# Math 273a: Optimization Basic concepts

Save this PDF as:

Size: px
Start display at page:

Download "Math 273a: Optimization Basic concepts" ## Transcription

1 Math 273a: Optimization Basic concepts Instructor: Wotao Yin Department of Mathematics, UCLA Spring 2015 slides based on Chong-Zak, 4th Ed.

2 Goals of this lecture The general form of optimization: minimize f(x) subject to x Ω We study the following topics: terminology types of minimizers optimality conditions

3 Unconstrained vs constrained optimization minimize f(x) subject to x Ω Suppose x R n, Ω is called the feasible set. if Ω = R n, then the problem is called unconstrained. otherwise, the problem is called constrained. In general, more sophisticated techniques are needed to solve constrained problems.

4 (off the topic) Later, we will study some nonsmooth analysis and algorithms that allow f to have the extended value,. Then, we can write any constrained problem in the unconstrained form where the indicator function minimize f(x) + ι Ω (x) ι Ω (x) = { 0, x Ω,, x Ω. The objective function f(x) + ι Ω (x) is nonsmooth.

5 Types of solutions x is a local minimizer if there is ɛ > 0 such that f(x) f(x ) for all x Ω \ {x } and x x < ɛ x is a global minimizer if f(x) f(x ) for all x Ω \ {x } If is replaced with >, then they are strict local minimizer and strict global minimizer, respectively. x 1: strict global minimizer; x 2: strict local minimizer; x 3: local minimizer

6 Convexity and global minimizers A set Ω is convex if λx + (1 λ)y Ω for any x, y Ω and λ [0, 1]. A function is convex if f(λx + (1 λ)y) λf(x) + (1 λ)f(y) for any x, y Ω and λ [0, 1]. A function is convex if and only if its graph is convex. An optimization problem is convex if both the objective function and feasible set are convex. Theorem: Any local minimizer of a convex optimization problem is a global minimizer.

7 Derivatives First-order derivative: row vector Gradient of f: f = (Df) T, which is a column vector. A gradient represents the slope of the tangent of the graph of function. It gives the linear approximation of f at a point. It points toward the greatest rate of increase.

8 Hessian (i.e., second-derivative) of f: which is a symmetric matrix. (F (x)) ij = 2 f x i x j = 2 f x j x i. For one-dimensional function f(x) where x R, it reduces to f (x). F (x) is the Jacobian of f(x), that is, F (x) = J( f(x)). Alternative notation: H(x) and 2 f(x) are also used for Hessian. A Hessian gives a quadratic approximation of f at a point. Gradient and Hessian are local properties that help us recognize local solutions and determine a direction to move at toward the next point.

9 Example Consider Then, and f(x 1, x 2) = x x 2 1 x 1x 2 + x x 1 + 8x [ ] 3x x 1 x f(x) = R 2 x 1 + 2x [ ] 6x F (x) = R Observation: if f is a quadratic function (remove x 3 1 in the above example), f(x) is a linear vector and F (x) is a symmetric constant matrix for any x.

10 Taylor expansion Suppose φ C m (m times continuously differentiable). The Taylor expansion of φ at a point a is φ(a + h) = φ(a) + φ (a)h + φ (a) 2! There are other ways to write the last two terms. h φm (a) h m + o(h m ). 1 m! Example: Consider x, d R n and f C 2. Define φ(α) = f(x + αd). Then, φ (α) = f(x + αd) T d φ (α) = d T F (x + αd) T d Hence, f(x + αd) = f(x) + ( f(x) T d ) α + o(α) = f(x) + ( f(x) T d ) α + dt F (x) T d α 2 + o(α 2 ). 2 1 o(α) collects the term(s) that is asymptotically smaller than α near 0, that is, o(α) α 0, as α 0.

