Lecture 18: Optimization Programming

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Lecture 18: Optimization Programming"

Transcription

1 Fall, 2016

2 Outline Unconstrained Optimization 1 Unconstrained Optimization 2 Equality-constrained Optimization Inequality-constrained Optimization Mixture-constrained Optimization 3 Quadratic Programming (QP), Linear Programming (LP)

3 Unconstrained Optimization Consider the minimization problem min x f (x), where x R d, and f is continuously differentiable. Necessary condition for local minimums If x is a local minimum of f (x), then f (x ) = 0. A point satisfying the necessary condition is called a stationary point, which can be a local minimum, local maximum, or saddle point. Sufficient conditions for x be a local minimum are: f (x ) = 0, 2 f (x ) is positive definite.

4 Equality- Inequality- Mixture- Consider the minimization problem min x f (x) subject to g i (x) 0, i = 1,, m. h j (x) = 0, j = 1,, k. Assume x R d. f : R d R is called the objective function g i : R d R, i = 1,, m are inequality constraints h j : R d R, j = 1,, k are equality constraints. Assume f, g i s, h j s are all continuously differentiable.

5 Single Equality Constraint Equality- Inequality- Mixture- Consider the minimization problem Introduce the Lagrange function: Lagrange extreme value theory: min x f (x) (1) subject to h(x) = 0. (2) L(x, λ) = f (x) + λh(x) If x is a local minimum under the constraint, then there exist some λ such that x L(x, λ ) = 0, λ L(x, λ ) = 0.

6 Interpretations Unconstrained Optimization Equality- Inequality- Mixture- Above conditions imply that (x, λ ) is a stationary point of L. The above equations are equivalent to x f (x ) + λ x h(x ) = 0, h(x ) = 0. Note the first condition gives d equations.

7 Multiple Equality Constraints Equality- Inequality- Mixture- Consider the minimization problem min x f (x) (3) subject to h j (x) = 0, j = 1,, k. (4) Introduce multiple Lagrange multipliers: λ = (λ 1,, λ k ) L(x, λ) = f (x) + k λ j h j (x) Lagrange extreme theory: If x is a local minimum under the constraint, then there exist some λ = (λ 1,, λ k ) such that j=1 x L(x, λ ) = 0, λ L(x, λ ) = 0. In other words, (x, λ ) is a stationary point of L.

8 Equality- Inequality- Mixture- Example: Entropy Maximization Problem Consider the minimization problem n min p i log(p i ), subject to p 1,,p n i=1 i=1 n p i = 1. i=1 The Lagrange function is n n L(p, λ) = p i log(p i ) + λ( p i 1). L p i = log(p i ) λ = 0, log(p i ) = 1 λ, i = 1,, n. Together with n i=1 p i = 1, we have p 1 = = p n = 1 n. This problem is known as maximizing information entropy. i=1

9 Generalized Lagrange Methods Consider the minimization problem Equality- Inequality- Mixture- min x f (x) subject to g i (x) 0, i = 1,, m. Introduce multiple non-negative Lagrange multipliers: µ 1 0,, µ m 0. m L(x, µ) = f (x) µ i g i (x). µ i s are also called dual variables or slack variables. Correspondingly, we call x as primal variables. We would like to minimize L with respect to x and maximize L with respect to µ. i=1

10 KKT Necessary Optimality Conditions Equality- Inequality- Mixture- If x is a local minimum under the above inequality constraints, then there exist some µ = (µ 1,, µ m) such that Stationarity: x L(x, µ ) = 0, i.e., x f (x ) m µ k xg i (x ) = 0. i=1 Primal feasibility: g i (x ) 0 for i = 1,, m. Dual feasibility: µ i 0 for i = 1,, m. Complementary slackness: KKT = Karush-Kuhn-Tucker µ i g i (x ) = 0, i = 1,, m.

11 Sufficient Optimality Conditions Equality- Inequality- Mixture- The above necessary conditions are also sufficient for optimality, if f, g i s are continuously differentiable f and g i s are convex functions. Under this case, There exists a solution x satisfying the KKT condition, x is actually a global optimum. Examples: SVM problems

12 Alternative Constraints Equality- Inequality- Mixture- Consider the minimization problem min x f (x) subject to g i (x) 0, i = 1,, m. Introduce multiple non-negative Lagrange multipliers: µ 1 0,, µ m 0. m L(x, µ) = f (x) + µ i g i (x). µ i s are also called dual variables or slack variables. Correspondingly, we call x as primal variables. We would like to minimize L with respect to x and maximize L with respect to µ. i=1

13 KKT Necessary Optimality Conditions Equality- Inequality- Mixture- If x is a local minimum under the above inequality constraints, then there exist some µ = (µ 1,, µ m) such that Stationarity: x L(x, µ ) = 0, i.e., x f (x ) + m µ k xg i (x ) = 0. i=1 Primal feasibility: g i (x ) 0 for i = 1,, m. Dual feasibility: µ i 0 for i = 1,, m. Complementary slackness: µ i g i (x ) = 0, i = 1,, m.

14 Generalized Lagrange Methods Consider the minimization problem Equality- Inequality- Mixture- min x f (x) subject to g i (x) 0, i = 1,, m. h j (x) = 0, j = 1,, k. Introduce two sets of Lagrange multipliers: Define the Lagrange function: µ 1 0,, µ m 0, λ 1,, λ k. L(x, µ, λ) = f (x) m µ i g i (x) + i=1 k λ j h j (x) j=1 µ i s and λ j s are dual variables or slack variables.

