Lecture 25: Subgradient Method and Bundle Methods April 24
|
|
- Dominick Webster
- 5 years ago
- Views:
Transcription
1 IE 51: Convex Optimization Spring 017, UIUC Lecture 5: Subgradient Method and Bundle Methods April 4 Instructor: Niao He Scribe: Shuanglong Wang Courtesy warning: hese notes do not necessarily cover everything discussed in the class. Please A swang157@illinois.edu) if you find any typos or mistakes. In this lecture, we cover the following topics Subgradient Method Bundle Method Kelley cutting plane method Level method Reference: Nesterov.004. Chapter 3..3, Subgradient Method Recall that subgradient method works as follows x t+1 = Π X x t γ t g t ), t = 1,,... where g t fx t ), γ t > 0 and Π X x) = arg min y X y x is the projection operator. Note that the projection on X is easy to compute when X is simple, e.g. X is a ball, box, simplex, polyhedron, etc. Lemma 5.1 Projection) x inr n, z X, x z x Π X x) + z Π X x) Proof: When x X, the inequality immediately hold true. Let x X. By definition. Π X x) = arg min z x By optimality condition, this implies Π X x) x) z Π X x)) 0, z X. Hence, x z = x Π X x) + Π X x) z x Π X x) + Π X x) z 5-1
2 Lecture 5: Subgradient Method and Bundle Methods April 4 5- Lemma 5. Key relation) For the subgradient method, we have Proof: x t+1 x x t x γ t fx t ) f ) + γ t g t ) x t+1 x = Π X x t γ t g t ) x x t γ t g t x = x t x γ t g t x t x ) + γ t g t Due to convexity of f, we have f fx t ) + g t x x t ), i.e. g t x t x ) > fx t ) f Combing there two inequalities leads to the desired result. Remark: Note that when f is known, we can choose the optimal γ t by minimizing the right hand side of ): γt = fxt) f, which is the Polyak s stepsize. g t In fact, knowing f is not a problem sometimes. For instance, when the goal is to solve the convex feasibility problem: Find x X, s.t. g i x) 0, i = 1,..., m. We can formulate this as m max g ix) or min maxg i x), 0) 1 i m min he optimal value f is known to be 0 in this case. i=1 If f is not known, one can replace f by its online estimate. heorem 5.3 Convergence) Suppose fx) is convex and Lipschitz continuous on X: fx) fy) M f x y, where M f < +. hen the subgradient method satisfies: where ˆx = γ t) 1 γ tx t ) x, y X fˆx ) f x 1 x + γ t M f γ t Proof: he Lipschitz continuity implies that g t M f, t. Summing up the key relation ) from t = 1 to t =, we obtain γ t fx t ) f ) x 1 x x +1 x + By convexity of f: γ t)fˆx ) γ tfx t ) his further leads to γ t )[fˆx ) f ] 1 x 1 x + M and concludes the proof. γt Mf γ t
3 Lecture 5: Subgradient Method and Bundle Methods April Convergence under various stepsize Assume D X = max x,y x y is the diameter of the set X. It is interesting to see how the bounds in the above theorem would imply the convergence and even the convergence rate with different choices of stepsizes. By abuse of notation, we denote both min fx t) f and fˆx ) f as ɛ. 1 t 1. Constant stepsize: with γ t γ, ɛ D X + γ M f γ = D X 1 γ + M f γ M f γ. It is worth noticing that the error upper-bound does not diminish to zero as grows to infinity, which shows one of the drawbacks of using arbitrary constant stepsizes. In addition, to optimize the upper bound, we can select the optimal stepsize γ to obtain: γ = D X M f ɛ D XM f. It is shown that under this optimal choice ɛ O D XM f ). However, this exhibits another drawback of constant stepsize that in practice is not known in prior for evaluating the optimal γ.. Scaled stepsize: with γ t = γ gx t), DX ɛ + γ M γ f 1/ gx t ) Ω 1 γ + 1 ) γ M f γ. Similarly, we can select the optimal γ by minimizing the right hand side, i.e. γ = D X : γ t = D X gxt ) ɛ D XM. he same convergence rate is achieved while the same drawback about not knowing in prior still exists in choosing γ t. 3. Non-summable but diminishing stepsize: ) / ) ɛ DX + γt Mf γ t ) / 1 DX + γt Mf ) γ t + Mf γt t= 1 +1 / t= 1 +1 where 1 1. When, select large 1 and the first term on the right hand side 0 since γ t is non-summable. he second term also 0 because γ t always approaches zero faster than γ t. Consequently, we know that γ t ɛ 0.