11 Feasible direction A vector d R n is a feasible direction at x Ω if d 0 and x + αd Ω for some small α > 0. (It is possible that d is an infeasible step, that is, x + d Ω. But if there is some room in Ω to move from x toward d, then d is a feasible direction.) d 1 is feasible, d 2 is infeasible If Ω = R n or x lies in the interior of Ω, then any d R n \ {0} is a feasible direction Feasible directions are introduced to establish optimality conditions, especially for points on the boundary of a constrained problem

12 First-order necessary condition Let C 1 be the set of continuously differentiable functions. Proof: Let d by any feasible direction. First-order Taylor expansion: f(x + αd) = f(x ) + αd T f(x ) + o(α). If d T f(x ) < 0, which does not depend on α, then f(x + αd) < f(x ) for all sufficiently small α > 0 (that is, all α (0, ᾱ) for some ᾱ > 0). This is a contradiction since x is a local minimizer.

13 Proof: Since any d R n \ {0} is a feasible direction, we can set d = f(x ). From Theorem 6.1, we have d T f(x ) = f(x ) 2 0. Since f(x ) 2 0, we have f(x ) 2 = 0 and thus f(x ) = 0. Comment: This condition also reduces the problem minimize f(x) to solving the equation f(x ) = 0.

14 x 1 fails to satisfy the FONC; x 2 satisfies the FONC

15 Second-order necessary condition In FONC, there are two possibilities d T f(x ) > 0; d T f(x ) = 0. In the first case, f(x + αd) > f(x ) for all sufficiently small α > 0. In the second case, the vanishing d T f(x ) allows us to check higher-order derivatives.

16 Let C 2 be the set of twice continuously differentiable functions. Proof: Assume that a feasible direction d with d T f(x ) = 0 and d T F (x )d < 0. By 2nd-order Taylor expansion (with a vanishing 1st order term), we have f(x + αd) = f(x ) + dt F (x )d α 2 + o(α 2 ), 2 where by our assumption d T F (x )d < 0. Hence, for all sufficiently small α > 0, we have f(x + αd) < f(x ), which contradicts that x is a local minimizer.

17

18 The necessary conditions are not sufficient Counter examples f(x) = x 3, f (x) = 3x 2, f (x) = 6x f(x) = x 2 1 x is a saddle point: f(0) = 0 but neither a local minimizer nor maximizer By SONC, 0 is not a local minimizer!

19 Second-order sufficient condition Comments: part 2 states F (x ) is positive definite: x T F (x )x > 0 for x 0. the condition is not necessary for strict local minimizer. Proof: For any d 0 and d = 1, we have d T F (x )d λ min(f (x )) > 0. Use the 2nd order Taylor expansion f(x +αd) = f(x )+ α2 2 dt F (x )d+o(α 2 ) f(x )+ α2 2 λmin(f (x ))+o(α 2 ). Then, ᾱ > 0, regardless of d, such that f(x + αd) > f(x ), α (0, ᾱ).

20 Graph of f(x) = x x 2 2 The point 0 satisfies the SOSC.

21 Roles of optimality conditions Recognize a solution: given a candidate solution, check optimality conditions to verify it is a solution. Measure the quality of an approximate solution: measure how close a point is to being a solution Develop algorithms: reduce an optimization problem to solving a (nonlinear) equation (finding a root of the gradient). Later, we will see other forms of optimality conditions and how they lead to equivalent subproblems, as well as algorithms

22 Quiz questions 1. Show that for Ω = {x R n : Ax = b}, d 0 is a feasible direction at x Ω if and only if Ad = Show that for any unconstrained quadratic program, which has the form minimize f(x) := 1 2 xt Qx b T x, if x satisfies the second-order necessary condition, then x is a global minimizer. 3. Show that for any unconstrained quadratic program with Q 0 (Q is symmetric and positive semi-definite), x is a global minimizer if and only if x satisfies the first-order necessary condition. That is, the problem is equivalent to solving Qx = b. 4. Consider minimize c T x, subject to x Ω. Suppose that c 0 and the problem has a global minimizer. Can the minimizer lie in the interior of Ω?

