ECE580 Exam 1 October 4, Please do not write on the back of the exam pages. Extra paper is available from the instructor.
|
|
- Maud Butler
- 5 years ago
- Views:
Transcription
1 ECE580 Exam 1 October 4, 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. I do not want the decimal equivalents. Cell phones and other electronic communication devices must be turned off and stowed under your desk. Please do not write on the back of the exam pages. Extra paper is available from the instructor
2 ECE580 Exam 1 October 4, (8 points) Consider the following matrices. A 1 = ], A 2 = ], A 3 = ], A 4 = ]. (a) State the definition of positive definiteness of a quadratic form x T Ax. The quadratic form x T Ax is positive definite if for all x, x T Ax 0 and x T Ax = 0 iff x = 0. (b) Which (if any) of the matrices listed above are negative definite (and why)? A 4 is negative definite because it is symmetric and all of its eigenvalues are negative. (c) Which (if any) of the matrices are positive semidefinite (and why)? A 2 and A 3 are positive semidefinite because they are symmetric and all of its eigenvalues are nonnegative. (d) Which matrices (if any) are indefinite (and why)? A 1 is indefinite because it is symmetric and has both positive and negative eigenvalues.
3 ECE580 Exam 1 October 4, (12 points) Determine whether f(x) has one or more local minimizers x, and, if it does, determine the x. f(x 1, x 2 ) = x x 1 x 2 + 3x x 2 We are given no constraints on feasible values so Ω = R 2 and all points of Ω are interior points. Thus, in order to satisfy the first order necessary condition (FONC), we need f(x ) = 0. f(x ) = 12(x 1) 2 + 2x x 1 + 6x 2 requires (second row) x 2 = x 1/3, and (first row) Substituting for x 2, we have 12(x 1) 2 + 2x = 0. 12(x 1) 2 2x 1/3 + 1 = 0. The roots of this quadratic equation are x 1 = 2/3 ± 4/9 48, 24 which are complex, so f(x) has no real minimizers.
4 ECE580 Exam 1 October 4, (8 points) Suppose we wish to use a fixed step gradient method to find the minimum of the function f(x) = x T Ax where (a) Give the formula for x (k+1). A = x (k+1) = x (k) α f(x (k) ). (b) What is the upper bound on the step size α to guarantee convergence? The upper bound is given by α < 2 λ max (A) = 1.
5 ECE580 Exam 1 October 4, (12 points) Given the function f(x 1, x 2 ) = x x 2 1 x (a) Find any points that satisfy the first order necessary condition. There are no restrictions on the set of feasible points Ω so Ω = R 2. Thus all points are interior points of Ω and the appropriate FONC requires that f(x ) = 0. The gradient, evaluated at x is f(x ) = 2x 1 3(x 2) 2 1 so we need x 1 = 0 and x 2 = ± 1/3., (b) Find any points that satisfy the second order sufficient condition. The second order necessary condition for interior points requires that in addition to satisfying the FONC, that the Hessian F (x ) be positive semidefinite. Here F (x ) = x 2 so we need x 2 0 which is satisfied only by the point x = (0, 1/3). (c) Identify any local minimizers. The second order sufficient condition requires the Hessian evaluated at x to be positive definite, which it is not, so the function has no minimizers.
6 ECE580 Exam 1 October 4, (12 points) If a constraint set Ω is bounded below by y = x 3 and above by y = x 2 + 2, determine whether each of the following points is an interior point, a boundary point, or is not in Ω. (Justify your answers.) (a) (1, 3) 1 3 = 2 < 3 and = 1 < 3. This means that the point (1, 3) is above both curves, thus not in Ω. (b) (0, 3) 0 3 = 3 = 3 and = 2 > 3. This means that the point (0, 2) is on the line y = x 3 and below y = x 2 +2, so the point is a boundary point of Ω. (c) ( 1, 3) 1 3 = 4 < 3 and = 1 > 3. This means that the point ( 1, 3) is above the line and below y = x 2 + 2, so the point is an interior point of Ω.
