March 8, 2010 MATH 408 FINAL EXAM SAMPLE

Size: px
Start display at page:

Download "March 8, 2010 MATH 408 FINAL EXAM SAMPLE"

Transcription

1 March 8, 200 MATH 408 FINAL EXAM SAMPLE EXAM OUTLINE The final exam for this course takes place in the regular course classroom (MEB 238) on Monday, March 2, 8:30-0:20 am. You may bring two-sided 8 page of notes to the final exam. This exam consists of 5 question areas each worth 60 points for a total of 300 points. The exam will be graded on a curve as will your total point score for this course. The content of each question is as follows. Question : In this question you will concern first- and second-order optimality conditions for unconstrained optimization problems as well as such related concepts as coercivity and convexity. Question 2: The question concerns some property or derivation for quadratic functions q : IR n IR, e.g. q(x) = α + c T x + 2 xt Qx, q(x) = α + g T (x x) + 2 (x x)t H(x x), or q(y) = f(x) + f(x) T (y x) + 2 (y x)t 2 f(x)(y x), where in the last possibility f : IR n IR is a twice continously differentiable (possibly non-quadratic) function. In this final case the expression given for q(y) is called the second-order Taylor expansion of f about x. If it so happens that f is quadratic, then q(y) = f(y) for all y regardless of the choice of x. In addition, this question may also concern properties of symmetric matrices such as their eigenvalue decomposition, Cholesky factorization, and the relationships between the quadratic form detered by a symmetric matrix and its eigenvalues. It may also concern the optimization of a quadratic function, e.g. convexity, coercivity, critical points, optimal solution sets, optimal value, uniqueness of solutions,... etc. In that last instance above, how is the associated optimization problem related to Newton s method. Question 3: This question concerns some property or derivation for linear least squares functions, e.g. l(x) = 2 Ax b 2 2 l(x) = 2 A(x x) + c 2 2, l Q (x) = 2 Ax b 2 Q = 2 (Ax b)t Q(Ax b), where Q is positive definite, or l(y) = 2 F (x) + F (x)(y x) 2 2, where F : IRn IR m is smooth. In addition, this question may also concerning the optimization of such functions, e.g. convexity, coercivity, critical points, optimal solution sets, optimal value, uniqueness of solutions,... etc. What is the relationship between quadratic functions and linear least squares functions? For example, it is easily shown that every linear least squares function can be written as a quadratic function, but under what conditions can a quadratic function be written as a linear least squares function? In the last instance above, how is the associated solution related to the Gauss-Newton method, and what is the Gauss-Newton method?

2 Question 4: This question concerns Newton, Newton-like, and Gauss-Newton methods for equation solving and optimization. In particular, how are these algorithms designed. What are appropriate line search methods? What are the details of these line search methods. In the context of Newton-Like methods, what are the motivations behind the derivations and properties of associated matrix secant methods (e.g. the Broyden and BFGS updates). I will NOT require that you memorize the various matrix secant formulas, but I may give them to you and base a question on them. Also, you need NOT memorize the Sherman-Morrison-Woodbury formula. If it is needed for any reason, I will give it to you. Question 5: This question will focus on the first- and second-order optimality conditions for constrained optimization problems. Of particular importance will be Lagrangian duality and the ability to derive and apply the KKT conditions. Particular cases include linear and quadratic programg. But you should also be able to compute KKT points for concrete instances of low dimensional optimization problems. SAMPLE QUESTIONS See the sample questions for the midterm exam, but also consider the following questions. Obviously, a final exam would not contain questions of this depth and scope for each question. That is, the final will contain some easy and some hard questions. The questions listed below are all on the hard side with some very hard.. (a) Is the function f(x, x 2 ) = e (x +x 2 ) 2 coercive? Is it convex? Justify your answers. (b) For what values of the parameters α, β IR is the function f(x, x 2 ) = x 2 + 2αx x 2 + βx 2 2 a convex function. For what values is it coercive? Justify your answers. (c) Compute and classify all critical points of the function f(x, x 2, x 3 ) = 2(ax ) 2 + (x x 2 2) 2 + (ax 2 + x 3 ) 2, where a > 0 is a given positive real number. 2. Consider the function f(x) = 2 xt Qx + c T x, where Q IR n n is symmetric and c IR m. (a) Under what condition on the matrix Q IR n n is f convex? Justify your answer. (b) If Q is symmetric and positive definite, show that there is a nonsingular matrix L such that Q = LL T. (c) With Q and L as defined in the part (b), show that f(x) = 2 LT x + L c ct Q c. (d) If f is convex, under what conditions is x IR n f(x) =? (e) Assume that Q is symmetric and positive definite and let S be a subspace of IR n. Show that x solves the problem x S f(x) if and only if f( x) S.