4 Lecture 5: Subgradient Method and Bundle Methods April An example choice of the stepsize is γ t = O ) 1 t with q 0, 1]. As in the above cases, if we q choose γ t = Ω M, then t γ t = D ) X DX M f ln ) ɛ O. M f t In fact, if we choose the averaging from instead of 1, we have ) min fx Mf D X t) f O. / t 4. Non-summable but square-summable stepsize: It is obvious that ) / ɛ Ω + M ) γt γ t 0. A typical choice of γ t = 1 t 1+q, q > 0 also result in the rate of O 1 ). 5. Polyak stepsize: he stepsize yields x t+1 x x t x fx t) f ) gx t ), 5.1) which guarantees x t x decreases each step. Applying 5.1) recursively, we obtain fx t ) f ) DX M f <. herefore we have ɛ 0 as and ɛ O 1 ). Corollary 5.4 When is known, setting γ t fˆx ) f D XM f D X M f, in particular, we have Remark: Subgradient method converges sublinearly. number of iterations. For an accuracy ɛ > 0, need O D X M f ɛ ) 5. Bundle Method When running the subgradient method, we obtain a bundle of affine underestimate of fx): fx t ) + g t x x t ), t = 1,,...
5 Lecture 5: Subgradient Method and Bundle Methods April Definition 5.5 he piecewise linear function: { f t x) = max fxi ) + gi x x i ) } 1 i t where g i fx i ) is called the t-th model of convex function f. Note 1. f t x) fx), x X. f t x i ) = fx i ), 1 i t 3. f 1 x) f x)... f t x)... fx) 5..1 Kelley method Kelley, 1960) he Kelley method works as follows: x t+1 = arg min f tx) Obviously, the above algorithm converges so long as X is compact. he auxiliary problem is not so disturbing reduces to LP) when X is polyhedron. However, the issue is that x t is not unique and Kelley method can be very unstable. Indeed, the worst-case complexity of Kelley method is at least O 1 ɛ n 1)/ ). Remedy: o prevent the instability issue, a possible remedy is update x t+1 by with properly selected α t > 0. { x t+1 = arg min f t x) + α } t x x t 5.. Level Method Lemarchal, Nemirovski, Nesterov, 1995) Denote = min ft f tx) f t = min fx i) 1 i t minimal value of the model) record value of the model) we have 1... f f f... f f 1 Denote the level set L t = { x : f t x) l t := 1 α) f t + α f } t Note that L t is nonempty, convex and closed, and doesn t contain the search points {x 1,..., x t }
6 Lecture 5: Subgradient Method and Bundle Methods April he Level method works as follows { x t+1 = Π Lt x t ) = arg min x xt } : f t x) l t Note when α = 0, reduces to Kelley method. α = 1, there will be no progress. he auxiliary problem reduces to a quadratic program when X is polyhedron. heorem 5.6 When t > 1 1 α) α α) ) Mf D X, ɛ we have f t f t ɛ where M f is the Lipschitz constant and D X is the diameter of set X. Remark: Level method achieves same complexity as the subgradient method which indeed is unimprovable), but can perform much better than subgradient method in practice.
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 informationSubgradient Method. Ryan Tibshirani Convex Optimization
Subgradient Method Ryan Tibshirani Convex Optimization 10-725 Consider the problem Last last time: gradient descent min x f(x) for f convex and differentiable, dom(f) = R n. Gradient descent: choose initial
More informationLecture 1: Convex Sets January 23
IE 521: Convex Optimization Instructor: Niao He Lecture 1: Convex Sets January 23 Spring 2017, UIUC Scribe: Niao He Courtesy warning: These notes do not necessarily cover everything discussed in the class.