### Lecture 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 information

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

More information

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

More information

### 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

### CE 191: Civil and Environmental Engineering Systems Analysis. LEC 05 : Optimality Conditions CE 191: Civil and Environmental Engineering Systems Analysis LEC : Optimality Conditions Professor Scott Moura Civil & Environmental Engineering University of California, Berkeley Fall 214 Prof. Moura

More information

### ECE580 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 information

### Math 273a: Optimization Netwon s methods Math 273a: Optimization Netwon s methods Instructor: Wotao Yin Department of Mathematics, UCLA Fall 2015 some material taken from Chong-Zak, 4th Ed. Main features of Newton s method Uses both first derivatives

More information

### Math (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 information

### Chapter 1. Optimality Conditions: Unconstrained Optimization. 1.1 Differentiable Problems Chapter 1 Optimality Conditions: Unconstrained Optimization 1.1 Differentiable Problems Consider the problem of minimizing the function f : R n R where f is twice continuously differentiable on R n : P

More information

### Mathematical Economics. Lecture Notes (in extracts) Prof. Dr. Frank Werner Faculty of Mathematics Institute of Mathematical Optimization (IMO) http://math.uni-magdeburg.de/ werner/math-ec-new.html Mathematical Economics Lecture Notes (in extracts) Winter

More information

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

More information

### Lecture Unconstrained optimization. In this lecture we will study the unconstrained problem. minimize f(x), (2.1) Lecture 2 In this lecture we will study the unconstrained problem minimize f(x), (2.1) where x R n. Optimality conditions aim to identify properties that potential minimizers need to satisfy in relation

More information

### 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

### MATHEMATICAL ECONOMICS: OPTIMIZATION. Contents MATHEMATICAL ECONOMICS: OPTIMIZATION JOÃO LOPES DIAS Contents 1. Introduction 2 1.1. Preliminaries 2 1.2. Optimal points and values 2 1.3. The optimization problems 3 1.4. Existence of optimal points 4

More information

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

More information

### 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 information

### OPER 627: Nonlinear Optimization Lecture 2: Math Background and Optimality Conditions OPER 627: Nonlinear Optimization Lecture 2: Math Background and Optimality Conditions Department of Statistical Sciences and Operations Research Virginia Commonwealth University Aug 28, 2013 (Lecture 2)

More information

### SWFR ENG 4TE3 (6TE3) COMP SCI 4TE3 (6TE3) Continuous Optimization Algorithm. Convex Optimization. Computing and Software McMaster University SWFR ENG 4TE3 (6TE3) COMP SCI 4TE3 (6TE3) Continuous Optimization Algorithm Convex Optimization Computing and Software McMaster University General NLO problem (NLO : Non Linear Optimization) (N LO) min

More information

### CHAPTER 2: QUADRATIC PROGRAMMING CHAPTER 2: QUADRATIC PROGRAMMING Overview Quadratic programming (QP) problems are characterized by objective functions that are quadratic in the design variables, and linear constraints. In this sense,

More information

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

More information

### Nonlinear Programming (NLP) Natalia Lazzati Mathematics for Economics (Part I) Note 6: Nonlinear Programming - Unconstrained Optimization Note 6 is based on de la Fuente (2000, Ch. 7), Madden (1986, Ch. 3 and 5) and Simon and Blume

More information

### LMI Methods in Optimal and Robust Control LMI Methods in Optimal and Robust Control Matthew M. Peet Arizona State University Lecture 02: Optimization (Convex and Otherwise) What is Optimization? An Optimization Problem has 3 parts. x F f(x) :

More information

### 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

### Lecture 4: Convex Functions, Part I February 1 IE 521: Convex Optimization Instructor: Niao He Lecture 4: Convex Functions, Part I February 1 Spring 2017, UIUC Scribe: Shuanglong Wang Courtesy warning: These notes do not necessarily cover everything