3 3. Let A IR m n and b IR m and consider the function (a) Show that Ran(A T A) = Ran(A T ). f(x) = 2 Ax b 2 2. (b) Show that for every b IR m there exists a solution to the equation A T Ax = A T b. (c) Show that there always exists a solution to the problem LS and describe the set of solutions to this problem. x IR n 2 Ax b 2 2, (d) Under what conditions on A is the solution to the problem LS unique. (e) Using the condition from part (d) derive a formula for the unique solution to LS. (f) If A T A is invertible, show that the matrix P = A(A T A) A T is the orthogonal projection onto Ran(A), that is, show that i. P 2 = P ii. P T = P iii. Ran(P ) = Ran(A). 4. (a) Matrix secant questions. i. Let u, v IR n with u 0 v. Show that the matrix uv T IR n n is rank. ii. Let B IR n n and s, y IR n. For what values of u, v IR n is it true that the matrix B + = B + uv T satisfies B + s = y? Justify your answer. iii. Let H IR n n be symmetric and s, y IR n. Under what conditions do there exist vectors u, v IR n such that the matrix H + = H + uv T is also symmetric and satisfies H + s = y? Justify your answer. iv. Let M IR n n be symmetric and positive definite and let s, y IR n be such that s T y > 0. The BFGS applied to M is M + = M + yyt s T y MssT M s T Ms. Show that M + can be rewritten in the form M + = M + UD U T, where U IR n 2 and D IR 2 2 is an invertible diagonal matrix, by deriving formulas for both U and D. (b) (This problem is over the top. But parts might be trimmed down for a suitable final exam question.) Let F : IR n IR m be continuously differentiable, and let be any norm on IR m. In this problem we consider the function f(x) = F (x) and properties of the the associated Gauss- Newton direction for imizing f. Recall that x IR n is a first-order stationary point for f if 0 f ( x; d) for all d IR n. i. Given x, d IR n and t > 0 show that F (x + td) F (x) + tf (x)d F (x + td) (F (x) + tf (x)d). ii. Why is it true that F (x + td) (F (x) + tf (x)d) lim = 0? t 0 t Hint: What is the definition of F (x)?

4 iii. Use parts i and ii to show that f (x; d) = lim t 0 F (x) + tf (x)d F (x) t iv. Use part iii and the convexity of the norm to show that f (x; d) F (x) + F (x)d F (x). Hint: F (x) + tf (x)d = ( t)f (x) + t(f (x) + F (x)d) v. Use parts iii and iv to show that 0 f (x; d) for all d if and only if F (x) F (x) + F (x)d for all d. vi. If x is not a first-order stationary point for f and d solves GN F (x) + F (x)d, d IR n show that d is a descent direction for f at x by showning that vii. Show that the problem f (x; d) F (x) + F (x)d F (x) < 0. d IR n F (x) + F (x)d can be written as a linear program. viii. Suppose at a given point x IR n one solves the problem GN in Part vi above to obtain a direction d for which F (x) + F (x) d < F (x). Show that the following backtracking line search procedure is finitely terating in the sense that the solution t is not zero. Line Search: Let 0 < c < and 0 < γ < and set t := γ s s.t. s {0,, 2, 3,...} and F (x + γ s d) ( γ s c) F (x) + cγ s F (x) + F (x) d. Hint: ( γ s c) F (x) + cγ s F (x) + F (x) d = F (x) + cγ s [ F (x) + F (x) d F (x) ]. Then use Part vi. 5. (a) Locate all of the KKT points for the following problem. Are these points local solutions? Are they global solutions? imize x 2 + x2 2 4x 4x 2 subject to x 2 x 2 x + x 2 2 (b) Let Q IR n n be symmetric and positive definite, g IR n, b IR m, and A IR m n with Nul(A T ) = {0}. i. Show that the matrix AQ A T is nonsingular. ii. Show that x = Q A T (AQ A T ) b [I Q A T (AQ A T ) A]Q c solves the problem imize 2 xt Qx + c T x subject to Ax = b Hint: Use the Lagrangian L(x, y) = f(x) + y T (b Ax) and show that the Lagrange multiplier ȳ must satisfy x = Q ( c + A T ȳ). Then multiply this expression through by A and solve for ȳ...