More information5. Subgradient method
L. Vandenberghe EE236C (Spring 2016) 5. Subgradient method subgradient method convergence analysis optimal step size when f is known alternating projections optimality 5-1 Subgradient method to minimize
More informationLecture 7: September 17
10-725: Optimization Fall 2013 Lecture 7: September 17 Lecturer: Ryan Tibshirani Scribes: Serim Park,Yiming Gu 7.1 Recap. The drawbacks of Gradient Methods are: (1) requires f is differentiable; (2) relatively
More informationSubgradient Method. Guest Lecturer: Fatma Kilinc-Karzan. Instructors: Pradeep Ravikumar, Aarti Singh Convex Optimization /36-725
Subgradient Method Guest Lecturer: Fatma Kilinc-Karzan Instructors: Pradeep Ravikumar, Aarti Singh Convex Optimization 10-725/36-725 Adapted from slides from Ryan Tibshirani Consider the problem Recall:
More informationMath 273a: Optimization Subgradients of convex functions
Math 273a: Optimization Subgradients of convex functions Made by: Damek Davis Edited by Wotao Yin Department of Mathematics, UCLA Fall 2015 online discussions on piazza.com 1 / 42 Subgradients Assumptions
More informationMath 273a: Optimization Subgradient Methods
Math 273a: Optimization Subgradient Methods Instructor: Wotao Yin Department of Mathematics, UCLA Fall 2015 online discussions on piazza.com Nonsmooth convex function Recall: For ˉx R n, f(ˉx) := {g R
More informationLecture 6: Conic Optimization September 8
IE 598: Big Data Optimization Fall 2016 Lecture 6: Conic Optimization September 8 Lecturer: Niao He Scriber: Juan Xu Overview In this lecture, we finish up our previous discussion on optimality conditions
More information1 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 informationLecture 6: September 12
10-725: Optimization Fall 2013 Lecture 6: September 12 Lecturer: Ryan Tibshirani Scribes: Micol Marchetti-Bowick Note: LaTeX template courtesy of UC Berkeley EECS dept. Disclaimer: These notes have not
More informationPrimal/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 informationLecture 24 November 27
EE 381V: Large Scale Optimization Fall 01 Lecture 4 November 7 Lecturer: Caramanis & Sanghavi Scribe: Jahshan Bhatti and Ken Pesyna 4.1 Mirror Descent Earlier, we motivated mirror descent as a way to improve
More informationLecture 14 Ellipsoid method
S. Boyd EE364 Lecture 14 Ellipsoid method idea of localization methods bisection on R center of gravity algorithm ellipsoid method 14 1 Localization f : R n R convex (and for now, differentiable) problem:
More information1. Gradient method. gradient method, first-order methods. quadratic bounds on convex functions. analysis of gradient method
L. Vandenberghe EE236C (Spring 2016) 1. Gradient method gradient method, first-order methods quadratic bounds on convex functions analysis of gradient method 1-1 Approximate course outline First-order
More informationSelected Topics in Optimization. Some slides borrowed from
Selected Topics in Optimization Some slides borrowed from http://www.stat.cmu.edu/~ryantibs/convexopt/ Overview Optimization problems are almost everywhere in statistics and machine learning. Input Model
More information10. Ellipsoid method
10. Ellipsoid method EE236C (Spring 2008-09) ellipsoid method convergence proof inequality constraints 10 1 Ellipsoid method history developed by Shor, Nemirovski, Yudin in 1970s used in 1979 by Khachian
More information6. Proximal gradient method
L. Vandenberghe EE236C (Spring 2016) 6. Proximal gradient method motivation proximal mapping proximal gradient method with fixed step size proximal gradient method with line search 6-1 Proximal mapping
More informationOne Mirror Descent Algorithm for Convex Constrained Optimization Problems with Non-Standard Growth Properties
One Mirror Descent Algorithm for Convex Constrained Optimization Problems with Non-Standard Growth Properties Fedor S. Stonyakin 1 and Alexander A. Titov 1 V. I. Vernadsky Crimean Federal University, Simferopol,
More informationPrimal-dual Subgradient Method for Convex Problems with Functional Constraints
