Introduction to unconstrained optimization - direct search methods
|
|
- Camron Brooks
- 5 years ago
- Views:
Transcription
1 Introduction to unconstrained optimization - direct search methods Jussi Hakanen Post-doctoral researcher jussi.hakanen@jyu.fi
2 Structure of optimization methods Typically Constraint handling converts the problem to (a series of) unconstrained problems In unconstrained optimization a search direction is determined at each iteration The best solution in the search direction is found with line search Constraint handling method Unconstrained optimization Line search
3 Group discussion 1. What kind of optimality conditions there exist for unconstrained optimization (x R n )? 2. List methods for unconstrained optimization? what are their general ideas? Discuss in small groups (3-4) for minutes Each group has a secretary who writes down the answers of the group At the end, we summarize what each group found
4 Reminder: gradient and hessian Definition: If function f: R n R is differentiable, then the gradient f(x) consists of the partial derivatives f(x) i.e. f x = f(x),, f(x) x 1 x n Definition: If f is twice differentiable, then the matrix 2 f(x) 2 f(x) x 1 x 1 x 1 x n H x = 2 f(x) 2 f(x) x n x 1 x n x n is called the Hessian (matrix) of f at x Result: If f is twice continuously differentiable, then 2 f(x) x i x j = 2 f(x) x j x i T x i
5 Reminder: Definite Matrices Definition: A symmetric n n matrix H is positive semidefinite if x R n x T Hx 0. Definition: A symmetric n n matrix H is positive definite if x T Hx > 0 0 x R n Note: If (> <), then H is negative semidefinite (definite). If H is neither positive nor negative semidefinite, then it is indefinite. Result: Let S R n be open convex set and f: S R twice differentiable in S. Function f is convex if and only if H(x ) is positive semidefinite for all x S.
6 Unconstraint problem min f x, s. t. x R n Necessary conditions: Let f be twice differentiable in x. If x is a local minimizer, then f x = 0 (that is, x is a critical point of f) and H x is positive semidefinite. Sufficient conditions: Let f be twice differentiable in x. If f x = 0 and H(x ) is positive definite, then x is a strict local minimizer. Result: Let f: R n R is twice differentiable in x. If f x = 0 and H(x ) is indefinite, then x is a saddle point.
7 Unconstraint problem Adopted from Prof. L.T. Biegler (Carnegie Mellon University)
8 Descent direction Definition: Let f: R n R. A vector d R n is a descent direction for f in x R n if δ > 0 s.t. f x + λd < f(x ) λ (0, δ]. Result: Let f: R n R be differentiable in x. If d R n s.t. f x T d < 0 then d is a descent direction for f in x.
9 Model algorithm for unconstrained minimization Let x h be the current estimate for x 1) [Test for convergence.] If conditions are satisfied, stop. The solution is x h. 2) [Compute a search direction.] Compute a nonzero vector d h R n which is the search direction. 3) [Compute a step length.] Compute α h > 0, the step length, for which it holds that f x h + α h d h < f(x h ). 4) [Update the estimate for minimum.] Set x h+1 = x h + α h d h, h = h + 1 and go to step 1. From Gill et al., Practical Optimization, 1981, Academic Press
10 On convergence Iterative method: a sequence {x h } s.t. x h x when h Definition: A method converges linearly if α [0,1) and M 0 s.t. h M x h+1 x α x h x, superlinearly if M 0 and for some sequence α h 0 it holds that h M x h+1 x α h x h x, with degree p if α 0, p > 0 and M 0 s.t. h M x h+1 x α x h x p. If p = 2 (p = 3), the convergence is quadratic (cubic).
11 Summary of group discussion for methods 1. Newton s method 1. Utilizes tangent 2. Golden section method 1. For line search 3. Downhill Simplex 4. Cyclic coordinate method 1. One coordinate at a time 5. Polytopy search (Nelder-Mead) 1. Idea based on geometry 6. Gradient descent (steepest descent) 1. Based on gradient information
12 Direct search methods Univariate search, coordinate descent, cyclic coordinate search Hooke and Jeeves Powell s method
13 From Miettinen: Nonlinear optimization, 2007 (in Finnish) Coordinate descent f x = 2x x 1 x 2 + x x 1 x 2
14 From Miettinen: Nonlinear optimization, 2007 (in Finnish) Idea of pattern search
15 From Miettinen: Nonlinear optimization, 2007 (in Finnish) Hooke and Jeeves f x = x x 1 2x 2 2
16 From Miettinen: Nonlinear optimization, 2007 (in Finnish) Hooke and Jeeves with fixed step length f x = x x 1 2x 2 2
17 Powell s method Most efficient pattern search method Differs from Hooke and Jeeves so that for each pattern search step one of the coordinate directions is replaced with previous pattern search direction.