5 (c) Let Q IR n n be symmetric and positive definite, and let c IR n. Consider the optimization problem 0 x 2 xt Qx + c T x. i. What is the Lagrangian function for this problem? ii. Show that the Lagrangian dual is the problem max y c 2 yt Q y = y c 2 yt Q y. iii. Show that if x, ȳ IR n satisfy ȳ = Q x, then x solves the primal problem if and only if ȳ solves the dual problem, and the optimal values in the primal and dual coincide. (d) (A harder duality problem) Let Q IR n n be symmetric and positive definite. optimization problem P imize 2 xt Qx + c T x subject to x. i. Show that this problem is equivalent to the problem ˆP imize 2 xt Qx subject to e x e, where e is the vector of all ones. ii. What is the Lagrangian for ˆP? iii. Show that the Lagrangian dual for ˆP is the problem Consider the D max 2 (y c)t Q (y c) y = 2 (y c)t Q (y c) + y. This is also the Lagrangian dual for cp. iv. Show that if x, ȳ IR n satisfy ȳ = Q x + c, then x solves P if and only if ȳ solves D, and the optimal values in P and D coincide.

March 5, 2012 MATH 408 FINAL EXAM SAMPLE

March 5, 2012 MATH 408 FINAL EXAM SAMPLE March 5, 202 MATH 408 FINAL EXAM SAMPLE Partial Solutions to Sample Questions (in progress) See the sample questions for the midterm exam, but also consider the following questions. Obviously, a final

More information

May 9, 2014 MATH 408 MIDTERM EXAM OUTLINE. Sample Questions

May 9, 2014 MATH 408 MIDTERM EXAM OUTLINE. Sample Questions May 9, 24 MATH 48 MIDTERM EXAM OUTLINE This exam will consist of two parts and each part will have multipart questions. Each of the 6 questions is worth 5 points for a total of points. The two part of

More information

Nonlinear Optimization: What s important?

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

More information

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

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

More information

5 Quasi-Newton Methods

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

More information

MATH 4211/6211 Optimization Quasi-Newton Method

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

More information

Additional Exercises for Introduction to Nonlinear Optimization Amir Beck March 16, 2017

Additional Exercises for Introduction to Nonlinear Optimization Amir Beck March 16, 2017 Additional Exercises for Introduction to Nonlinear Optimization Amir Beck March 16, 2017 Chapter 1 - Mathematical Preliminaries 1.1 Let S R n. (a) Suppose that T is an open set satisfying T S. Prove that

More information

Conditional Gradient (Frank-Wolfe) Method

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

More information

Optimization and Root Finding. Kurt Hornik

Optimization 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 information

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

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

More information

COMP 558 lecture 18 Nov. 15, 2010

COMP 558 lecture 18 Nov. 15, 2010 Least squares We have seen several least squares problems thus far, and we will see more in the upcoming lectures. For this reason it is good to have a more general picture of these problems and how to

More information

Lecture 18: Optimization Programming

Lecture 18: Optimization Programming Fall, 2016 Outline Unconstrained Optimization 1 Unconstrained Optimization 2 Equality-constrained Optimization Inequality-constrained Optimization Mixture-constrained Optimization 3 Quadratic Programming

More information

10-725/ Optimization Midterm Exam

10-725/ Optimization Midterm Exam 10-725/36-725 Optimization Midterm Exam November 6, 2012 NAME: ANDREW ID: Instructions: This exam is 1hr 20mins long Except for a single two-sided sheet of notes, no other material or discussion is permitted

More information

Constrained Optimization

Constrained Optimization 1 / 22 Constrained Optimization ME598/494 Lecture Max Yi Ren Department of Mechanical Engineering, Arizona State University March 30, 2015 2 / 22 1. Equality constraints only 1.1 Reduced gradient 1.2 Lagrange

More information

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

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

More information

Lecture 14: October 17

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

More information

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

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

More information

Quasi-Newton methods for minimization

Quasi-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 information

Quasi-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 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 information

Optimality Conditions for Constrained Optimization

Optimality Conditions for Constrained Optimization 72 CHAPTER 7 Optimality Conditions for Constrained Optimization 1. First Order Conditions In this section we consider first order optimality conditions for the constrained problem P : minimize f 0 (x)