Primal-dual Subgradient Method for Convex Problems with Functional Constraints Yurii Nesterov, CORE/INMA (UCL) Workshop on embedded optimization EMBOPT2014 September 9, 2014 (Lucca) Yu. Nesterov Primal-dual
More informationLMI Methods in Optimal and Robust Control
LMI Methods in Optimal and Robust Control Matthew M. Peet Arizona State University Lecture 15: Nonlinear Systems and Lyapunov Functions Overview Our next goal is to extend LMI s and optimization to nonlinear
More informationPrimal Solutions and Rate Analysis for Subgradient Methods
Primal Solutions and Rate Analysis for Subgradient Methods Asu Ozdaglar Joint work with Angelia Nedić, UIUC Conference on Information Sciences and Systems (CISS) March, 2008 Department of Electrical Engineering
More informationCoordinate 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 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 informationNonlinear Systems Theory
Nonlinear Systems Theory Matthew M. Peet Arizona State University Lecture 2: Nonlinear Systems Theory Overview Our next goal is to extend LMI s and optimization to nonlinear systems analysis. Today we
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 methods
Optimization methods Optimization-Based Data Analysis http://www.cims.nyu.edu/~cfgranda/pages/obda_spring16 Carlos Fernandez-Granda /8/016 Introduction Aim: Overview of optimization methods that Tend to
More informationOptimization methods
Lecture notes 3 February 8, 016 1 Introduction Optimization methods In these notes we provide an overview of a selection of optimization methods. We focus on methods which rely on first-order information,
More informationSubgradients. subgradients and quasigradients. subgradient calculus. optimality conditions via subgradients. directional derivatives
Subgradients subgradients and quasigradients subgradient calculus optimality conditions via subgradients directional derivatives Prof. S. Boyd, EE392o, Stanford University Basic inequality recall basic
More information15-859E: 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 informationLecture 5: Gradient Descent. 5.1 Unconstrained minimization problems and Gradient descent
10-725/36-725: Convex Optimization Spring 2015 Lecturer: Ryan Tibshirani Lecture 5: Gradient Descent Scribes: Loc Do,2,3 Disclaimer: These notes have not been subjected to the usual scrutiny reserved for
More informationCubic regularization of Newton s method for convex problems with constraints
CORE DISCUSSION PAPER 006/39 Cubic regularization of Newton s method for convex problems with constraints Yu. Nesterov March 31, 006 Abstract In this paper we derive efficiency estimates of the regularized
More informationAccelerated primal-dual methods for linearly constrained convex problems
Accelerated primal-dual methods for linearly constrained convex problems Yangyang Xu SIAM Conference on Optimization May 24, 2017 1 / 23 Accelerated proximal gradient For convex composite problem: minimize
More informationLecture 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 informationarxiv: v1 [math.oc] 1 Jul 2016
Convergence Rate of Frank-Wolfe for Non-Convex Objectives Simon Lacoste-Julien INRIA - SIERRA team ENS, Paris June 8, 016 Abstract arxiv:1607.00345v1 [math.oc] 1 Jul 016 We give a simple proof that the
More informationd(x n, x) d(x n, x nk ) + d(x nk, x) where we chose any fixed k > N
Problem 1. Let f : A R R have the property that for every x A, there exists ɛ > 0 such that f(t) > ɛ if t (x ɛ, x + ɛ) A. If the set A is compact, prove there exists c > 0 such that f(x) > c for all x
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 informationDuality (Continued) min f ( x), X R R. Recall, the general primal problem is. The Lagrangian is a function. defined by
Duality (Continued) Recall, the general primal problem is min f ( x), xx g( x) 0 n m where X R, f : X R, g : XR ( X). he Lagrangian is a function L: XR R m defined by L( xλ, ) f ( x) λ g( x) Duality (Continued)
More informationLecture 6: September 17
10-725/36-725: Convex Optimization Fall 2015 Lecturer: Ryan Tibshirani Lecture 6: September 17 Scribes: Scribes: Wenjun Wang, Satwik Kottur, Zhiding Yu Note: LaTeX template courtesy of UC Berkeley EECS