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 informationOptimization: Nonlinear Optimization without Constraints. Nonlinear Optimization without Constraints 1 / 23
Optimization: Nonlinear Optimization without Constraints Nonlinear Optimization without Constraints 1 / 23 Nonlinear optimization without constraints Unconstrained minimization min x f(x) where f(x) is
More 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 informationEAD 115. Numerical Solution of Engineering and Scientific Problems. David M. Rocke Department of Applied Science
EAD 115 Numerical Solution of Engineering and Scientific Problems David M. Rocke Department of Applied Science Multidimensional Unconstrained Optimization Suppose we have a function f() of more than one
More 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 informationECE580 Exam 1 October 4, Please do not write on the back of the exam pages. Extra paper is available from the instructor.
ECE580 Exam 1 October 4, 2012 1 Name: Solution Score: /100 You must show ALL of your work for full credit. This exam is closed-book. Calculators may NOT be used. Please leave fractions as fractions, etc.
More informationNonlinearOptimization
1/35 NonlinearOptimization Pavel Kordík Department of Computer Systems Faculty of Information Technology Czech Technical University in Prague Jiří Kašpar, Pavel Tvrdík, 2011 Unconstrained nonlinear optimization,
More informationNonlinear Optimization: What s important?
Nonlinear Optimization: What s important? Julian Hall 10th May 2012 Convexity: convex problems A local minimizer is a global minimizer A solution of f (x) = 0 (stationary point) is a minimizer A global
More informationOptimization Concepts and Applications in Engineering
Optimization Concepts and Applications in Engineering Ashok D. Belegundu, Ph.D. Department of Mechanical Engineering The Pennsylvania State University University Park, Pennsylvania Tirupathi R. Chandrupatia,
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 informationStatistics 580 Optimization Methods
Statistics 580 Optimization Methods Introduction Let fx be a given real-valued function on R p. The general optimization problem is to find an x ɛ R p at which fx attain a maximum or a minimum. It is of
More informationECE580 Fall 2015 Solution to Midterm Exam 1 October 23, Please leave fractions as fractions, but simplify them, etc.
ECE580 Fall 2015 Solution to Midterm Exam 1 October 23, 2015 1 Name: Solution Score: /100 This exam is closed-book. You must show ALL of your work for full credit. Please read the questions carefully.
More informationUnconstrained optimization
Chapter 4 Unconstrained optimization An unconstrained optimization problem takes the form min x Rnf(x) (4.1) for a target functional (also called objective function) f : R n R. In this chapter and throughout
More informationNumerical 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 informationComputational Optimization. Augmented Lagrangian NW 17.3
Computational Optimization Augmented Lagrangian NW 17.3 Upcoming Schedule No class April 18 Friday, April 25, in class presentations. Projects due unless you present April 25 (free extension until Monday
More informationLecture 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 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 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 informationNumerisches 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, b = 0. (2) 1 2 The eigenvectors of A corresponding to the eigenvalues λ 1 = 1, λ 2 = 3 are
Quadratic forms We consider the quadratic function f : R 2 R defined by f(x) = 2 xt Ax b T x with x = (x, x 2 ) T, () where A R 2 2 is symmetric and b R 2. We will see that, depending on the eigenvalues
More information14. Nonlinear equations
L. Vandenberghe ECE133A (Winter 2018) 14. Nonlinear equations Newton method for nonlinear equations damped Newton method for unconstrained minimization Newton method for nonlinear least squares 14-1 Set
More informationScientific Computing: An Introductory Survey
Scientific Computing: An Introductory Survey Chapter 6 Optimization Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign Copyright c 2002. Reproduction permitted
More informationScientific Computing: An Introductory Survey
Scientific Computing: An Introductory Survey Chapter 6 Optimization Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign Copyright c 2002. Reproduction permitted
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 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 informationOn prediction. Jussi Hakanen Post-doctoral researcher. TIES445 Data mining (guest lecture)
On prediction Jussi Hakanen Post-doctoral researcher jussi.hakanen@jyu.fi Learning outcomes To understand the basic principles of prediction To understand linear regression in prediction To be aware of
More informationConstrained 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 informationHigher-Order Methods
Higher-Order Methods Stephen J. Wright 1 2 Computer Sciences Department, University of Wisconsin-Madison. PCMI, July 2016 Stephen Wright (UW-Madison) Higher-Order Methods PCMI, July 2016 1 / 25 Smooth
More information1 Numerical optimization
Contents 1 Numerical optimization 5 1.1 Optimization of single-variable functions............ 5 1.1.1 Golden Section Search................... 6 1.1. Fibonacci Search...................... 8 1. Algorithms
More informationOutline. Scientific Computing: An Introductory Survey. Optimization. Optimization Problems. Examples: Optimization Problems
Outline Scientific Computing: An Introductory Survey Chapter 6 Optimization 1 Prof. Michael. Heath Department of Computer Science University of Illinois at Urbana-Champaign Copyright c 2002. Reproduction
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 informationCE 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 informationOptimization. Totally not complete this is...don't use it yet...