More information

1. Search Directions In this chapter we again focus on the unconstrained optimization problem. lim sup ν

1. Search Directions In this chapter we again focus on the unconstrained optimization problem. lim sup ν 1 Search Directions In this chapter we again focus on the unconstrained optimization problem P min f(x), x R n where f : R n R is assumed to be twice continuously differentiable, and consider the selection

More information

5 Handling Constraints

5 Handling Constraints 5 Handling Constraints Engineering design optimization problems are very rarely unconstrained. Moreover, the constraints that appear in these problems are typically nonlinear. This motivates our interest

More information

Written Examination

Written 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 information

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

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

More information

1 Computing with constraints

1 Computing with constraints Notes for 2017-04-26 1 Computing with constraints Recall that our basic problem is minimize φ(x) s.t. x Ω where the feasible set Ω is defined by equality and inequality conditions Ω = {x R n : c i (x)

More information

minimize x subject to (x 2)(x 4) u,

minimize x subject to (x 2)(x 4) u, Math 6366/6367: Optimization and Variational Methods Sample Preliminary Exam Questions 1. Suppose that f : [, L] R is a C 2 -function with f () on (, L) and that you have explicit formulae for

More information

SF2822 Applied nonlinear optimization, final exam Saturday December

SF2822 Applied nonlinear optimization, final exam Saturday December SF2822 Applied nonlinear optimization, final exam Saturday December 5 27 8. 3. Examiner: Anders Forsgren, tel. 79 7 27. Allowed tools: Pen/pencil, ruler and rubber; plus a calculator provided by the department.

More information

Duality in Linear Programs. Lecturer: Ryan Tibshirani Convex Optimization /36-725

Duality in Linear Programs. Lecturer: Ryan Tibshirani Convex Optimization /36-725 Duality in Linear Programs Lecturer: Ryan Tibshirani Convex Optimization 10-725/36-725 1 Last time: proximal gradient descent Consider the problem x g(x) + h(x) with g, h convex, g differentiable, and

More information

Quasi-Newton Methods. Javier Peña Convex Optimization /36-725

Quasi-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 information

MATH 4211/6211 Optimization Constrained Optimization

MATH 4211/6211 Optimization Constrained Optimization MATH 4211/6211 Optimization Constrained Optimization Xiaojing Ye Department of Mathematics & Statistics Georgia State University Xiaojing Ye, Math & Stat, Georgia State University 0 Constrained optimization

More information

MA/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 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

Convex Optimization. Newton s method. ENSAE: Optimisation 1/44

Convex Optimization. Newton s method. ENSAE: Optimisation 1/44 Convex Optimization Newton s method ENSAE: Optimisation 1/44 Unconstrained minimization minimize f(x) f convex, twice continuously differentiable (hence dom f open) we assume optimal value p = inf x f(x)

More information

Examination paper for TMA4180 Optimization I

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

More information

Optimization. Yuh-Jye Lee. March 28, Data Science and Machine Intelligence Lab National Chiao Tung University 1 / 40

Optimization. Yuh-Jye Lee. March 28, Data Science and Machine Intelligence Lab National Chiao Tung University 1 / 40 Optimization Yuh-Jye Lee Data Science and Machine Intelligence Lab National Chiao Tung University March 28, 2017 1 / 40 The Key Idea of Newton s Method Let f : R n R be a twice differentiable function

More information

UC Berkeley Department of Electrical Engineering and Computer Science. EECS 227A Nonlinear and Convex Optimization. Solutions 5 Fall 2009

UC Berkeley Department of Electrical Engineering and Computer Science. EECS 227A Nonlinear and Convex Optimization. Solutions 5 Fall 2009 UC Berkeley Department of Electrical Engineering and Computer Science EECS 227A Nonlinear and Convex Optimization Solutions 5 Fall 2009 Reading: Boyd and Vandenberghe, Chapter 5 Solution 5.1 Note that

More information

Lectures 9 and 10: Constrained optimization problems and their optimality conditions

Lectures 9 and 10: Constrained optimization problems and their optimality conditions Lectures 9 and 10: Constrained optimization problems and their optimality conditions Coralia Cartis, Mathematical Institute, University of Oxford C6.2/B2: Continuous Optimization Lectures 9 and 10: Constrained

More information

CS711008Z Algorithm Design and Analysis

CS711008Z Algorithm Design and Analysis CS711008Z Algorithm Design and Analysis Lecture 8 Linear programming: interior point method Dongbo Bu Institute of Computing Technology Chinese Academy of Sciences, Beijing, China 1 / 31 Outline Brief

