Unconstrained Geometric Programming
|
|
- Augustus Gregory
- 5 years ago
- Views:
Transcription
1 Jim Lambers MAT 49/59 Summer Session 20-2 Lecture 8 Notes These notes correspond to Section 2.5 in the text. Unconstrained Geometric Programming Previously, we learned how to use the A-G Inequality to solve an unconstrained minimization problem. We now formalize the procedure for solving such problems, in the case where the obective function to be minimized has the following particular form. Definition Let D R m be the convex) subset of R m defined by A function g : D R m of the form D = {t, t 2,..., t m ) R m t > 0, =, 2,..., m}. gt) = m c i t α i, where c i > 0 for i =, 2,..., n and α i R for i =, 2,..., n, =, 2,..., m, is called a posynomial. We now investigate how the A-G Inequality can be used to find the minimum of a given posynomial gt) on D, if one exists. That is, we will be solving the primal geometric program GP) We denote the ith term of gt) by Minimize gt) = n c i subect to t, t 2,..., t m > 0. g i t) = c i m m tα i t α i, x i = g it), i =, 2,..., n, where δ, δ 2,..., δ n > 0 and Then, the A-G Inequality yields gt) = =. x i
2 x i δ i δ i ) δi n m ) δi n m ) δi m t t α i t α i n α i. Because this is an unconstrained minimization problem, we need the quantity on the low side of the A-G Inequality to be a constant. It follows that the exponents must satisfy α i = 0, =, 2,..., m. If a vector δ = δ, δ 2,..., δ n ) can be found that satisfies this condition, as well as the previous conditions we have imposed on the s, then we have a candidate for a solution to the dual geometric program DGP) Maximize vδ) = ) δi n subect to δ, δ 2,..., δ n > 0 Positivity Condition) n = Normality Condition) n α i = 0, =, 2,..., m Orthogonality Condition). A vector δ that satisfies all three of the above conditions is said to be a feasible vector of the DGP. By the A-G Inequality, we have, for each feasible vector δ, gt) vδ), t D. This inequality is known as the Primal-Dual Inequality. If, in addition, δ is a global maximizer of vδ), and is therefore a solution of the DGP, then vδ ) is at least a lower bound for the minimum value of gt) on D. It can be shown using the criterion gt) = 0 that if t is a global minimizer of gt) on D, and therefore is a solution of the GP, then gt ) = vδ ) for some feasible vector δ. That is, the Primal-Dual Inequality actually becomes an equality. 2
3 Therefore, if δ is a solution of the DGP, then, by the A-G Inequality, the solution to the GP t = t, t 2,..., t m) can be found from the relations or x = x 2 = = x n = vδ ), g i t ) = vδ )δ i, i =, 2,..., n. Note that the s indicate the relative contributions of each term g i t ) of gt ) to the minimum. By taking the natural logarithm of both sides of these equations, we obtain a system of linear equations for the unknowns z = ln t, =, 2,..., m. Spefically, we can solve m [ vδ )δ ] i α i z = ln, i =, 2,..., n. c i Exponentiating the z s yields the components t, =, 2,..., m, of the minimizer t. This leads to the following method for solving the GP, known as Unconstrained Geometric Programming:. Find all feasible vectors δ for the corresponding DGP. 2. If no feasible vectors can be found, then the DGP, and therefore the GP, have no solution. 3. Compute the value of vδ) for each feasible vector δ. Each vector δ that maximizes the value of vδ) is a solution to the DGP. 4. To obtain the solution t to the GP, solve the system of equations g i t ) = vδ )δ i, i =, 2,..., n for t, t 2,..., t m, which can be reduced to a system of linear equations as described above. Example We will solve the GP Minimize gt) = t t 2 t 3 + 2t 2 t 3 + 3t t 3 + 4t t 2 subect to t, t 2, t 3 > 0. This leads to the DGP Maximize ) δ ) δ2 ) δ3 ) δ4 vδ) = δ δ2 δ3 δ4 subect to δ, δ 2, δ 3, δ 4 > 0 Positivity Condition) δ + δ 2 + δ 3 + δ 4 = Normality Condition) δ + δ 3 + δ 4 = 0, δ + δ 2 + δ 4 = 0, Orthogonality Condition) δ + δ 2 + δ 3 = 0 3
