 January 29, 2007 Overview Recap Recap selected concepts from last time (with examples) Use of quadratic functions Tests for positive
More informationlecture 2 and 3: algorithms for linear algebra
lecture 2 and 3: algorithms for linear algebra STAT 545: Introduction to computational statistics Vinayak Rao Department of Statistics, Purdue University August 27, 2018 Solving a system of linear equations
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 informationComputational Intelligence Winter Term 2017/18
Computational Intelligence Winter Term 2017/18 Prof. Dr. Günter Rudolph Lehrstuhl für Algorithm Engineering (LS 11) Fakultät für Informatik TU Dortmund mutation: Y = X + Z Z ~ N(0, C) multinormal distribution
More information