More information

Homework 5. Convex Optimization /36-725

Homework 5. Convex Optimization /36-725 Homework 5 Convex Optimization 10-725/36-725 Due Tuesday November 22 at 5:30pm submitted to Christoph Dann in Gates 8013 (Remember to a submit separate writeup for each problem, with your name at the top)

More information

QUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday September 21, 2004 (Day 1)

QUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday September 21, 2004 (Day 1) QUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday September 21, 2004 (Day 1) Each of the six questions is worth 10 points. 1) Let H be a (real or complex) Hilbert space. We say

More information

ORIE 6326: Convex Optimization. Quasi-Newton Methods

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

More information

Linear Systems. Class 27. c 2008 Ron Buckmire. TITLE Projection Matrices and Orthogonal Diagonalization CURRENT READING Poole 5.4

Linear Systems. Class 27. c 2008 Ron Buckmire. TITLE Projection Matrices and Orthogonal Diagonalization CURRENT READING Poole 5.4 Linear Systems Math Spring 8 c 8 Ron Buckmire Fowler 9 MWF 9: am - :5 am http://faculty.oxy.edu/ron/math//8/ Class 7 TITLE Projection Matrices and Orthogonal Diagonalization CURRENT READING Poole 5. Summary

More information

Unconstrained minimization of smooth functions

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

More information

Lecture 18: November Review on Primal-dual interior-poit methods

Lecture 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 information

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

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

More information

10 Numerical methods for constrained problems

10 Numerical methods for constrained problems 10 Numerical methods for constrained problems min s.t. f(x) h(x) = 0 (l), g(x) 0 (m), x X The algorithms can be roughly divided the following way: ˆ primal methods: find descent direction keeping inside

More information

There are six more problems on the next two pages

There 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 information

Preliminary/Qualifying Exam in Numerical Analysis (Math 502a) Spring 2012

Preliminary/Qualifying Exam in Numerical Analysis (Math 502a) Spring 2012 Instructions Preliminary/Qualifying Exam in Numerical Analysis (Math 502a) Spring 2012 The exam consists of four problems, each having multiple parts. You should attempt to solve all four problems. 1.

More information

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

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

More information

SECTION: CONTINUOUS OPTIMISATION LECTURE 4: QUASI-NEWTON METHODS

SECTION: CONTINUOUS OPTIMISATION LECTURE 4: QUASI-NEWTON METHODS SECTION: CONTINUOUS OPTIMISATION LECTURE 4: QUASI-NEWTON METHODS HONOUR SCHOOL OF MATHEMATICS, OXFORD UNIVERSITY HILARY TERM 2005, DR RAPHAEL HAUSER 1. The Quasi-Newton Idea. In this lecture we will discuss

More information

Support Vector Machines

Support Vector Machines Support Vector Machines Support vector machines (SVMs) are one of the central concepts in all of machine learning. They are simply a combination of two ideas: linear classification via maximum (or optimal

More information

Search Directions for Unconstrained Optimization

Search Directions for Unconstrained Optimization 8 CHAPTER 8 Search Directions for Unconstrained Optimization In this chapter we study the choice of search directions used in our basic updating scheme x +1 = x + t d. for solving P min f(x). x R n All

More information

ISM206 Lecture Optimization of Nonlinear Objective with Linear Constraints

ISM206 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 information

E5295/5B5749 Convex optimization with engineering applications. Lecture 5. Convex programming and semidefinite programming

E5295/5B5749 Convex optimization with engineering applications. Lecture 5. Convex programming and semidefinite programming E5295/5B5749 Convex optimization with engineering applications Lecture 5 Convex programming and semidefinite programming A. Forsgren, KTH 1 Lecture 5 Convex optimization 2006/2007 Convex quadratic program

More information

2. Quasi-Newton methods

2. 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 information

Frank-Wolfe Method. Ryan Tibshirani Convex Optimization

Frank-Wolfe Method. Ryan Tibshirani Convex Optimization Frank-Wolfe Method Ryan Tibshirani Convex Optimization 10-725 Last time: ADMM For the problem min x,z f(x) + g(z) subject to Ax + Bz = c we form augmented Lagrangian (scaled form): L ρ (x, z, w) = f(x)

More information

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

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

More information

Convex Optimization. Lecture 12 - Equality Constrained Optimization. Instructor: Yuanzhang Xiao. Fall University of Hawaii at Manoa