4 The Normality Condition and Orthogonality Condition, together, form a system of 4 equations with 4 unknowns whose coeffient matrix is nonsingular, so the system has the unique solution 2 δ = 5, 5, 5, ), 5 which also satisfies the Positivity Condition, so it is feasible. As it is the only feasible vector for the DGP, it is also the solution to the DGP. It follows that the maximum value of vδ), which is also the minimum value of gt), is ) 5 2/5 vδ ) = 0 /5 5 /5 20 / To find the minimizer t, we can solve the equations t t 2 t 3 From these equations, we obtain the relations which yields the solutions = δ vδ ) = 2 5 vδ ) 2t 2t 3 = δ 2 vδ ) = 5 vδ ) 3t t 3 = δ 3 vδ ) = 5 vδ ) 4t t 2 = δ 4 vδ ) = 5 vδ ). 3t 3 = 4t 2, 2t 3 = 4t, 2t 2 = 3t, t 5 = 3 6vδ ), t 2 = vδ ), t 5 3 = 2 3 6vδ ). 3t )3 = 2 5 vδ ) Substituting these values into gt) yields the value of vδ ), as expected. Example We now consider the GP Minimize gt) = 2 t t 2 + t t 2 + t subect to t, t 2 > 0. 4
5 This leads to the DGP Maximize ) δ ) δ2 ) δ3 vδ) = 2 δ δ2 δ3 subect to δ, δ 2, δ 3 > 0 Positivity Condition) δ + δ 2 + δ 3 = Normality Condition) δ + δ 2 + δ 3 = 0, δ + δ 2 = 0 Orthogonality Condition) Unfortunately, the only values of δ, δ 2, δ 3 that satisfy the Normality Condition and the Orthogonality Condition are δ = 2, δ 2 = 2, δ 3 = 0. These values do not satisfy the Positivity Condition, so there are no feasible vectors for the DGP. We conclude that there is no solution to the GP. Exerses. Chapter 2, Exerse 8 2. Chapter 2, Exerse 2 3. Chapter 2, Exerse Chapter 2, Exerse 26 5
Functions of Several Variables
Jim Lambers MAT 419/519 Summer Session 2011-12 Lecture 2 Notes These notes correspond to Section 1.2 in the text. Functions of Several Variables We now generalize the results from the previous section,
More informationi.e., into a monomial, using the Arithmetic-Geometric Mean Inequality, the result will be a posynomial approximation!
Dennis L. Bricker Dept of Mechanical & Industrial Engineering The University of Iowa i.e., 1 1 1 Minimize X X X subject to XX 4 X 1 0.5X 1 Minimize X X X X 1X X s.t. 4 1 1 1 1 4X X 1 1 1 1 0.5X X X 1 1
More informationLogarithmic and Exponential Equations and Change-of-Base
Logarithmic and Exponential Equations and Change-of-Base MATH 101 College Algebra J. Robert Buchanan Department of Mathematics Summer 2012 Objectives In this lesson we will learn to solve exponential equations
More informationJim Lambers MAT 419/519 Summer Session Lecture 13 Notes
Jim Lambers MAT 419/519 Summer Session 2011-12 Lecture 13 Notes These notes correspond to Section 4.1 in the text. Least Squares Fit One of the most fundamental problems in science and engineering is data
More informationTRINITY COLLEGE DUBLIN THE UNIVERSITY OF DUBLIN. School of Mathematics
JS and SS Mathematics JS and SS TSM Mathematics TRINITY COLLEGE DUBLIN THE UNIVERSITY OF DUBLIN School of Mathematics MA3484 Methods of Mathematical Economics Trinity Term 2015 Saturday GOLDHALL 09.30
More informationQuiz Discussion. IE417: Nonlinear Programming: Lecture 12. Motivation. Why do we care? Jeff Linderoth. 16th March 2006
Quiz Discussion IE417: Nonlinear Programming: Lecture 12 Jeff Linderoth Department of Industrial and Systems Engineering Lehigh University 16th March 2006 Motivation Why do we care? We are interested in
More informationMODIFIED GEOMETRIC PROGRAMMING PROBLEM AND ITS APPLICATIONS
J. Appl. Math. & Computing Vol. 172005), No. 1-2, pp. 121-144 MODIFIED GEOMETRIC PROGRAMMING PROBLEM AND ITS APPLICATIONS SAHIDUL ISLAM AND TAPAN KUMAR ROY Abstract. In this paper, we propose unconstrained