More informationShiqian Ma, MAT-258A: Numerical Optimization 1. Chapter 4. Subgradient
Shiqian Ma, MAT-258A: Numerical Optimization 1 Chapter 4 Subgradient Shiqian Ma, MAT-258A: Numerical Optimization 2 4.1. Subgradients definition subgradient calculus duality and optimality conditions Shiqian
More informationLecture 15: October 15
10-725: Optimization Fall 2012 Lecturer: Barnabas Poczos Lecture 15: October 15 Scribes: Christian Kroer, Fanyi Xiao Note: LaTeX template courtesy of UC Berkeley EECS dept. Disclaimer: These notes have
More informationLecture 6 : Projected Gradient Descent
Lecture 6 : Projected Gradient Descent EE227C. Lecturer: Professor Martin Wainwright. Scribe: Alvin Wan Consider the following update. x l+1 = Π C (x l α f(x l )) Theorem Say f : R d R is (m, M)-strongly
More informationLecture 23: Online convex optimization Online convex optimization: generalization of several algorithms
EECS 598-005: heoretical Foundations of Machine Learning Fall 2015 Lecture 23: Online convex optimization Lecturer: Jacob Abernethy Scribes: Vikas Dhiman Disclaimer: hese notes have not been subjected
More informationMore First-Order Optimization Algorithms
More First-Order Optimization Algorithms Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A. http://www.stanford.edu/ yyye Chapters 3, 8, 3 The SDM
More informationProximal and First-Order Methods for Convex Optimization
Proximal and First-Order Methods for Convex Optimization John C Duchi Yoram Singer January, 03 Abstract We describe the proximal method for minimization of convex functions We review classical results,
More informationLECTURE 25: REVIEW/EPILOGUE LECTURE OUTLINE
LECTURE 25: REVIEW/EPILOGUE LECTURE OUTLINE CONVEX ANALYSIS AND DUALITY Basic concepts of convex analysis Basic concepts of convex optimization Geometric duality framework - MC/MC Constrained optimization
More informationBarrier Method. Javier Peña Convex Optimization /36-725
Barrier Method Javier Peña Convex Optimization 10-725/36-725 1 Last time: Newton s method For root-finding F (x) = 0 x + = x F (x) 1 F (x) For optimization x f(x) x + = x 2 f(x) 1 f(x) Assume f strongly
More informationLecture 8. Strong Duality Results. September 22, 2008
Strong Duality Results September 22, 2008 Outline Lecture 8 Slater Condition and its Variations Convex Objective with Linear Inequality Constraints Quadratic Objective over Quadratic Constraints Representation
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 informationmin f(x). (2.1) Objectives consisting of a smooth convex term plus a nonconvex regularization term;
Chapter 2 Gradient Methods The gradient method forms the foundation of all of the schemes studied in this book. We will provide several complementary perspectives on this algorithm that highlight the many
More informationAlgorithms for Nonsmooth Optimization
Algorithms for Nonsmooth Optimization Frank E. Curtis, Lehigh University presented at Center for Optimization and Statistical Learning, Northwestern University 2 March 2018 Algorithms for Nonsmooth Optimization
More informationLecture 23: November 21
10-725/36-725: Convex Optimization Fall 2016 Lecturer: Ryan Tibshirani Lecture 23: November 21 Scribes: Yifan Sun, Ananya Kumar, Xin Lu Note: LaTeX template courtesy of UC Berkeley EECS dept. Disclaimer:
More informationSECTION C: CONTINUOUS OPTIMISATION LECTURE 9: FIRST ORDER OPTIMALITY CONDITIONS FOR CONSTRAINED NONLINEAR PROGRAMMING
Nf SECTION C: CONTINUOUS OPTIMISATION LECTURE 9: FIRST ORDER OPTIMALITY CONDITIONS FOR CONSTRAINED NONLINEAR PROGRAMMING f(x R m g HONOUR SCHOOL OF MATHEMATICS, OXFORD UNIVERSITY HILARY TERM 5, DR RAPHAEL
More informationCO 250 Final Exam Guide
Spring 2017 CO 250 Final Exam Guide TABLE OF CONTENTS richardwu.ca CO 250 Final Exam Guide Introduction to Optimization Kanstantsin Pashkovich Spring 2017 University of Waterloo Last Revision: March 4,
More informationShiqian Ma, MAT-258A: Numerical Optimization 1. Chapter 9. Alternating Direction Method of Multipliers