Convex Optimization. Lecture 12 - Equality Constrained Optimization. Instructor: Yuanzhang Xiao. Fall University of Hawaii at Manoa Convex Optimization Lecture 12 - Equality Constrained Optimization Instructor: Yuanzhang Xiao University of Hawaii at Manoa Fall 2017 1 / 19 Today s Lecture 1 Basic Concepts 2 for Equality Constrained

More information

Final Exam - Take Home Portion Math 211, Summer 2017

Final Exam - Take Home Portion Math 211, Summer 2017 Final Exam - Take Home Portion Math 2, Summer 207 Name: Directions: Complete a total of 5 problems. Problem must be completed. The remaining problems are categorized in four groups. Select one problem

More information

Miscellaneous Nonlinear Programming Exercises

Miscellaneous Nonlinear Programming Exercises Miscellaneous Nonlinear Programming Exercises Henry Wolkowicz 2 08 21 University of Waterloo Department of Combinatorics & Optimization Waterloo, Ontario N2L 3G1, Canada Contents 1 Numerical Analysis Background

More information

Determination of Feasible Directions by Successive Quadratic Programming and Zoutendijk Algorithms: A Comparative Study

Determination of Feasible Directions by Successive Quadratic Programming and Zoutendijk Algorithms: A Comparative Study International Journal of Mathematics And Its Applications Vol.2 No.4 (2014), pp.47-56. ISSN: 2347-1557(online) Determination of Feasible Directions by Successive Quadratic Programming and Zoutendijk Algorithms:

More information

Math 164-1: Optimization Instructor: Alpár R. Mészáros

Math 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 information

Karush-Kuhn-Tucker Conditions. Lecturer: Ryan Tibshirani Convex Optimization /36-725

Karush-Kuhn-Tucker Conditions. Lecturer: Ryan Tibshirani Convex Optimization /36-725 Karush-Kuhn-Tucker Conditions Lecturer: Ryan Tibshirani Convex Optimization 10-725/36-725 1 Given a minimization problem Last time: duality min x subject to f(x) h i (x) 0, i = 1,... m l j (x) = 0, j =

More information

Constrained Optimization and Lagrangian Duality

Constrained Optimization and Lagrangian Duality CIS 520: Machine Learning Oct 02, 2017 Constrained Optimization and Lagrangian Duality Lecturer: Shivani Agarwal Disclaimer: These notes are designed to be a supplement to the lecture. They may or may

More information

Optimization for Machine Learning

Optimization for Machine Learning Optimization for Machine Learning (Problems; Algorithms - A) SUVRIT SRA Massachusetts Institute of Technology PKU Summer School on Data Science (July 2017) Course materials http://suvrit.de/teaching.html

More information

Algorithms for Constrained Optimization

Algorithms 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 information

Math 118, Fall 2014 Final Exam

Math 118, Fall 2014 Final Exam Math 8, Fall 4 Final Exam True or false Please circle your choice; no explanation is necessary True There is a linear transformation T such that T e ) = e and T e ) = e Solution Since T is linear, if T

More information

Optimality, Duality, Complementarity for Constrained Optimization

Optimality, Duality, Complementarity for Constrained Optimization Optimality, Duality, Complementarity for Constrained Optimization Stephen Wright University of Wisconsin-Madison May 2014 Wright (UW-Madison) Optimality, Duality, Complementarity May 2014 1 / 41 Linear

More information

Course Notes for EE227C (Spring 2018): Convex Optimization and Approximation

Course Notes for EE227C (Spring 2018): Convex Optimization and Approximation Course Notes for EE227C (Spring 2018): Convex Optimization and Approximation Instructor: Moritz Hardt Email: hardt+ee227c@berkeley.edu Graduate Instructor: Max Simchowitz Email: msimchow+ee227c@berkeley.edu

More information

A First Course in Optimization: Answers to Selected Exercises

A First Course in Optimization: Answers to Selected Exercises A First Course in Optimization: Answers to Selected Exercises Charles L. Byrne Department of Mathematical Sciences University of Massachusetts Lowell Lowell, MA 01854 December 2, 2009 (The most recent

More information

Lecture 1: Introduction. Outline. B9824 Foundations of Optimization. Fall Administrative matters. 2. Introduction. 3. Existence of optima