More informationGeneralization 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 informationLinear Programming. Operations Research. Anthony Papavasiliou 1 / 21
1 / 21 Linear Programming Operations Research Anthony Papavasiliou Contents 2 / 21 1 Primal Linear Program 2 Dual Linear Program Table of Contents 3 / 21 1 Primal Linear Program 2 Dual Linear Program Linear
More informationNotes taken by Graham Taylor. January 22, 2005
CSC4 - Linear Programming and Combinatorial Optimization Lecture : Different forms of LP. The algebraic objects behind LP. Basic Feasible Solutions Notes taken by Graham Taylor January, 5 Summary: We first
More informationJim Lambers MAT 169 Fall Semester Lecture 6 Notes. a n. n=1. S = lim s k = lim. n=1. n=1
Jim Lambers MAT 69 Fall Semester 2009-0 Lecture 6 Notes These notes correspond to Section 8.3 in the text. The Integral Test Previously, we have defined the sum of a convergent infinite series to be the
More informationIntroduction to Mathematical Programming IE406. Lecture 10. Dr. Ted Ralphs
Introduction to Mathematical Programming IE406 Lecture 10 Dr. Ted Ralphs IE406 Lecture 10 1 Reading for This Lecture Bertsimas 4.1-4.3 IE406 Lecture 10 2 Duality Theory: Motivation Consider the following
More informationConstrained 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 informationGaussian Elimination and Back Substitution
Jim Lambers MAT 610 Summer Session 2009-10 Lecture 4 Notes These notes correspond to Sections 31 and 32 in the text Gaussian Elimination and Back Substitution The basic idea behind methods for solving
More informationJim Lambers MAT 419/519 Summer Session Lecture 11 Notes
Jim Lambers MAT 49/59 Summer Session 20-2 Lecture Notes These notes correspond to Section 34 in the text Broyden s Method One of the drawbacks of using Newton s Method to solve a system of nonlinear equations
More informationLecture #21. c T x Ax b. maximize subject to
COMPSCI 330: Design and Analysis of Algorithms 11/11/2014 Lecture #21 Lecturer: Debmalya Panigrahi Scribe: Samuel Haney 1 Overview In this lecture, we discuss linear programming. We first show that the
More informationOptimization for Communications and Networks. Poompat Saengudomlert. Session 4 Duality and Lagrange Multipliers
Optimization for Communications and Networks Poompat Saengudomlert Session 4 Duality and Lagrange Multipliers P Saengudomlert (2015) Optimization Session 4 1 / 14 24 Dual Problems Consider a primal convex
More informationJim Lambers MAT 610 Summer Session Lecture 2 Notes
Jim Lambers MAT 610 Summer Session 2009-10 Lecture 2 Notes These notes correspond to Sections 2.2-2.4 in the text. Vector Norms Given vectors x and y of length one, which are simply scalars x and y, the
More informationToday: Linear Programming (con t.)
Today: Linear Programming (con t.) COSC 581, Algorithms April 10, 2014 Many of these slides are adapted from several online sources Reading Assignments Today s class: Chapter 29.4 Reading assignment for
More informationMVE165/MMG631 Linear and integer optimization with applications Lecture 13 Overview of nonlinear programming. Ann-Brith Strömberg
MVE165/MMG631 Overview of nonlinear programming Ann-Brith Strömberg 2015 05 21 Areas of applications, examples (Ch. 9.1) Structural optimization Design of aircraft, ships, bridges, etc Decide on the material
More informationLagrange duality. The Lagrangian. We consider an optimization program of the form
Lagrange duality Another way to arrive at the KKT conditions, and one which gives us some insight on solving constrained optimization problems, is through the Lagrange dual. The dual is a maximization
More information2.6 Logarithmic Functions. Inverse Functions. Question: What is the relationship between f(x) = x 2 and g(x) = x?