7 ECE580 Exam 1 October 4, (12 points) Given a point x = 1 0 ] T and the function f(x 1, x 2 ) = x x 2 1 x (a) Find the gradient at the given point. f 1 0 = 2(1) 3(0) 2 1 = 2 1 (b) Find the rate of increase of the function at the given point. The rate of increase of the function at the given point is f(x) T d d = 2d 1 d 2 d d 2 2 (c) Find the equation of the plane tangent to the surface at the given point. In the case that f(x) 0, equation of the tangent plane at a point x 0 = 1 0 ] T is f(x 0 ) T (x x 0 ) = 0. Thus the equation for the tangent plane at x ] x 1 1 x 2 = 2(x 1 1) x 2 = 0. is
8 ECE580 Exam 1 October 4, (8 points) Given initial point x (0) = 2 and the function f(x) = x 3 x 2 x + 1, find x (1) using Newton s method for finding a minimizer of f. In order to apply Newton s method, we will need the first and second derivatives of f evaluated at x 0 : f (x 0 ) = 3x 2 0 2x 0 1 = = 7, f (x 0 ) = 6x 0 2 = 12 2 = 10. Now applying Newton s method we obtain x (1) = x (0) f (x (0) ) f (x (0) ) = 2 7/10 = 13/10.
9 ECE580 Exam 1 October 4, (8 points) Given initial point x (0) = 2 and the equation f(x) = x 3 x 2 x + 1 = 0, find x (1) using Newton s method of tangents for finding a root of f. The formula for the update in the method of tangents is x (k+1) = x (k) f(x(k) ) f (x (k) ). The derivative is f (x) = 3x 2 2x 1. Thus x (1) = x (0) (2) 2 2(2) 1 = 2 3/7 = 11/7.
10 ECE580 Exam 1 October 4, (8 points) Given initial points x (0) = 2 and x ( 1) = 1 and the equation f(x) = x 3 x 2 x + 1, find x (1) using the secant method for finding a minimizer of f. The secant method update algorithm is The derivative is Thus x (1) x (k+1) = x (k) x (k) x (k 1) f (x (k) ) f (x (k 1) ) f (x (k) ). f (x) = 3x 2 2x = 2 (3(2) 2 2(2) 1) (3(1) 2 2(1) 1) (3(2)2 2(2) 1) = = 2 1 = 1.
11 ECE580 Exam 1 October 4, (12 points) Given an initial point x (0) = 1 1 f(x 1, x 2 ) = x x 2 1 x 2 + 1, ] T and the equation find x (1) using the steepest descent method for finding a minimizer of f. Gradient descent methods have the form x (k+1) = x (k) α k f(x (k) ). The method of steepest descent uses α k = arg min α 0 f ( x (k) α f(x (k) ) ). The gradient evaluated at x (0) f x=x (0) = is 2x 1 3x x=x (0) = 2 2. Evaluating the right-hand side of the expression for α k yields α k = arg min α 0 8α 2 8α 2 8α + 2. Taking the derivative and equating it to zero we find that the critical points occur at 1± 2 so the expression has no minimum. Thus we need to try a different x (0) or a different method.
ECE580 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 informationECE580 Exam 2 November 01, Name: Score: / (20 points) You are given a two data sets
ECE580 Exam 2 November 01, 2011 1 Name: 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. I do
More informationECE569 Exam 1 October 28, Name: Score: /100. Please leave fractions as fractions, but simplify them, etc.
ECE569 Exam 1 October 28, 2015 1 Name: Score: /100 This exam is closed-book. You must show ALL of your work for full credit. Please read the questions carefully. Please check your answers carefully. Calculators
More informationECE 680 Modern Automatic Control. Gradient and Newton s Methods A Review
ECE 680Modern Automatic Control p. 1/1 ECE 680 Modern Automatic Control Gradient and Newton s Methods A Review Stan Żak October 25, 2011 ECE 680Modern Automatic Control p. 2/1 Review of the Gradient Properties
More informationECE580 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 informationECE580 Final Exam December 14, Please leave fractions as fractions, I do not want the decimal equivalents.