Lecture 1: Introduction. Outline. B9824 Foundations of Optimization. Fall Administrative matters. 2. Introduction. 3. Existence of optima B9824 Foundations of Optimization Lecture 1: Introduction Fall 2009 Copyright 2009 Ciamac Moallemi Outline 1. Administrative matters 2. Introduction 3. Existence of optima 4. Local theory of unconstrained

More information

DO NOT OPEN THIS QUESTION BOOKLET UNTIL YOU ARE TOLD TO DO SO

DO 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 information

A Brief Review on Convex Optimization

A Brief Review on Convex Optimization A Brief Review on Convex Optimization 1 Convex set S R n is convex if x,y S, λ,µ 0, λ+µ = 1 λx+µy S geometrically: x,y S line segment through x,y S examples (one convex, two nonconvex sets): A Brief Review

More information

You should be able to...

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

More information

Lagrange duality. The Lagrangian. We consider an optimization program of the form

Lagrange duality. The Lagrangian. We consider an optimization program of the form Lagrange duality Another way to arrive at the KKT conditions, and one which gives us some insight on solving constrained optimization problems, is through the Lagrange dual. The dual is a maximization

More information

Exam in TMA4180 Optimization Theory

Exam in TMA4180 Optimization Theory Norwegian University of Science and Technology Department of Mathematical Sciences Page 1 of 11 Contact during exam: Anne Kværnø: 966384 Exam in TMA418 Optimization Theory Wednesday May 9, 13 Tid: 9. 13.

More information

1. Introduction. Consider the following quadratically constrained quadratic optimization problem:

1. Introduction. Consider the following quadratically constrained quadratic optimization problem: ON LOCAL NON-GLOBAL MINIMIZERS OF QUADRATIC OPTIMIZATION PROBLEM WITH A SINGLE QUADRATIC CONSTRAINT A. TAATI AND M. SALAHI Abstract. In this paper, we consider the nonconvex quadratic optimization problem

More information

Structural and Multidisciplinary Optimization. P. Duysinx and P. Tossings

Structural and Multidisciplinary Optimization. P. Duysinx and P. Tossings Structural and Multidisciplinary Optimization P. Duysinx and P. Tossings 2018-2019 CONTACTS Pierre Duysinx Institut de Mécanique et du Génie Civil (B52/3) Phone number: 04/366.91.94 Email: P.Duysinx@uliege.be

More information

Duality. Geoff Gordon & Ryan Tibshirani Optimization /

Duality. Geoff Gordon & Ryan Tibshirani Optimization / Duality Geoff Gordon & Ryan Tibshirani Optimization 10-725 / 36-725 1 Duality in linear programs Suppose we want to find lower bound on the optimal value in our convex problem, B min x C f(x) E.g., consider

More information

Unconstrained optimization

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

More information

Convex Optimization Boyd & Vandenberghe. 5. Duality

Convex Optimization Boyd & Vandenberghe. 5. Duality 5. Duality Convex Optimization Boyd & Vandenberghe Lagrange dual problem weak and strong duality geometric interpretation optimality conditions perturbation and sensitivity analysis examples generalized

More information

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

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

More information

Computation. For QDA we need to calculate: Lets first consider the case that

Computation. For QDA we need to calculate: Lets first consider the case that Computation For QDA we need to calculate: δ (x) = 1 2 log( Σ ) 1 2 (x µ ) Σ 1 (x µ ) + log(π ) Lets first consider the case that Σ = I,. This is the case where each distribution is spherical, around the

More information

CSCI : Optimization and Control of Networks. Review on Convex Optimization

CSCI : Optimization and Control of Networks. Review on Convex Optimization CSCI7000-016: Optimization and Control of Networks Review on Convex Optimization 1 Convex set S R n is convex if x,y S, λ,µ 0, λ+µ = 1 λx+µy S geometrically: x,y S line segment through x,y S examples (one

More information

Preliminary Examination, Numerical Analysis, August 2016

Preliminary Examination, Numerical Analysis, August 2016 Preliminary Examination, Numerical Analysis, August 2016 Instructions: This exam is closed books and notes. The time allowed is three hours and you need to work on any three out of questions 1-4 and any

More information

IE 5531 Midterm #2 Solutions

IE 5531 Midterm #2 Solutions IE 5531 Midterm #2 s Prof. John Gunnar Carlsson November 9, 2011 Before you begin: This exam has 9 pages and a total of 5 problems. Make sure that all pages are present. To obtain credit for a problem,