Inverse Functions Question: What is the relationship between f(x) = x 3 and g(x) = 3 x? Question: What is the relationship between f(x) = x 2 and g(x) = x? Definition (One-to-One Function) A function f
More informationThe Eigenvalue Problem: Perturbation Theory
Jim Lambers MAT 610 Summer Session 2009-10 Lecture 13 Notes These notes correspond to Sections 7.2 and 8.1 in the text. The Eigenvalue Problem: Perturbation Theory The Unsymmetric Eigenvalue Problem Just
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 informationNonlinear Optimization for Optimal Control
Nonlinear Optimization for Optimal Control Pieter Abbeel UC Berkeley EECS Many slides and figures adapted from Stephen Boyd [optional] Boyd and Vandenberghe, Convex Optimization, Chapters 9 11 [optional]
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 informationgpcvx A Matlab Solver for Geometric Programs in Convex Form
gpcvx A Matlab Solver for Geometric Programs in Convex Form Kwangmoo Koh deneb1@stanford.edu Almir Mutapcic almirm@stanford.edu Seungjean Kim sjkim@stanford.edu Stephen Boyd boyd@stanford.edu May 22, 2006
More informationELE539A: Optimization of Communication Systems Lecture 6: Quadratic Programming, Geometric Programming, and Applications
ELE539A: Optimization of Communication Systems Lecture 6: Quadratic Programming, Geometric Programming, and Applications Professor M. Chiang Electrical Engineering Department, Princeton University February
More informationprinceton univ. F 13 cos 521: Advanced Algorithm Design Lecture 17: Duality and MinMax Theorem Lecturer: Sanjeev Arora
princeton univ F 13 cos 521: Advanced Algorithm Design Lecture 17: Duality and MinMax Theorem Lecturer: Sanjeev Arora Scribe: Today we first see LP duality, which will then be explored a bit more in the
More informationICS-E4030 Kernel Methods in Machine Learning
ICS-E4030 Kernel Methods in Machine Learning Lecture 3: Convex optimization and duality Juho Rousu 28. September, 2016 Juho Rousu 28. September, 2016 1 / 38 Convex optimization Convex optimisation This
More informationLecture Note 5: Semidefinite Programming for Stability Analysis
ECE7850: Hybrid Systems:Theory and Applications Lecture Note 5: Semidefinite Programming for Stability Analysis Wei Zhang Assistant Professor Department of Electrical and Computer Engineering Ohio State
More information4. Algebra and Duality
4-1 Algebra and Duality P. Parrilo and S. Lall, CDC 2003 2003.12.07.01 4. Algebra and Duality Example: non-convex polynomial optimization Weak duality and duality gap The dual is not intrinsic The cone
More informationConvex Optimization & Lagrange Duality
Convex Optimization & Lagrange Duality Chee Wei Tan CS 8292 : Advanced Topics in Convex Optimization and its Applications Fall 2010 Outline Convex optimization Optimality condition Lagrange duality KKT
More informationLecture 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 informationApplications 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 informationLECTURE 13 LECTURE OUTLINE
LECTURE 13 LECTURE OUTLINE Problem Structures Separable problems Integer/discrete problems Branch-and-bound Large sum problems Problems with many constraints Conic Programming Second Order Cone Programming
More informationLP Duality: outline. Duality theory for Linear Programming. alternatives. optimization I Idea: polyhedra
LP Duality: outline I Motivation and definition of a dual LP I Weak duality I Separating hyperplane theorem and theorems of the alternatives I Strong duality and complementary slackness I Using duality
More informationEE 227A: Convex Optimization and Applications October 14, 2008
EE 227A: Convex Optimization and Applications October 14, 2008 Lecture 13: SDP Duality Lecturer: Laurent El Ghaoui Reading assignment: Chapter 5 of BV. 13.1 Direct approach 13.1.1 Primal problem Consider
More informationUtility, Fairness and Rate Allocation
Utility, Fairness and Rate Allocation Laila Daniel and Krishnan Narayanan 11th March 2013 Outline of the talk A rate allocation example Fairness criteria and their formulation as utilities Convex optimization