Optimization Totally not complete this is...don't use it yet... Bisection? Doing a root method is akin to doing a optimization method, but bi-section would not be an effective method - can detect sign
More informationnonrobust estimation The n measurement vectors taken together give the vector X R N. The unknown parameter vector is P R M.
Introduction to nonlinear LS estimation R. I. Hartley and A. Zisserman: Multiple View Geometry in Computer Vision. Cambridge University Press, 2ed., 2004. After Chapter 5 and Appendix 6. We will use x
More informationOptimization Methods
Optimization Methods Categorization of Optimization Problems Continuous Optimization Discrete Optimization Combinatorial Optimization Variational Optimization Common Optimization Concepts in Computer Vision
More informationScientific Computing: Optimization
Scientific Computing: Optimization Aleksandar Donev Courant Institute, NYU 1 donev@courant.nyu.edu 1 Course MATH-GA.2043 or CSCI-GA.2112, Spring 2012 March 8th, 2011 A. Donev (Courant Institute) Lecture
More informationComparative study of Optimization methods for Unconstrained Multivariable Nonlinear Programming Problems
International Journal of Scientific and Research Publications, Volume 3, Issue 10, October 013 1 ISSN 50-3153 Comparative study of Optimization methods for Unconstrained Multivariable Nonlinear Programming
More informationIE 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 informationLecture V. Numerical Optimization
Lecture V Numerical Optimization Gianluca Violante New York University Quantitative Macroeconomics G. Violante, Numerical Optimization p. 1 /19 Isomorphism I We describe minimization problems: to maximize
More informationOptimization. Escuela de Ingeniería Informática de Oviedo. (Dpto. de Matemáticas-UniOvi) Numerical Computation Optimization 1 / 30
Optimization Escuela de Ingeniería Informática de Oviedo (Dpto. de Matemáticas-UniOvi) Numerical Computation Optimization 1 / 30 Unconstrained optimization Outline 1 Unconstrained optimization 2 Constrained
More informationLecture Notes: Geometric Considerations in Unconstrained Optimization
Lecture Notes: Geometric Considerations in Unconstrained Optimization James T. Allison February 15, 2006 The primary objectives of this lecture on unconstrained optimization are to: Establish connections
More informationOptimization. Next: Curve Fitting Up: Numerical Analysis for Chemical Previous: Linear Algebraic and Equations. Subsections
Next: Curve Fitting Up: Numerical Analysis for Chemical Previous: Linear Algebraic and Equations Subsections One-dimensional Unconstrained Optimization Golden-Section Search Quadratic Interpolation Newton's
More information5 Quasi-Newton Methods
Unconstrained Convex Optimization 26 5 Quasi-Newton Methods If the Hessian is unavailable... Notation: H = Hessian matrix. B is the approximation of H. C is the approximation of H 1. Problem: Solve min
More 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 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 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 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 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 informationConvex Optimization CMU-10725
Convex Optimization CMU-10725 Quasi Newton Methods Barnabás Póczos & Ryan Tibshirani Quasi Newton Methods 2 Outline Modified Newton Method Rank one correction of the inverse Rank two correction of the
More informationQuasi-Newton Methods
Newton s Method Pros and Cons Quasi-Newton Methods MA 348 Kurt Bryan Newton s method has some very nice properties: It s extremely fast, at least once it gets near the minimum, and with the simple modifications
More information1 Numerical optimization
Contents Numerical optimization 5. Optimization of single-variable functions.............................. 5.. Golden Section Search..................................... 6.. Fibonacci Search........................................