ECE580 Final Exam December 14, 2012 1 Name: Score: /100 This exam is closed-book. Calculators may NOT be used. Please leave fractions as fractions, I do not want the decimal equivalents. Cell phones and
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 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 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 informationMath 51 Second Exam May 18, 2017
Math 51 Second Exam May 18, 2017 Name: SUNet ID: ID #: Complete the following problems. In order to receive full credit, please show all of your work and justify your answers. You do not need to simplify
More informationMA 113 Calculus I Fall 2017 Exam 1 Tuesday, 19 September Multiple Choice Answers. Question
MA 113 Calculus I Fall 2017 Exam 1 Tuesday, 19 September 2017 Name: Section: Last 4 digits of student ID #: This exam has 12 multiple choice questions (five points each) and 4 free response questions (ten
More informationMTH 132 Solutions to Exam 2 Nov. 23rd 2015
Name: Section: Instructor: READ THE FOLLOWING INSTRUCTIONS. Do not open your exam until told to do so. No calculators, cell phones or any other electronic devices can be used on this exam. Clear your desk
More informationAssignment #10: Diagonalization of Symmetric Matrices, Quadratic Forms, Optimization, Singular Value Decomposition. Name:
Assignment #10: Diagonalization of Symmetric Matrices, Quadratic Forms, Optimization, Singular Value Decomposition Due date: Friday, May 4, 2018 (1:35pm) Name: Section Number Assignment #10: Diagonalization
More informationECE580 Solution to Problem Set 3: Applications of the FONC, SONC, and SOSC
ECE580 Spring 2016 Solution to Problem Set 3 February 8, 2016 1 ECE580 Solution to Problem Set 3: Applications of the FONC, SONC, and SOSC These problems are from the textbook by Chong and Zak, 4th edition,
More informationChapter 8 Gradient Methods
Chapter 8 Gradient Methods An Introduction to Optimization Spring, 2014 Wei-Ta Chu 1 Introduction Recall that a level set of a function is the set of points satisfying for some constant. Thus, a point
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 informationMA 113 Calculus I Fall 2015 Exam 3 Tuesday, 17 November Multiple Choice Answers. Question
MA 11 Calculus I Fall 2015 Exam Tuesday, 17 November 2015 Name: Section: Last 4 digits of student ID #: This exam has ten multiple choice questions (five points each) and five free response questions (ten
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 informationPerformance Surfaces and Optimum Points
CSC 302 1.5 Neural Networks Performance Surfaces and Optimum Points 1 Entrance Performance learning is another important class of learning law. Network parameters are adjusted to optimize the performance
More informationIntegration - Past Edexcel Exam Questions
Integration - Past Edexcel Exam Questions 1. (a) Given that y = 5x 2 + 7x + 3, find i. - ii. - (b) ( 1 + 3 ) x 1 x dx. [4] 2. Question 2b - January 2005 2. The gradient of the curve C is given by The point
More informationMath 164-1: Optimization Instructor: Alpár R. Mészáros
Math 164-1: Optimization Instructor: Alpár R. Mészáros Final Exam, June 9, 2016 Name (use a pen): Student ID (use a pen): Signature (use a pen): Rules: Duration of the exam: 180 minutes. By writing your
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 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 informationMA 113 Calculus I Fall 2015 Exam 1 Tuesday, 22 September Multiple Choice Answers. Question
MA 113 Calculus I Fall 2015 Exam 1 Tuesday, 22 September 2015 Name: Section: Last 4 digits of student ID #: This exam has ten multiple choice questions (five points each) and five free response questions
More informationMath 241 Final Exam, Spring 2013