More informationMotivating examples Introduction to algorithms Simplex algorithm. On a particular example General algorithm. Duality An application to game theory
Instructor: Shengyu Zhang 1 LP Motivating examples Introduction to algorithms Simplex algorithm On a particular example General algorithm Duality An application to game theory 2 Example 1: profit maximization
More informationIntroduction to Machine Learning Lecture 7. Mehryar Mohri Courant Institute and Google Research
Introduction to Machine Learning Lecture 7 Mehryar Mohri Courant Institute and Google Research mohri@cims.nyu.edu Convex Optimization Differentiation Definition: let f : X R N R be a differentiable function,
More informationLearning Module 1 - Basic Algebra Review (Appendix A)
Learning Module 1 - Basic Algebra Review (Appendix A) Element 1 Real Numbers and Operations on Polynomials (A.1, A.2) Use the properties of real numbers and work with subsets of the real numbers Determine
More informationLecture 9 Sequential unconstrained minimization
S. Boyd EE364 Lecture 9 Sequential unconstrained minimization brief history of SUMT & IP methods logarithmic barrier function central path UMT & SUMT complexity analysis feasibility phase generalized inequalities
More information1 Review Session. 1.1 Lecture 2
1 Review Session Note: The following lists give an overview of the material that was covered in the lectures and sections. Your TF will go through these lists. If anything is unclear or you have questions
More informationLagrangian Duality and Convex Optimization
Lagrangian Duality and Convex Optimization David Rosenberg New York University February 11, 2015 David Rosenberg (New York University) DS-GA 1003 February 11, 2015 1 / 24 Introduction Why Convex Optimization?
More informationORIE 6300 Mathematical Programming I August 25, Lecture 2
ORIE 6300 Mathematical Programming I August 25, 2016 Lecturer: Damek Davis Lecture 2 Scribe: Johan Bjorck Last time, we considered the dual of linear programs in our basic form: max(c T x : Ax b). We also
More informationMidterm Review. Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A.
Midterm Review Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A. http://www.stanford.edu/ yyye (LY, Chapter 1-4, Appendices) 1 Separating hyperplane
More informationOptimality 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 informationExtreme Abridgment of Boyd and Vandenberghe s Convex Optimization
Extreme Abridgment of Boyd and Vandenberghe s Convex Optimization Compiled by David Rosenberg Abstract Boyd and Vandenberghe s Convex Optimization book is very well-written and a pleasure to read. The
More informationWeek 3 Linear programming duality
Week 3 Linear programming duality This week we cover the fascinating topic of linear programming duality. We will learn that every minimization program has associated a maximization program that has the
More informationPrimal-Dual Interior-Point Methods for Linear Programming based on Newton s Method
Primal-Dual Interior-Point Methods for Linear Programming based on Newton s Method Robert M. Freund March, 2004 2004 Massachusetts Institute of Technology. The Problem The logarithmic barrier approach
More informationUC Berkeley Department of Electrical Engineering and Computer Science. EECS 227A Nonlinear and Convex Optimization. Solutions 5 Fall 2009
UC Berkeley Department of Electrical Engineering and Computer Science EECS 227A Nonlinear and Convex Optimization Solutions 5 Fall 2009 Reading: Boyd and Vandenberghe, Chapter 5 Solution 5.1 Note that
More informationJim Lambers MAT 610 Summer Session Lecture 1 Notes
Jim Lambers MAT 60 Summer Session 2009-0 Lecture Notes Introduction This course is about numerical linear algebra, which is the study of the approximate solution of fundamental problems from linear algebra
More information4. Convex optimization problems
Convex Optimization Boyd & Vandenberghe 4. Convex optimization problems optimization problem in standard form convex optimization problems quasiconvex optimization linear optimization quadratic optimization
More informationThe dual simplex method with bounds