More informationComputational 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 informationAdvanced Mathematical Programming IE417. Lecture 24. Dr. Ted Ralphs
Advanced Mathematical Programming IE417 Lecture 24 Dr. Ted Ralphs IE417 Lecture 24 1 Reading for This Lecture Sections 11.2-11.2 IE417 Lecture 24 2 The Linear Complementarity Problem Given M R p p and
More informationContents. Preface. 1 Introduction Optimization view on mathematical models NLP models, black-box versus explicit expression 3
Contents Preface ix 1 Introduction 1 1.1 Optimization view on mathematical models 1 1.2 NLP models, black-box versus explicit expression 3 2 Mathematical modeling, cases 7 2.1 Introduction 7 2.2 Enclosing
More informationReview of Classical Optimization
Part II Review of Classical Optimization Multidisciplinary Design Optimization of Aircrafts 51 2 Deterministic Methods 2.1 One-Dimensional Unconstrained Minimization 2.1.1 Motivation Most practical optimization
More information2. Quasi-Newton methods
L. Vandenberghe EE236C (Spring 2016) 2. Quasi-Newton methods variable metric methods quasi-newton methods BFGS update limited-memory quasi-newton methods 2-1 Newton method for unconstrained minimization
More 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 informationLine Search Methods for Unconstrained Optimisation
Line Search Methods for Unconstrained Optimisation Lecture 8, Numerical Linear Algebra and Optimisation Oxford University Computing Laboratory, MT 2007 Dr Raphael Hauser (hauser@comlab.ox.ac.uk) The Generic
More 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 informationMathematical optimization
Optimization Mathematical optimization Determine the best solutions to certain mathematically defined problems that are under constrained determine optimality criteria determine the convergence of the
More informationNumerical 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 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 informationEAD 115. Numerical Solution of Engineering and Scientific Problems. David M. Rocke Department of Applied Science
EAD 115 Numerical Solution of Engineering and Scientific Problems David M. Rocke Department of Applied Science Taylor s Theorem Can often approximate a function by a polynomial The error in the approximation
More informationStatic unconstrained optimization
Static unconstrained optimization 2 In unconstrained optimization an objective function is minimized without any additional restriction on the decision variables, i.e. min f(x) x X ad (2.) with X ad R
More informationFunctions of Several Variables
Functions of Several Variables The Unconstrained Minimization Problem where In n dimensions the unconstrained problem is stated as f() x variables. minimize f()x x, is a scalar objective function of vector
More informationIntroduction. New Nonsmooth Trust Region Method for Unconstraint Locally Lipschitz Optimization Problems
New Nonsmooth Trust Region Method for Unconstraint Locally Lipschitz Optimization Problems Z. Akbari 1, R. Yousefpour 2, M. R. Peyghami 3 1 Department of Mathematics, K.N. Toosi University of Technology,
More informationGradient Descent. Ryan Tibshirani Convex Optimization /36-725
Gradient Descent Ryan Tibshirani Convex Optimization 10-725/36-725 Last time: canonical convex programs Linear program (LP): takes the form min x subject to c T x Gx h Ax = b Quadratic program (QP): like
More informationComputational 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 informationECS550NFB Introduction to Numerical Methods using Matlab Day 2
ECS550NFB Introduction to Numerical Methods using Matlab Day 2 Lukas Laffers lukas.laffers@umb.sk Department of Mathematics, University of Matej Bel June 9, 2015 Today Root-finding: find x that solves
More informationLagrange Multipliers
Lagrange Multipliers (Com S 477/577 Notes) Yan-Bin Jia Nov 9, 2017 1 Introduction We turn now to the study of minimization with constraints. More specifically, we will tackle the following problem: minimize
More informationIntroduction to Scientific Computing
Introduction to Scientific Computing Benson Muite benson.muite@ut.ee http://kodu.ut.ee/ benson https://courses.cs.ut.ee/2018/isc/spring 26 March 2018 [Public Domain,https://commons.wikimedia.org/wiki/File1
More informationUnconstrained minimization of smooth functions
Unconstrained minimization of smooth functions We want to solve min x R N f(x), where f is convex. In this section, we will assume that f is differentiable (so its gradient exists at every point), and
More informationIntroduction to Optimization
Introduction to Optimization Konstantin Tretyakov (kt@ut.ee) MTAT.03.227 Machine Learning So far Machine learning is important and interesting The general concept: Fitting models to data So far Machine
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 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 informationScientific Data Computing: Lecture 3
Scientific Data Computing: Lecture 3 Benson Muite benson.muite@ut.ee 23 April 2018 Outline Monday 10-12, Liivi 2-207 Monday 12-14, Liivi 2-205 Topics Introduction, statistical methods and their applications
More informationNumerical Optimization of Partial Differential Equations
Numerical Optimization of Partial Differential Equations Part I: basic optimization concepts in R n Bartosz Protas Department of Mathematics & Statistics McMaster University, Hamilton, Ontario, Canada
More informationConstrained optimization. Unconstrained optimization. One-dimensional. Multi-dimensional. Newton with equality constraints. Active-set method.