Math 241 Final Exam, Spring 2013 Name: Section number: Instructor: Read all of the following information before starting the exam. Question Points Score 1 5 2 5 3 12 4 10 5 17 6 15 7 6 8 12 9 12 10 14
More informationMath 164-1: Optimization Instructor: Alpár R. Mészáros
Math 164-1: Optimization Instructor: Alpár R. Mészáros First Midterm, April 20, 2016 Name (use a pen): Student ID (use a pen): Signature (use a pen): Rules: Duration of the exam: 50 minutes. By writing
More informationConstrained 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 informationMath 164 (Lec 1): Optimization Instructor: Alpár R. Mészáros
Math 164 (Lec 1): Optimization Instructor: Alpár R. Mészáros Midterm, October 6, 016 Name (use a pen): Student ID (use a pen): Signature (use a pen): Rules: Duration of the exam: 50 minutes. By writing
More information6.252 NONLINEAR PROGRAMMING LECTURE 10 ALTERNATIVES TO GRADIENT PROJECTION LECTURE OUTLINE. Three Alternatives/Remedies for Gradient Projection
6.252 NONLINEAR PROGRAMMING LECTURE 10 ALTERNATIVES TO GRADIENT PROJECTION LECTURE OUTLINE Three Alternatives/Remedies for Gradient Projection Two-Metric Projection Methods Manifold Suboptimization Methods
More informationMTH 132 Solutions to Exam 2 Apr. 13th 2015
MTH 13 Solutions to Exam Apr. 13th 015 Name: Section: Instructor: READ THE FOLLOWING INSTRUCTIONS. Do not open your exam until told to do so. No calculators, cell phones or any other electronic devices
More informationMTH132 Exam 1 Covers: Page Total. Max
Name: PID: A Section #: Instructor: Page 3 4 5 6 7 8 Total Score Max 4 4 4 4 1 150 Instructions 1. You will be given exactly 90 minutes for this exam.. No calculators, phones, or any electronic devices.
More informationLecture 3: Basics of set-constrained and unconstrained optimization
Lecture 3: Basics of set-constrained and unconstrained optimization (Chap 6 from textbook) Xiaoqun Zhang Shanghai Jiao Tong University Last updated: October 9, 2018 Optimization basics Outline Optimization
More informationMA 113 Calculus I Fall 2012 Exam 3 13 November Multiple Choice Answers. Question
MA 113 Calculus I Fall 2012 Exam 3 13 November 2012 Name: Section: Last 4 digits of student ID #: This exam has ten multiple choice questions (five points each) and five free response questions (ten points
More informationMATH 251 Final Examination August 10, 2011 FORM A. Name: Student Number: Section:
MATH 251 Final Examination August 10, 2011 FORM A Name: Student Number: Section: This exam has 10 questions for a total of 150 points. In order to obtain full credit for partial credit problems, all work
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 informationMA 125 CALCULUS I FALL 2006 December 08, 2006 FINAL EXAM. Name (Print last name first):... Instructor:... Section:... PART I
CALCULUS I, FINAL EXAM 1 MA 125 CALCULUS I FALL 2006 December 08, 2006 FINAL EXAM Name (Print last name first):............................................. Student ID Number:...........................
More informationMath 125 Final Examination Autumn 2015
Math 125 Final Examination Autumn 2015 Your Name Your Signature Student ID # Quiz Section Professor s Name TA s Name Turn off all cell phones, pagers, radios, mp3 players, and other similar devices. This
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 informationORF 363/COS 323 Final Exam, Fall 2018
Name: Princeton University ORF 363/COS 323 Final Exam, Fall 2018 January 16, 2018 Instructor: A.A. Ahmadi AIs: Dibek, Duan, Gong, Khadir, Mirabelli, Pumir, Tang, Yu, Zhang 1. Please write out and sign
More informationECE302 Exam 2 Version A April 21, You must show ALL of your work for full credit. Please leave fractions as fractions, but simplify them, etc.