Shiqian Ma, MAT-258A: Numerical Optimization 1 Chapter 9 Alternating Direction Method of Multipliers Shiqian Ma, MAT-258A: Numerical Optimization 2 Separable convex optimization a special case is min f(x)
More information6. Proximal gradient method
L. Vandenberghe EE236C (Spring 2013-14) 6. Proximal gradient method motivation proximal mapping proximal gradient method with fixed step size proximal gradient method with line search 6-1 Proximal mapping
More informationNonlinear equations. Norms for R n. Convergence orders for iterative methods
Nonlinear equations Norms for R n Assume that X is a vector space. A norm is a mapping X R with x such that for all x, y X, α R x = = x = αx = α x x + y x + y We define the following norms on the vector
More informationEstimate sequence methods: extensions and approximations
Estimate sequence methods: extensions and approximations Michel Baes August 11, 009 Abstract The approach of estimate sequence offers an interesting rereading of a number of accelerating schemes proposed
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 informationDesign and Analysis of Algorithms Lecture Notes on Convex Optimization CS 6820, Fall Nov 2 Dec 2016
Design and Analysis of Algorithms Lecture Notes on Convex Optimization CS 6820, Fall 206 2 Nov 2 Dec 206 Let D be a convex subset of R n. A function f : D R is convex if it satisfies f(tx + ( t)y) tf(x)
More informationarxiv: v1 [math.oc] 5 Dec 2014
FAST BUNDLE-LEVEL TYPE METHODS FOR UNCONSTRAINED AND BALL-CONSTRAINED CONVEX OPTIMIZATION YUNMEI CHEN, GUANGHUI LAN, YUYUAN OUYANG, AND WEI ZHANG arxiv:141.18v1 [math.oc] 5 Dec 014 Abstract. It has been
More informationLecture 3: Lagrangian duality and algorithms for the Lagrangian dual problem
Lecture 3: Lagrangian duality and algorithms for the Lagrangian dual problem Michael Patriksson 0-0 The Relaxation Theorem 1 Problem: find f := infimum f(x), x subject to x S, (1a) (1b) where f : R n R
More informationConvex Optimization. (EE227A: UC Berkeley) Lecture 15. Suvrit Sra. (Gradient methods III) 12 March, 2013
Convex Optimization (EE227A: UC Berkeley) Lecture 15 (Gradient methods III) 12 March, 2013 Suvrit Sra Optimal gradient methods 2 / 27 Optimal gradient methods We saw following efficiency estimates for
More informationIFT Lecture 6 Nesterov s Accelerated Gradient, Stochastic Gradient Descent
IFT 6085 - Lecture 6 Nesterov s Accelerated Gradient, Stochastic Gradient Descent This version of the notes has not yet been thoroughly checked. Please report any bugs to the scribes or instructor. Scribe(s):
More informationSubgradients. subgradients. strong and weak subgradient calculus. optimality conditions via subgradients. directional derivatives
Subgradients subgradients strong and weak subgradient calculus optimality conditions via subgradients directional derivatives Prof. S. Boyd, EE364b, Stanford University Basic inequality recall basic inequality
More informationIntroduction 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 informationHow hard is this function to optimize?
How hard is this function to optimize? John Duchi Based on joint work with Sabyasachi Chatterjee, John Lafferty, Yuancheng Zhu Stanford University West Coast Optimization Rumble October 2016 Problem minimize
More informationConverse Lyapunov theorem and Input-to-State Stability
Converse Lyapunov theorem and Input-to-State Stability April 6, 2014 1 Converse Lyapunov theorem In the previous lecture, we have discussed few examples of nonlinear control systems and stability concepts
More informationKaisa Joki Adil M. Bagirov Napsu Karmitsa Marko M. Mäkelä. New Proximal Bundle Method for Nonsmooth DC Optimization
Kaisa Joki Adil M. Bagirov Napsu Karmitsa Marko M. Mäkelä New Proximal Bundle Method for Nonsmooth DC Optimization TUCS Technical Report No 1130, February 2015 New Proximal Bundle Method for Nonsmooth
More information1 Convexity, concavity and quasi-concavity. (SB )
UNIVERSITY OF MARYLAND ECON 600 Summer 2010 Lecture Two: Unconstrained Optimization 1 Convexity, concavity and quasi-concavity. (SB 21.1-21.3.) For any two points, x, y R n, we can trace out the line of