More information

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

More information

### Module 04 Optimization Problems KKT Conditions & Solvers Module 04 Optimization Problems KKT Conditions & Solvers Ahmad F. Taha EE 5243: Introduction to Cyber-Physical Systems Email: ahmad.taha@utsa.edu Webpage: http://engineering.utsa.edu/ taha/index.html September

More information

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

More information

### Symmetric Matrices and Eigendecomposition Symmetric Matrices and Eigendecomposition Robert M. Freund January, 2014 c 2014 Massachusetts Institute of Technology. All rights reserved. 1 2 1 Symmetric Matrices and Convexity of Quadratic Functions

More information

### ECE580 Solution to Problem Set 6 ECE580 Fall 2015 Solution to Problem Set 6 December 23 2015 1 ECE580 Solution to Problem Set 6 These problems are from the textbook by Chong and Zak 4th edition which is the textbook for the ECE580 Fall

More information

### Numerical Optimization Unconstrained Optimization Computer Science and Automation Indian Institute of Science Bangalore 560 01, India. NPTEL Course on Unconstrained Minimization Let f : R n R. Consider the optimization problem:

More information

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

More information

### ECE580 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 information

### 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

### Gradient 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 information

### Convex Optimization and Modeling Convex Optimization and Modeling Convex Optimization Fourth lecture, 05.05.2010 Jun.-Prof. Matthias Hein Reminder from last time Convex functions: first-order condition: f(y) f(x) + f x,y x, second-order

More information

### Nonlinear Programming Models Nonlinear Programming Models Fabio Schoen 2008 http://gol.dsi.unifi.it/users/schoen Nonlinear Programming Models p. Introduction Nonlinear Programming Models p. NLP problems minf(x) x S R n Standard form:

More information

### Week 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 information

### Introduction to Nonlinear Stochastic Programming School of Mathematics T H E U N I V E R S I T Y O H F R G E D I N B U Introduction to Nonlinear Stochastic Programming Jacek Gondzio Email: J.Gondzio@ed.ac.uk URL: http://www.maths.ed.ac.uk/~gondzio SPS

More information

### Computational Finance Department of Mathematics at University of California, San Diego Computational Finance Optimization Techniques [Lecture 2] Michael Holst January 9, 2017 Contents 1 Optimization Techniques 3 1.1 Examples

More information

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

More information

### 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

### Chapter 4: Unconstrained nonlinear optimization Chapter 4: Unconstrained nonlinear optimization Edoardo Amaldi DEIB Politecnico di Milano edoardo.amaldi@polimi.it Website: http://home.deib.polimi.it/amaldi/opt-15-16.shtml Academic year 2015-16 Edoardo

More information

### Constrained 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 information

### CE 191: Civil & Environmental Engineering Systems Analysis. LEC 17 : Final Review CE 191: Civil & Environmental Engineering Systems Analysis LEC 17 : Final Review Professor Scott Moura Civil & Environmental Engineering University of California, Berkeley Fall 2014 Prof. Moura UC Berkeley

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 A. Forsgren, KTH 1 Lecture 8 Convex optimization 2006/2007 Unconstrained

More information

### The general programming problem is the nonlinear programming problem where a given function is maximized subject to a set of inequality constraints. 1 Optimization Mathematical programming refers to the basic mathematical problem of finding a maximum to a function, f, subject to some constraints. 1 In other words, the objective is to find a point,

More information

### Constrained Optimization Theory Constrained Optimization Theory Stephen J. Wright 1 2 Computer Sciences Department, University of Wisconsin-Madison. IMA, August 2016 Stephen Wright (UW-Madison) Constrained Optimization Theory IMA, August

More information

### ECE 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 information

### Coordinate Update Algorithm Short Course Subgradients and Subgradient Methods Coordinate Update Algorithm Short Course Subgradients and Subgradient Methods Instructor: Wotao Yin (UCLA Math) Summer 2016 1 / 30 Notation f : H R { } is a closed proper convex function domf := {x R n

More information

### 1 Strict local optimality in unconstrained optimization ORF 53 Lecture 14 Spring 016, Princeton University Instructor: A.A. Ahmadi Scribe: G. Hall Thursday, April 14, 016 When in doubt on the accuracy of these notes, please cross check with the instructor s

More information

### Chapter 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 information

### Optimality Conditions Chapter 2 Optimality Conditions 2.1 Global and Local Minima for Unconstrained Problems When a minimization problem does not have any constraints, the problem is to find the minimum of the objective function.