ECE32 Exam 2 Version A April 21, 214 1 Name: Solution Score: /1 This exam is closed-book. You must show ALL of your work for full credit. Please read the questions carefully. Please check your answers
More informationWeek 4: Calculus and Optimization (Jehle and Reny, Chapter A2)
Week 4: Calculus and Optimization (Jehle and Reny, Chapter A2) Tsun-Feng Chiang *School of Economics, Henan University, Kaifeng, China September 27, 2015 Microeconomic Theory Week 4: Calculus and Optimization
More informationMath 273a: Optimization Basic concepts
Math 273a: Optimization Basic concepts Instructor: Wotao Yin Department of Mathematics, UCLA Spring 2015 slides based on Chong-Zak, 4th Ed. Goals of this lecture The general form of optimization: minimize
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 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 informationMath (P)refresher Lecture 8: Unconstrained Optimization
Math (P)refresher Lecture 8: Unconstrained Optimization September 2006 Today s Topics : Quadratic Forms Definiteness of Quadratic Forms Maxima and Minima in R n First Order Conditions Second Order Conditions
More information2.098/6.255/ 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 informationWithout fully opening the exam, check that you have pages 1 through 11.
MTH 33 Solutions to Final Exam May, 8 Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through. Show
More informationProblem Point Value Points
Math 70 TUFTS UNIVERSITY October 12, 2015 Linear Algebra Department of Mathematics Sections 1 and 2 Exam I Instructions: No notes or books are allowed. All calculators, cell phones, or other electronic
More informationMore First-Order Optimization Algorithms
More First-Order Optimization Algorithms Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A. http://www.stanford.edu/ yyye Chapters 3, 8, 3 The SDM
More informationAlgorithms 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 informationMath 31A Differential and Integral Calculus. Final
Math 31A Differential and Integral Calculus Final Instructions: You have 3 hours to complete this exam. There are eight questions, worth a total of??? points. This test is closed book and closed notes.
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 informationCS 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 informationWritten Examination
Division of Scientific Computing Department of Information Technology Uppsala University Optimization Written Examination 202-2-20 Time: 4:00-9:00 Allowed Tools: Pocket Calculator, one A4 paper with notes
More informationFinal Exam Practice Problems Part II: Sequences and Series Math 1C: Calculus III
Name : c Jeffrey A. Anderson Class Number:. Final Exam Practice Problems Part II: Sequences and Series Math C: Calculus III What are the rules of this exam? PLEASE DO NOT TURN THIS PAGE UNTIL TOLD TO DO
More informationOptimization. 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 informationMA 125 CALCULUS I SPRING 2007 April 27, 2007 FINAL EXAM. Name (Print last name first):... Student ID Number (last four digits):...
CALCULUS I, FINAL EXAM 1 MA 125 CALCULUS I SPRING 2007 April 27, 2007 FINAL EXAM Name (Print last name first):............................................. Student ID Number (last four digits):........................
More informationDO NOT OPEN THIS QUESTION BOOKLET UNTIL YOU ARE TOLD TO DO SO
QUESTION BOOKLET EECS 227A Fall 2009 Midterm Tuesday, Ocotober 20, 11:10-12:30pm DO NOT OPEN THIS QUESTION BOOKLET UNTIL YOU ARE TOLD TO DO SO You have 80 minutes to complete the midterm. The midterm consists
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 informationMath 251 December 14, 2005 Final Exam. 1 18pt 2 16pt 3 12pt 4 14pt 5 12pt 6 14pt 7 14pt 8 16pt 9 20pt 10 14pt Total 150pt
Math 251 December 14, 2005 Final Exam Name Section There are 10 questions on this exam. Many of them have multiple parts. The point value of each question is indicated either at the beginning of each question
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 informationMath 2400, Midterm 2
Math 24, Midterm 2 October 22, 218 PRINT your name: PRINT instructor s name: Mark your section/instructor: Section 1 Kevin Berg 8: 8:5 Section 2 Philip Kopel 8: 8:5 Section 3 Daniel Martin 8: 8:5 Section
More informationMA 113 Calculus I Spring 2013 Exam 3 09 April Multiple Choice Answers VERSION 1. Question