More informationNonlinear Control Lecture 5: Stability Analysis II
Nonlinear Control Lecture 5: Stability Analysis II Farzaneh Abdollahi Department of Electrical Engineering Amirkabir University of Technology Fall 2010 Farzaneh Abdollahi Nonlinear Control Lecture 5 1/41
More informationCOS 511: Theoretical Machine Learning. Lecturer: Rob Schapire Lecture 24 Scribe: Sachin Ravi May 2, 2013
COS 5: heoretical Machine Learning Lecturer: Rob Schapire Lecture 24 Scribe: Sachin Ravi May 2, 203 Review of Zero-Sum Games At the end of last lecture, we discussed a model for two player games (call
More informationFlat Chain and Flat Cochain Related to Koch Curve
ISSN 1479-3889 (print), 1479-3897 (online) International Journal of Nonlinear Science Vol. 3 (2007) No. 2, pp. 144-149 Flat Chain and Flat Cochain Related to Koch Curve Lifeng Xi Institute of Mathematics,
More informationLecture 5: September 12
10-725/36-725: Convex Optimization Fall 2015 Lecture 5: September 12 Lecturer: Lecturer: Ryan Tibshirani Scribes: Scribes: Barun Patra and Tyler Vuong Note: LaTeX template courtesy of UC Berkeley EECS
More informationThe Proximal Gradient Method
Chapter 10 The Proximal Gradient Method Underlying Space: In this chapter, with the exception of Section 10.9, E is a Euclidean space, meaning a finite dimensional space endowed with an inner product,
More informationConvex Optimization. Ofer Meshi. Lecture 6: Lower Bounds Constrained Optimization
Convex Optimization Ofer Meshi Lecture 6: Lower Bounds Constrained Optimization Lower Bounds Some upper bounds: #iter μ 2 M #iter 2 M #iter L L μ 2 Oracle/ops GD κ log 1/ε M x # ε L # x # L # ε # με f
More informationAgenda. Fast proximal gradient methods. 1 Accelerated first-order methods. 2 Auxiliary sequences. 3 Convergence analysis. 4 Numerical examples
Agenda Fast proximal gradient methods 1 Accelerated first-order methods 2 Auxiliary sequences 3 Convergence analysis 4 Numerical examples 5 Optimality of Nesterov s scheme Last time Proximal gradient method
More informationSolving Dual Problems
Lecture 20 Solving Dual Problems We consider a constrained problem where, in addition to the constraint set X, there are also inequality and linear equality constraints. Specifically the minimization problem
More informationAn Accelerated Hybrid Proximal Extragradient Method for Convex Optimization and its Implications to Second-Order Methods
An Accelerated Hybrid Proximal Extragradient Method for Convex Optimization and its Implications to Second-Order Methods Renato D.C. Monteiro B. F. Svaiter May 10, 011 Revised: May 4, 01) Abstract This
More informationThe Q-parametrization (Youla) Lecture 13: Synthesis by Convex Optimization. Lecture 13: Synthesis by Convex Optimization. Example: Spring-mass System
The Q-parametrization (Youla) Lecture 3: Synthesis by Convex Optimization controlled variables z Plant distubances w Example: Spring-mass system measurements y Controller control inputs u Idea for lecture
More informationLecture 1: Background on Convex Analysis
Lecture 1: Background on Convex Analysis John Duchi PCMI 2016 Outline I Convex sets 1.1 Definitions and examples 2.2 Basic properties 3.3 Projections onto convex sets 4.4 Separating and supporting hyperplanes
More informationMath 273a: Optimization Convex Conjugacy
Math 273a: Optimization Convex Conjugacy Instructor: Wotao Yin Department of Mathematics, UCLA Fall 2015 online discussions on piazza.com Convex conjugate (the Legendre transform) Let f be a closed proper
More informationLecture 23: Conditional Gradient Method
10-725/36-725: Conve Optimization Spring 2015 Lecture 23: Conditional Gradient Method Lecturer: Ryan Tibshirani Scribes: Shichao Yang,Diyi Yang,Zhanpeng Fang Note: LaTeX template courtesy of UC Berkeley
More informationLecture 2: Subgradient Methods
Lecture 2: Subgradient Methods John C. Duchi Stanford University Park City Mathematics Institute 206 Abstract In this lecture, we discuss first order methods for the minimization of convex functions. We