More information

### MA102: Multivariable Calculus MA102: Multivariable Calculus Rupam Barman and Shreemayee Bora Department of Mathematics IIT Guwahati Differentiability of f : U R n R m Definition: Let U R n be open. Then f : U R n R m is differentiable

More information

### 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

### MATH529 Fundamentals of Optimization Unconstrained Optimization II MATH529 Fundamentals of Optimization Unconstrained Optimization II Marco A. Montes de Oca Mathematical Sciences, University of Delaware, USA 1 / 31 Recap 2 / 31 Example Find the local and global minimizers

More information

### Fundamentals 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 information

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

More information

### 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

### 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

### EC /11. Math for Microeconomics September Course, Part II Lecture Notes. Course Outline LONDON SCHOOL OF ECONOMICS Professor Leonardo Felli Department of Economics S.478; x7525 EC400 20010/11 Math for Microeconomics September Course, Part II Lecture Notes Course Outline Lecture 1: Tools for

More information

### 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

### Computational Optimization. Constrained Optimization Part 2 Computational Optimization Constrained Optimization Part Optimality Conditions Unconstrained Case X* is global min Conve f X* is local min SOSC f ( *) = SONC Easiest Problem Linear equality constraints

More information

### Coordinate Update Algorithm Short Course Proximal Operators and Algorithms Coordinate Update Algorithm Short Course Proximal Operators and Algorithms Instructor: Wotao Yin (UCLA Math) Summer 2016 1 / 36 Why proximal? Newton s method: for C 2 -smooth, unconstrained problems allow

More information

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

More information

### Optimization: an Overview Optimization: an Overview Moritz Diehl University of Freiburg and University of Leuven (some slide material was provided by W. Bangerth and K. Mombaur) Overview of presentation Optimization: basic definitions

More information

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

More information

### 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 information

### MAT 419 Lecture Notes Transcribed by Eowyn Cenek 6/1/2012 (Homework 1: Chapter 1: Exercises 1-7, 9, 11, 19, due Monday June 11th See also the course website for lectures, assignments, etc) Note: today s lecture is primarily about definitions Lots of definitions

More information

### AM 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 information

### Math 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 information

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

More information

### Second Order Optimality Conditions for Constrained Nonlinear Programming Second Order Optimality Conditions for Constrained Nonlinear Programming Lecture 10, Continuous Optimisation Oxford University Computing Laboratory, HT 2006 Notes by Dr Raphael Hauser (hauser@comlab.ox.ac.uk)

More information

### 6.252 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 information

### Convex envelopes, cardinality constrained optimization and LASSO. An application in supervised learning: support vector machines (SVMs) ORF 523 Lecture 8 Princeton University Instructor: A.A. Ahmadi Scribe: G. Hall Any typos should be emailed to a a a@princeton.edu. 1 Outline Convexity-preserving operations Convex envelopes, cardinality

More information

### Applications of Linear Programming Applications of Linear Programming lecturer: András London University of Szeged Institute of Informatics Department of Computational Optimization Lecture 9 Non-linear programming In case of LP, the goal

More information

### Optimization Problems with Constraints - introduction to theory, numerical Methods and applications Optimization Problems with Constraints - introduction to theory, numerical Methods and applications Dr. Abebe Geletu Ilmenau University of Technology Department of Simulation and Optimal Processes (SOP)

More information

### Preliminary draft only: please check for final version ARE211, Fall2012 CALCULUS4: THU, OCT 11, 2012 PRINTED: AUGUST 22, 2012 (LEC# 15) Contents 3. Univariate and Multivariate Differentiation (cont) 1 3.6. Taylor s Theorem (cont) 2 3.7. Applying Taylor theory:

More information

### Introduction 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 information

### ARE202A, Fall 2005 CONTENTS. 1. Graphical Overview of Optimization Theory (cont) Separating Hyperplanes 1 AREA, Fall 5 LECTURE #: WED, OCT 5, 5 PRINT DATE: OCTOBER 5, 5 (GRAPHICAL) CONTENTS 1. Graphical Overview of Optimization Theory (cont) 1 1.4. Separating Hyperplanes 1 1.5. Constrained Maximization: One

More information

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

More information

### 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

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

More information

### 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

### Lecture 3. Optimization Problems and Iterative Algorithms Lecture 3 Optimization Problems and Iterative Algorithms January 13, 2016 This material was jointly developed with Angelia Nedić at UIUC for IE 598ns Outline Special Functions: Linear, Quadratic, Convex

More information

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

More information

### Mathematical Economics (ECON 471) Lecture 3 Calculus of Several Variables & Implicit Functions Mathematical Economics (ECON 471) Lecture 3 Calculus of Several Variables & Implicit Functions Teng Wah Leo 1 Calculus of Several Variables 11 Functions Mapping between Euclidean Spaces Where as in univariate

More information

### MATH2070 Optimisation MATH2070 Optimisation Nonlinear optimisation with constraints Semester 2, 2012 Lecturer: I.W. Guo Lecture slides courtesy of J.R. Wishart Review The full nonlinear optimisation problem with equality constraints

More information

### Mechanical Systems II. Method of Lagrange Multipliers Mechanical Systems II. Method of Lagrange Multipliers Rafael Wisniewski Aalborg University Abstract. So far our approach to classical mechanics was limited to finding a critical point of a certain functional.

More information

### Functions 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 information

### Deep Learning. Authors: I. Goodfellow, Y. Bengio, A. Courville. Chapter 4: Numerical Computation. Lecture slides edited by C. Yim. C. Chapter 4: Numerical Computation Deep Learning Authors: I. Goodfellow, Y. Bengio, A. Courville Lecture slides edited by 1 Chapter 4: Numerical Computation 4.1 Overflow and Underflow 4.2 Poor Conditioning

More information

### EC /11. Math for Microeconomics September Course, Part II Problem Set 1 with Solutions. a11 a 12. x 2 LONDON SCHOOL OF ECONOMICS Professor Leonardo Felli Department of Economics S.478; x7525 EC400 2010/11 Math for Microeconomics September Course, Part II Problem Set 1 with Solutions 1. Show that the general

More information

### Lecture 1. Stochastic Optimization: Introduction. January 8, 2018 Lecture 1 Stochastic Optimization: Introduction January 8, 2018 Optimization Concerned with mininmization/maximization of mathematical functions Often subject to constraints Euler (1707-1783): Nothing

More information

### Lecture 1: Entropy, convexity, and matrix scaling CSE 599S: Entropy optimality, Winter 2016 Instructor: James R. Lee Last updated: January 24, 2016 Lecture 1: Entropy, convexity, and matrix scaling CSE 599S: Entropy optimality, Winter 2016 Instructor: James R. Lee Last updated: January 24, 2016 1 Entropy Since this course is about entropy maximization,

More information

### 15-859E: Advanced Algorithms CMU, Spring 2015 Lecture #16: Gradient Descent February 18, 2015 5-859E: Advanced Algorithms CMU, Spring 205 Lecture #6: Gradient Descent February 8, 205 Lecturer: Anupam Gupta Scribe: Guru Guruganesh In this lecture, we will study the gradient descent algorithm and

More information

### Numerical Optimization Constrained Optimization Computer Science and Automation Indian Institute of Science Bangalore 560 012, India. NPTEL Course on Constrained Optimization Constrained Optimization Problem: min h j (x) 0,

More information

### 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

### 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

### 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