MA 113 Calculus I Spring 013 Exam 3 09 April 013 Multiple Choice Answers VERSION 1 Question Name: Section: Last 4digits ofstudent ID #: This exam has ten multiple choice questions (five points each) and
More informationMidterm exam CS 189/289, Fall 2015
Midterm exam CS 189/289, Fall 2015 You have 80 minutes for the exam. Total 100 points: 1. True/False: 36 points (18 questions, 2 points each). 2. Multiple-choice questions: 24 points (8 questions, 3 points
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 informationLecture 10: October 27, 2016
Mathematical Toolkit Autumn 206 Lecturer: Madhur Tulsiani Lecture 0: October 27, 206 The conjugate gradient method In the last lecture we saw the steepest descent or gradient descent method for finding
More informationPage Points Score Total: 100
Math 1130 Spring 2019 Sample Midterm 2a 2/28/19 Name (Print): Username.#: Lecturer: Rec. Instructor: Rec. Time: This exam contains 10 pages (including this cover page) and 9 problems. Check to see if any
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 information4 damped (modified) Newton methods
4 damped (modified) Newton methods 4.1 damped Newton method Exercise 4.1 Determine with the damped Newton method the unique real zero x of the real valued function of one variable f(x) = x 3 +x 2 using
More informationMA 113 Calculus I Fall 2013 Exam 3 Tuesday, 19 November Multiple Choice Answers. Question
MA 113 Calculus I Fall 2013 Exam 3 Tuesday, 19 November 2013 Name: Section: Last 4 digits of student ID #: This exam has ten multiple choice questions (five points each) and five free response questions
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 informationMath 131 Exam 1 October 4, :00-9:00 p.m.
Name (Last, First) My Solutions ID # Signature Lecturer Section (01, 02, 03, etc.) university of massachusetts amherst department of mathematics and statistics Math 131 Exam 1 October 4, 2017 7:00-9:00
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 informationComprehensive Exam in Real Analysis Fall 2006 Thursday September 14, :00-11:30am INSTRUCTIONS
Exam Packet # Comprehensive Exam in Real Analysis Fall 2006 Thursday September 14, 2006 9:00-11:30am Name (please print): Student ID: INSTRUCTIONS (1) The examination is divided into three sections to
More informationMTH 133 Final Exam Dec 8, 2014
Name: PID: Section: Recitation Instructor: DO NOT WRITE BELOW THIS LINE. GO ON TO THE NEXT PAGE. Page Problem Score Max Score 1 5 3 2 5 3a 5 3b 5 4 4 5 5a 5 5b 5 6 5 5 7a 5 7b 5 6 8 18 7 8 9 10 11 12 9a
More informationUniversity of Connecticut Department of Mathematics
University of Connecticut Department of Mathematics Math 1131 Sample Exam 2 Fall 2015 Name: Instructor Name: Section: TA Name: Discussion Section: This sample exam is just a guide to prepare for the actual
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 informationUniversity of Connecticut Department of Mathematics
University of Connecticut Department of Mathematics Math 1131 Sample Exam 1 Fall 2013 Name: This sample exam is just a guide to prepare for the actual exam. Questions on the actual exam may or may not
More informationMath 111 Exam 1. Instructions
Math 111 Exam 1 Instructions Please read all of these instructions thoroughly before beginning the exam. This exam has two parts. The first part must be done without the use of a calculator. When you are
More informationTwo hours. To be provided by Examinations Office: Mathematical Formula Tables. THE UNIVERSITY OF MANCHESTER. xx xxxx 2017 xx:xx xx.
Two hours To be provided by Examinations Office: Mathematical Formula Tables. THE UNIVERSITY OF MANCHESTER CONVEX OPTIMIZATION - SOLUTIONS xx xxxx 27 xx:xx xx.xx Answer THREE of the FOUR questions. If
More informationMark your answers ON THE EXAM ITSELF. If you are not sure of your answer you may wish to provide a brief explanation.
CS 189 Spring 2015 Introduction to Machine Learning Midterm You have 80 minutes for the exam. The exam is closed book, closed notes except your one-page crib sheet. No calculators or electronic items.
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 informationMath 104 Section 2 Midterm 2 November 1, 2013
Math 104 Section 2 Midterm 2 November 1, 2013 Name: Complete the following problems. In order to receive full credit, please provide rigorous proofs and show all of your work and justify your answers.