More informationNOTES ON FIRST-ORDER METHODS FOR MINIMIZING SMOOTH FUNCTIONS. 1. Introduction. We consider first-order methods for smooth, unconstrained
NOTES ON FIRST-ORDER METHODS FOR MINIMIZING SMOOTH FUNCTIONS 1. Introduction. We consider first-order methods for smooth, unconstrained optimization: (1.1) minimize f(x), x R n where f : R n R. We assume
More informationComposite nonlinear models at scale
Composite nonlinear models at scale Dmitriy Drusvyatskiy Mathematics, University of Washington Joint work with D. Davis (Cornell), M. Fazel (UW), A.S. Lewis (Cornell) C. Paquette (Lehigh), and S. Roy (UW)
More informationOptimization over Sparse Symmetric Sets via a Nonmonotone Projected Gradient Method
Optimization over Sparse Symmetric Sets via a Nonmonotone Projected Gradient Method Zhaosong Lu November 21, 2015 Abstract We consider the problem of minimizing a Lipschitz dierentiable function over a
More informationAccelerating the cubic regularization of Newton s method on convex problems
Accelerating the cubic regularization of Newton s method on convex problems Yu. Nesterov September 005 Abstract In this paper we propose an accelerated version of the cubic regularization of Newton s method
More informationConvex Analysis and Economic Theory AY Elementary properties of convex functions
Division of the Humanities and Social Sciences Ec 181 KC Border Convex Analysis and Economic Theory AY 2018 2019 Topic 6: Convex functions I 6.1 Elementary properties of convex functions We may occasionally
More information1 Dimension Reduction in Euclidean Space
CSIS0351/8601: Randomized Algorithms Lecture 6: Johnson-Lindenstrauss Lemma: Dimension Reduction Lecturer: Hubert Chan Date: 10 Oct 011 hese lecture notes are supplementary materials for the lectures.
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 informationGradient methods for minimizing composite functions
Gradient methods for minimizing composite functions Yu. Nesterov May 00 Abstract In this paper we analyze several new methods for solving optimization problems with the objective function formed as a sum
More informationSpring 2017 CO 250 Course Notes TABLE OF CONTENTS. richardwu.ca. CO 250 Course Notes. Introduction to Optimization
Spring 2017 CO 250 Course Notes TABLE OF CONTENTS richardwu.ca CO 250 Course Notes Introduction to Optimization Kanstantsin Pashkovich Spring 2017 University of Waterloo Last Revision: March 4, 2018 Table
More informationLecture 8: February 9
0-725/36-725: Convex Optimiation Spring 205 Lecturer: Ryan Tibshirani Lecture 8: February 9 Scribes: Kartikeya Bhardwaj, Sangwon Hyun, Irina Caan 8 Proximal Gradient Descent In the previous lecture, we
More information1. Nonlinear Equations. This lecture note excerpted parts from Michael Heath and Max Gunzburger. f(x) = 0
Numerical Analysis 1 1. Nonlinear Equations This lecture note excerpted parts from Michael Heath and Max Gunzburger. Given function f, we seek value x for which where f : D R n R n is nonlinear. f(x) =
More informationLecture 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 informationLecture 9: September 28
0-725/36-725: Convex Optimization Fall 206 Lecturer: Ryan Tibshirani Lecture 9: September 28 Scribes: Yiming Wu, Ye Yuan, Zhihao Li Note: LaTeX template courtesy of UC Berkeley EECS dept. Disclaimer: These
More informationOn Stochastic Subgradient Mirror-Descent Algorithm with Weighted Averaging
On Stochastic Subgradient Mirror-Descent Algorithm with Weighted Averaging arxiv:307.879v [math.oc] 7 Jul 03 Angelia Nedić and Soomin Lee July, 03 Dedicated to Paul Tseng Abstract This paper considers
More informationarxiv: v1 [stat.ml] 12 Nov 2015
Random Multi-Constraint Projection: Stochastic Gradient Methods for Convex Optimization with Many Constraints Mengdi Wang, Yichen Chen, Jialin Liu, Yuantao Gu arxiv:5.03760v [stat.ml] Nov 05 November 3,
More informationLecture 19: Follow The Regulerized Leader
COS-511: Learning heory Spring 2017 Lecturer: Roi Livni Lecture 19: Follow he Regulerized Leader Disclaimer: hese notes have not been subjected to the usual scrutiny reserved for formal publications. hey
More information