More information

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

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

More information

Generalization to inequality constrained problem. Maximize

Generalization to inequality constrained problem. Maximize Lecture 11. 26 September 2006 Review of Lecture #10: Second order optimality conditions necessary condition, sufficient condition. If the necessary condition is violated the point cannot be a local minimum

More information

Duality Uses and Correspondences. Ryan Tibshirani Convex Optimization

Duality Uses and Correspondences. Ryan Tibshirani Convex Optimization Duality Uses and Correspondences Ryan Tibshirani Conve Optimization 10-725 Recall that for the problem Last time: KKT conditions subject to f() h i () 0, i = 1,... m l j () = 0, j = 1,... r the KKT conditions

More information

University of Colorado Denver Department of Mathematical and Statistical Sciences Applied Linear Algebra Ph.D. Preliminary Exam June 10, 2011

University of Colorado Denver Department of Mathematical and Statistical Sciences Applied Linear Algebra Ph.D. Preliminary Exam June 10, 2011 University of Colorado Denver Department of Mathematical and Statistical Sciences Applied Linear Algebra PhD Preliminary Exam June 2 Name: Exam Rules: This is a closed book exam Once the exam begins you

More information

Primal/Dual Decomposition Methods

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

More information

Constrained optimization: direct methods (cont.)

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

More information

SF2822 Applied nonlinear optimization, final exam Wednesday June

SF2822 Applied nonlinear optimization, final exam Wednesday June SF2822 Applied nonlinear optimization, final exam Wednesday June 3 205 4.00 9.00 Examiner: Anders Forsgren, tel. 08-790 7 27. Allowed tools: Pen/pencil, ruler and eraser. Note! Calculator is not allowed.

More information

Lecture 1: Introduction. Outline. B9824 Foundations of Optimization. Fall Administrative matters. 2. Introduction. 3. Existence of optima

Lecture 1: Introduction. Outline. B9824 Foundations of Optimization. Fall Administrative matters. 2. Introduction. 3. Existence of optima B9824 Foundations of Optimization Lecture 1: Introduction Fall 2010 Copyright 2010 Ciamac Moallemi Outline 1. Administrative matters 2. Introduction 3. Existence of optima 4. Local theory of unconstrained

More information

Elementary Linear Algebra Review for Exam 2 Exam is Monday, November 16th.

Elementary Linear Algebra Review for Exam 2 Exam is Monday, November 16th. Elementary Linear Algebra Review for Exam Exam is Monday, November 6th. The exam will cover sections:.4,..4, 5. 5., 7., the class notes on Markov Models. You must be able to do each of the following. Section.4

More information

Mathematical Optimisation, Chpt 2: Linear Equations and inequalities

Mathematical Optimisation, Chpt 2: Linear Equations and inequalities Mathematical Optimisation, Chpt 2: Linear Equations and inequalities Peter J.C. Dickinson p.j.c.dickinson@utwente.nl http://dickinson.website version: 12/02/18 Monday 5th February 2018 Peter J.C. Dickinson

More information

MATH 315 Linear Algebra Homework #1 Assigned: August 20, 2018

MATH 315 Linear Algebra Homework #1 Assigned: August 20, 2018 Homework #1 Assigned: August 20, 2018 Review the following subjects involving systems of equations and matrices from Calculus II. Linear systems of equations Converting systems to matrix form Pivot entry

More information

5. Duality. Lagrangian

5. Duality. Lagrangian 5. Duality Convex Optimization Boyd & Vandenberghe Lagrange dual problem weak and strong duality geometric interpretation optimality conditions perturbation and sensitivity analysis examples generalized

More information

Math 353, Practice Midterm 1

Math 353, Practice Midterm 1 Math 353, Practice Midterm Name: This exam consists of 8 pages including this front page Ground Rules No calculator is allowed 2 Show your work for every problem unless otherwise stated Score 2 2 3 5 4

More information

Check that your exam contains 30 multiple-choice questions, numbered sequentially.

Check that your exam contains 30 multiple-choice questions, numbered sequentially. MATH EXAM SPRING VERSION A NAME STUDENT NUMBER INSTRUCTOR SECTION NUMBER On your scantron, write and bubble your PSU ID, Section Number, and Test Version. Failure to correctly code these items may result

More information

Matrix Secant Methods

Matrix Secant Methods Equation Solving g(x) = 0 Newton-Lie Iterations: x +1 := x J g(x ), where J g (x ). Newton-Lie Iterations: x +1 := x J g(x ), where J g (x ). 3700 years ago the Babylonians used the secant method in 1D:

More information