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 information5. Hand in the entire exam booklet and your computer score sheet.
WINTER 2016 MATH*2130 Final Exam Last name: (PRINT) First name: Student #: Instructor: M. R. Garvie 19 April, 2016 INSTRUCTIONS: 1. This is a closed book examination, but a calculator is allowed. The test
More informationA.P. Calculus BC Test Three Section Two Free-Response No Calculators Time 45 minutes Number of Questions 3
A.P. Calculus BC Test Three Section Two Free-Response No Calculators Time 45 minutes Number of Questions 3 Each of the three questions is worth 9 points. The maximum possible points earned on this section
More informationPage Problem Score Max Score a 8 12b a b 10 14c 6 6
Fall 2014 MTH 234 FINAL EXAM December 8, 2014 Name: PID: Section: Instructor: DO NOT WRITE BELOW THIS LINE. Go to the next page. Page Problem Score Max Score 1 5 2 5 1 3 5 4 5 5 5 6 5 7 5 2 8 5 9 5 10
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 informationMATH 120 THIRD UNIT TEST
MATH 0 THIRD UNIT TEST Friday, April 4, 009. NAME: Circle the recitation Tuesday, Thursday Tuesday, Thursday section you attend MORNING AFTERNOON A B Instructions:. Do not separate the pages of the exam.
More informationExamination 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 informationQ1 Q2 Q3 Q4 Tot Letr Xtra
Mathematics 54.1 Final Exam, 12 May 2011 180 minutes, 90 points NAME: ID: GSI: INSTRUCTIONS: You must justify your answers, except when told otherwise. All the work for a question should be on the respective
More informationCaculus 221. Possible questions for Exam II. March 19, 2002
Caculus 221 Possible questions for Exam II March 19, 2002 These notes cover the recent material in a style more like the lecture than the book. The proofs in the book are in section 1-11. At the end there
More informationLECTURE 22: SWARM INTELLIGENCE 3 / CLASSICAL OPTIMIZATION
15-382 COLLECTIVE INTELLIGENCE - S19 LECTURE 22: SWARM INTELLIGENCE 3 / CLASSICAL OPTIMIZATION TEACHER: GIANNI A. DI CARO WHAT IF WE HAVE ONE SINGLE AGENT PSO leverages the presence of a swarm: the outcome
More information5.) For each of the given sets of vectors, determine whether or not the set spans R 3. Give reasons for your answers.
Linear Algebra - Test File - Spring Test # For problems - consider the following system of equations. x + y - z = x + y + 4z = x + y + 6z =.) Solve the system without using your calculator..) Find the
More informationScientific Computing II
Technische Universität München ST 008 Institut für Informatik Dr. Miriam Mehl Scientific Computing II Final Exam, July, 008 Iterative Solvers (3 pts + 4 extra pts, 60 min) a) Steepest Descent and Conjugate
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 informationSample questions for Fundamentals of Machine Learning 2018
Sample questions for Fundamentals of Machine Learning 2018 Teacher: Mohammad Emtiyaz Khan A few important informations: In the final exam, no electronic devices are allowed except a calculator. Make sure
More informationPHYS 314 FIRST HOUR EXAM
PHYS 314 FIRST HOUR EXAM Spring 2017 This is a closed book, closed note exam. You will not need nor be allowed to use calculators or other electronic devices on this test. At this time, store all electronic
More information1. Background: The SVD and the best basis (questions selected from Ch. 6- Can you fill in the exercises?)
Math 35 Exam Review SOLUTIONS Overview In this third of the course we focused on linear learning algorithms to model data. summarize: To. Background: The SVD and the best basis (questions selected from
More informationSTUDENT NAME: STUDENT SIGNATURE: STUDENT ID NUMBER: SECTION NUMBER RECITATION INSTRUCTOR:
MA262 FINAL EXAM SPRING 2016 MAY 2, 2016 TEST NUMBER 01 INSTRUCTIONS: 1. Do not open the exam booklet until you are instructed to do so. 2. Before you open the booklet fill in the information below and
More information