Optimization Unconstrained optimization One-dimensional Multi-dimensional Newton s method Basic Newton Gauss- Newton Quasi- Newton Descent methods Gradient descent Conjugate gradient Constrained optimization
More informationConvex Optimization. Problem set 2. Due Monday April 26th
Convex Optimization Problem set 2 Due Monday April 26th 1 Gradient Decent without Line-search In this problem we will consider gradient descent with predetermined step sizes. That is, instead of determining
More informationNonlinear Programming
Nonlinear Programming Kees Roos e-mail: C.Roos@ewi.tudelft.nl URL: http://www.isa.ewi.tudelft.nl/ roos LNMB Course De Uithof, Utrecht February 6 - May 8, A.D. 2006 Optimization Group 1 Outline for week
More informationLecture 5: September 15
10-725/36-725: Convex Optimization Fall 2015 Lecture 5: September 15 Lecturer: Lecturer: Ryan Tibshirani Scribes: Scribes: Di Jin, Mengdi Wang, Bin Deng Note: LaTeX template courtesy of UC Berkeley EECS
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 informationOptimization 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 informationOptimization II: Unconstrained Multivariable
Optimization II: Unconstrained Multivariable CS 205A: Mathematical Methods for Robotics, Vision, and Graphics Justin Solomon CS 205A: Mathematical Methods Optimization II: Unconstrained Multivariable 1
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/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 informationCHAPTER 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 informationMethods for Unconstrained Optimization Numerical Optimization Lectures 1-2
Methods for Unconstrained Optimization Numerical Optimization Lectures 1-2 Coralia Cartis, University of Oxford INFOMM CDT: Modelling, Analysis and Computation of Continuous Real-World Problems Methods
More informationOptimization Methods. Lecture 19: Line Searches and Newton s Method
15.93 Optimization Methods Lecture 19: Line Searches and Newton s Method 1 Last Lecture Necessary Conditions for Optimality (identifies candidates) x local min f(x ) =, f(x ) PSD Slide 1 Sufficient Conditions
More informationNumerical Optimization. Review: Unconstrained Optimization
Numerical Optimization Finding the best feasible solution Edward P. Gatzke Department of Chemical Engineering University of South Carolina Ed Gatzke (USC CHE ) Numerical Optimization ECHE 589, Spring 2011
More informationComputational 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 informationMultidisciplinary System Design Optimization (MSDO)
Multidisciplinary System Design Optimization (MSDO) Numerical Optimization II Lecture 8 Karen Willcox 1 Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Today s Topics Sequential
More information4TE3/6TE3. Algorithms for. Continuous Optimization
4TE3/6TE3 Algorithms for Continuous Optimization (Algorithms for Constrained Nonlinear Optimization Problems) Tamás TERLAKY Computing and Software McMaster University Hamilton, November 2005 terlaky@mcmaster.ca
More informationUnconstrained Multivariate Optimization
Unconstrained Multivariate Optimization Multivariate optimization means optimization of a scalar function of a several variables: and has the general form: y = () min ( ) where () is a nonlinear scalar-valued
More 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 informationNumerical Optimization: Basic Concepts and Algorithms
May 27th 2015 Numerical Optimization: Basic Concepts and Algorithms R. Duvigneau R. Duvigneau - Numerical Optimization: Basic Concepts and Algorithms 1 Outline Some basic concepts in optimization Some
More informationShiqian Ma, MAT-258A: Numerical Optimization 1. Chapter 3. Gradient Method
Shiqian Ma, MAT-258A: Numerical Optimization 1 Chapter 3 Gradient Method Shiqian Ma, MAT-258A: Numerical Optimization 2 3.1. Gradient method Classical gradient method: to minimize a differentiable convex
More informationCoE 3SK3 Computer Aided Engineering Tutorial: Unconstrained Optimization
CoE 3SK3 Computer Aided Engineering Tutorial: Unconstrained Optimization Jie Cao caoj23@grads.ece.mcmaster.ca Department of Electrical and Computer Engineering McMaster University Feb. 2, 2010 Outline
More informationAn Iterative Descent Method
Conjugate Gradient: An Iterative Descent Method The Plan Review Iterative Descent Conjugate Gradient Review : Iterative Descent Iterative Descent is an unconstrained optimization process x (k+1) = x (k)
More informationOn the Local Quadratic Convergence of the Primal-Dual Augmented Lagrangian Method
Optimization Methods and Software Vol. 00, No. 00, Month 200x, 1 11 On the Local Quadratic Convergence of the Primal-Dual Augmented Lagrangian Method ROMAN A. POLYAK Department of SEOR and Mathematical
More information