The dual simplex method with bounds Linear programming basis. Let a linear programming problem be given by min s.t. c T x Ax = b x R n, (P) where we assume A R m n to be full row rank (we will see in the
More informationLecture Support Vector Machine (SVM) Classifiers
Introduction to Machine Learning Lecturer: Amir Globerson Lecture 6 Fall Semester Scribe: Yishay Mansour 6.1 Support Vector Machine (SVM) Classifiers Classification is one of the most important tasks in
More informationCS 6820 Fall 2014 Lectures, October 3-20, 2014
Analysis of Algorithms Linear Programming Notes CS 6820 Fall 2014 Lectures, October 3-20, 2014 1 Linear programming The linear programming (LP) problem is the following optimization problem. We are given
More informationExample Problem. Linear Program (standard form) CSCI5654 (Linear Programming, Fall 2013) Lecture-7. Duality
CSCI5654 (Linear Programming, Fall 013) Lecture-7 Duality Lecture 7 Slide# 1 Lecture 7 Slide# Linear Program (standard form) Example Problem maximize c 1 x 1 + + c n x n s.t. a j1 x 1 + + a jn x n b j
More information10-725/ Optimization Midterm Exam
10-725/36-725 Optimization Midterm Exam November 6, 2012 NAME: ANDREW ID: Instructions: This exam is 1hr 20mins long Except for a single two-sided sheet of notes, no other material or discussion is permitted
More informationLecture Notes on Support Vector Machine
Lecture Notes on Support Vector Machine Feng Li fli@sdu.edu.cn Shandong University, China 1 Hyperplane and Margin In a n-dimensional space, a hyper plane is defined by ω T x + b = 0 (1) where ω R n is
More information- Well-characterized problems, min-max relations, approximate certificates. - LP problems in the standard form, primal and dual linear programs
LP-Duality ( Approximation Algorithms by V. Vazirani, Chapter 12) - Well-characterized problems, min-max relations, approximate certificates - LP problems in the standard form, primal and dual linear programs
More informationLecture: Convex Optimization Problems
1/36 Lecture: Convex Optimization Problems http://bicmr.pku.edu.cn/~wenzw/opt-2015-fall.html Acknowledgement: this slides is based on Prof. Lieven Vandenberghe s lecture notes Introduction 2/36 optimization
More informationThe Fundamental Theorem of Linear Inequalities
The Fundamental Theorem of Linear Inequalities Lecture 8, Continuous Optimisation Oxford University Computing Laboratory, HT 2006 Notes by Dr Raphael Hauser (hauser@comlab.ox.ac.uk) Constrained Optimisation
More informationMachine Learning. Support Vector Machines. Manfred Huber
Machine Learning Support Vector Machines Manfred Huber 2015 1 Support Vector Machines Both logistic regression and linear discriminant analysis learn a linear discriminant function to separate the data
More informationEE364a Review Session 5
EE364a Review Session 5 EE364a Review announcements: homeworks 1 and 2 graded homework 4 solutions (check solution to additional problem 1) scpd phone-in office hours: tuesdays 6-7pm (650-723-1156) 1 Complementary
More informationDiscrete Optimization
Prof. Friedrich Eisenbrand Martin Niemeier Due Date: April 15, 2010 Discussions: March 25, April 01 Discrete Optimization Spring 2010 s 3 You can hand in written solutions for up to two of the exercises
More informationCS-E4830 Kernel Methods in Machine Learning
CS-E4830 Kernel Methods in Machine Learning Lecture 3: Convex optimization and duality Juho Rousu 27. September, 2017 Juho Rousu 27. September, 2017 1 / 45 Convex optimization Convex optimisation This
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 informationSection Notes 9. Midterm 2 Review. Applied Math / Engineering Sciences 121. Week of December 3, 2018
Section Notes 9 Midterm 2 Review Applied Math / Engineering Sciences 121 Week of December 3, 2018 The following list of topics is an overview of the material that was covered in the lectures and sections
More information2 Lecture Defining Optimization with Equality Constraints
2 Lecture 2 2.1 Defining Optimization with Equality Constraints So far we have been concentrating on an arbitrary set. Because of this, we could of course incorporate constrains directly into the set.
More informationNonlinear Programming (Hillier, Lieberman Chapter 13) CHEM-E7155 Production Planning and Control
Nonlinear Programming (Hillier, Lieberman Chapter 13) CHEM-E7155 Production Planning and Control 19/4/2012 Lecture content Problem formulation and sample examples (ch 13.1) Theoretical background Graphical
More informationGEOMETRIC PROGRAMMING APPROACHES OF RELIABILITY ALLOCATION
U.P.B. Sci. Bull., Series A, Vol. 79, Iss. 3, 2017 ISSN 1223-7027 GEOMETRIC PROGRAMMING APPROACHES OF RELIABILITY ALLOCATION Constantin Udrişte 1, Saad Abbas Abed 2 and Ionel Ţevy 3 One of the important
More information3. Linear Programming and Polyhedral Combinatorics
Massachusetts Institute of Technology 18.433: Combinatorial Optimization Michel X. Goemans February 28th, 2013 3. Linear Programming and Polyhedral Combinatorics Summary of what was seen in the introductory
More informationGame Theory: Lecture 3
Game Theory: Lecture 3 Lecturer: Pingzhong Tang Topic: Mixed strategy Scribe: Yuan Deng March 16, 2015 Definition 1 (Mixed strategy). A mixed strategy S i : A i [0, 1] assigns a probability S i ( ) 0 to
More informationEE/ACM Applications of Convex Optimization in Signal Processing and Communications Lecture 17
EE/ACM 150 - Applications of Convex Optimization in Signal Processing and Communications Lecture 17 Andre Tkacenko Signal Processing Research Group Jet Propulsion Laboratory May 29, 2012 Andre Tkacenko
More informationSEMIDEFINITE PROGRAM BASICS. Contents
SEMIDEFINITE PROGRAM BASICS BRIAN AXELROD Abstract. A introduction to the basics of Semidefinite programs. Contents 1. Definitions and Preliminaries 1 1.1. Linear Algebra 1 1.2. Convex Analysis (on R n
More informationCS269: Machine Learning Theory Lecture 16: SVMs and Kernels November 17, 2010
CS269: Machine Learning Theory Lecture 6: SVMs and Kernels November 7, 200 Lecturer: Jennifer Wortman Vaughan Scribe: Jason Au, Ling Fang, Kwanho Lee Today, we will continue on the topic of support vector
More informationLecture 9: Dantzig-Wolfe Decomposition
Lecture 9: Dantzig-Wolfe Decomposition (3 units) Outline Dantzig-Wolfe decomposition Column generation algorithm Relation to Lagrangian dual Branch-and-price method Generated assignment problem and multi-commodity
More informationChapter 1. Preliminaries
Introduction This dissertation is a reading of chapter 4 in part I of the book : Integer and Combinatorial Optimization by George L. Nemhauser & Laurence A. Wolsey. The chapter elaborates links between
More informationCOMP3121/9101/3821/9801 Lecture Notes. Linear Programming
COMP3121/9101/3821/9801 Lecture Notes Linear Programming LiC: Aleks Ignjatovic THE UNIVERSITY OF NEW SOUTH WALES School of Computer Science and Engineering The University of New South Wales Sydney 2052,
More informationLagrange Relaxation and Duality
Lagrange Relaxation and Duality As we have already known, constrained optimization problems are harder to solve than unconstrained problems. By relaxation we can solve a more difficult problem by a simpler
More informationI.3. LMI DUALITY. Didier HENRION EECI Graduate School on Control Supélec - Spring 2010
I.3. LMI DUALITY Didier HENRION henrion@laas.fr EECI Graduate School on Control Supélec - Spring 2010 Primal and dual For primal problem p = inf x g 0 (x) s.t. g i (x) 0 define Lagrangian L(x, z) = g 0
More informationBBM402-Lecture 20: LP Duality
BBM402-Lecture 20: LP Duality Lecturer: Lale Özkahya Resources for the presentation: https://courses.engr.illinois.edu/cs473/fa2016/lectures.html An easy LP? which is compact form for max cx subject to
More informationLinear and Combinatorial Optimization
Linear and Combinatorial Optimization The dual of an LP-problem. Connections between primal and dual. Duality theorems and complementary slack. Philipp Birken (Ctr. for the Math. Sc.) Lecture 3: Duality
More informationCONSTRAINED OPTIMALITY CRITERIA
5 CONSTRAINED OPTIMALITY CRITERIA In Chapters 2 and 3, we discussed the necessary and sufficient optimality criteria for unconstrained optimization problems. But most engineering problems involve optimization
More informationLast Revised: :19: (Fri, 12 Jan 2007)(Revision:
0-0 1 Demand Lecture Last Revised: 2007-01-12 16:19:03-0800 (Fri, 12 Jan 2007)(Revision: 67) a demand correspondence is a special kind of choice correspondence where the set of alternatives is X = { x
More informationELE539A: Optimization of Communication Systems Lecture 16: Pareto Optimization and Nonconvex Optimization
ELE539A: Optimization of Communication Systems Lecture 16: Pareto Optimization and Nonconvex Optimization Professor M. Chiang Electrical Engineering Department, Princeton University March 16, 2007 Lecture
More informationAlgebra and Trigonometry 2006 (Foerster) Correlated to: Washington Mathematics Standards, Algebra 2 (2008)
A2.1. Core Content: Solving problems The first core content area highlights the type of problems students will be able to solve by the end of, as they extend their ability to solve problems with additional
More informationsubject to (x 2)(x 4) u,
Exercises Basic definitions 5.1 A simple example. Consider the optimization problem with variable x R. minimize x 2 + 1 subject to (x 2)(x 4) 0, (a) Analysis of primal problem. Give the feasible set, the
More informationExample: feasibility. Interpretation as formal proof. Example: linear inequalities and Farkas lemma
4-1 Algebra and Duality P. Parrilo and S. Lall 2006.06.07.01 4. Algebra and Duality Example: non-convex polynomial optimization Weak duality and duality gap The dual is not intrinsic The cone of valid
More informationLecture 10: Linear programming. duality. and. The dual of the LP in standard form. maximize w = b T y (D) subject to A T y c, minimize z = c T x (P)
Lecture 10: Linear programming duality Michael Patriksson 19 February 2004 0-0 The dual of the LP in standard form minimize z = c T x (P) subject to Ax = b, x 0 n, and maximize w = b T y (D) subject to
More informationDuality of LPs and Applications
Lecture 6 Duality of LPs and Applications Last lecture we introduced duality of linear programs. We saw how to form duals, and proved both the weak and strong duality theorems. In this lecture we will
More informationChapter 33 MSMYM1 Mathematical Linear Programming
Chapter 33 MSMYM1 Mathematical Linear Programming 33.1 The Simplex Algorithm The Simplex method for solving linear programming problems has already been covered in Chapter??. A given problem may always
More informationDifferential Equations Practice: 2nd Order Linear: Nonhomogeneous Equations: Variation of Parameters Page 1
Differential Equations Practice: 2nd Order Linear: Nonhomogeneous Equations: Variation of Parameters Page Questions Example (3.6.) Find a particular solution of the differential equation y 5y + 6y = 2e
More informationThis research was partially supported by the Faculty Research and Development Fund of the University of North Carolina at Wilmington
LARGE SCALE GEOMETRIC PROGRAMMING: AN APPLICATION IN CODING THEORY Yaw O. Chang and John K. Karlof Mathematical Sciences Department The University of North Carolina at Wilmington This research was partially
More information12. Interior-point methods
12. Interior-point methods Convex Optimization Boyd & Vandenberghe inequality constrained minimization logarithmic barrier function and central path barrier method feasibility and phase I methods complexity
More informationCurriculum Map: Mathematics
Curriculum Map: Mathematics Course: Calculus Grade(s): 11/12 Unit 1: Prerequisites for Calculus This initial chapter, A Prerequisites for Calculus, is just that-a review chapter. This chapter will provide
More information15. Conic optimization
L. Vandenberghe EE236C (Spring 216) 15. Conic optimization conic linear program examples modeling duality 15-1 Generalized (conic) inequalities Conic inequality: a constraint x K where K is a convex cone
More informationChapter 2 Functions and Graphs
Chapter 2 Functions and Graphs Section 6 Logarithmic Functions Learning Objectives for Section 2.6 Logarithmic Functions The student will be able to use and apply inverse functions. The student will be
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 information