15 KKT Necessary Optimality Conditions Equality- Inequality- Mixture- If x is a local minimum under the above inequality constraints, then there exist some (µ, λ ) such that Stationarity: x L(x, µ, λ ) = 0, i.e., x f (x ) Primal feasibility: m µ i x g i (x ) + i=1 k λ j x h j (x ) = 0. j=1 g i (x ) 0, i = 1,, m; h j (x ) = 0, j = 1,, k. Dual feasibility: µ i 0 for i = 1,, m. Complementary slackness: µ i g i (x ) = 0, i = 1,, m.

16 Sufficient Optimality Conditions Equality- Inequality- Mixture- The above necessary conditions are also sufficient for optimality, if f, g i s are continuously differentiable. f and g i s are convex functions, h j s are linear constraints. Under this case, There exists a solution x satisfying the KKT condition, x is actually a global optimum.

17 Quadratic Programming (QP) Quadratic Programming (QP) The objective function is quadratic in x The constraints are linear in x. Consider the minimization problem min x f (x) = 1 2 xt Qx + c T x, subject to Ax b, Ex = d. where Q is a symmetric matrix of d d, A is a matrix of m d, E is a matrix of k d, c R d, b R m, and d R k. Example: SVM

18 Properties of QP Unconstrained Optimization Quadratic Programming (QP) If Q is a non-negative definite matrix, then f (x) is convex. If Q is a non-negative definite, the above QP has a global minimizer if there exists at least one x satisfying the constraints and f (x) is bounded below in the feasible region. If Q is a positive definite, the above QP has a unique global minimizer. If Q = 0, the problem becomes a linear programming (LP). From optimization theory, If Q is non-negative definite, the necessary and sufficient condition for a point x to be a global minimizer is for it to satisfy the KKT condition.

19 Solving QP Unconstrained Optimization Quadratic Programming (QP) If there are only equality constraints, then the QP can be solved by a linear system. In general, a variety of methods for solving the QP are commonly used, including Methods: ellipsoid, interior point, active set, conjugate gradient, etc Software packages: Matlab, CPLEX, AMPL, GAMS, etc. For positive definite Q, the ellipsoid method solves the problem in polynomial time. In other words, the running time is upper bounded by a polynomial in the size of the input for the algorithm, i.e., T = O(d k ) for some k.

20 Dual Problem for QP Quadratic Programming (QP) Consider the minimization problem The Lagrange function. min f (x) = 1 x 2 xt Qx, subject to Ax b. L(x, λ) = f (x) + λ T (Ax b). We need to minimize L with respect to x and maximize L with respect to λ. The dual function is defined as a function of dual variables only by solving x first with any fixed λ: G(λ) = inf x L(x, λ).

21 Dual Problem Derivation for QP Quadratic Programming (QP) With fixed λ, we solve the equation L x (x, λ) = 0 and obtain x = Q 1 A T λ. Plug this into L, we get the dual function The dual problem of the QP is G(λ) = 1 2 λt AQ 1 A T b T λ. max G(λ) = 1 λ 2 λt AQ 1 A T b T λ, subject to λ 0. We first find λ = arg max λ G(λ), and then compute x = Q 1 A T λ. The idea is similar to the profile method.

5. Duality. Lagrangian

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

More information

14. Duality. ˆ Upper and lower bounds. ˆ General duality. ˆ Constraint qualifications. ˆ Counterexample. ˆ Complementary slackness.

14. Duality. ˆ Upper and lower bounds. ˆ General duality. ˆ Constraint qualifications. ˆ Counterexample. ˆ Complementary slackness. CS/ECE/ISyE 524 Introduction to Optimization Spring 2016 17 14. Duality ˆ Upper and lower bounds ˆ General duality ˆ Constraint qualifications ˆ Counterexample ˆ Complementary slackness ˆ Examples ˆ Sensitivity

More information

Convex Optimization & Lagrange Duality

Convex 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 information

Convex Optimization M2

Convex Optimization M2 Convex Optimization M2 Lecture 3 A. d Aspremont. Convex Optimization M2. 1/49 Duality A. d Aspremont. Convex Optimization M2. 2/49 DMs DM par email: dm.daspremont@gmail.com A. d Aspremont. Convex Optimization

More information

Convex Optimization Boyd & Vandenberghe. 5. Duality

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

More information

ICS-E4030 Kernel Methods in Machine Learning

ICS-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 information

Lecture: Duality.

Lecture: Duality. Lecture: Duality http://bicmr.pku.edu.cn/~wenzw/opt-2016-fall.html Acknowledgement: this slides is based on Prof. Lieven Vandenberghe s lecture notes Introduction 2/35 Lagrange dual problem weak and strong

More information

Support Vector Machines: Maximum Margin Classifiers

Support Vector Machines: Maximum Margin Classifiers Support Vector Machines: Maximum Margin Classifiers Machine Learning and Pattern Recognition: September 16, 2008 Piotr Mirowski Based on slides by Sumit Chopra and Fu-Jie Huang 1 Outline What is behind

More information

Lecture: Duality of LP, SOCP and SDP

Lecture: Duality of LP, SOCP and SDP 1/33 Lecture: Duality of LP, SOCP and SDP Zaiwen Wen Beijing International Center For Mathematical Research Peking University http://bicmr.pku.edu.cn/~wenzw/bigdata2017.html wenzw@pku.edu.cn Acknowledgement:

More information

Duality. Lagrange dual problem weak and strong duality optimality conditions perturbation and sensitivity analysis generalized inequalities

Duality. Lagrange dual problem weak and strong duality optimality conditions perturbation and sensitivity analysis generalized inequalities Duality Lagrange dual problem weak and strong duality optimality conditions perturbation and sensitivity analysis generalized inequalities Lagrangian Consider the optimization problem in standard form

More information

Convex Optimization. Dani Yogatama. School of Computer Science, Carnegie Mellon University, Pittsburgh, PA, USA. February 12, 2014

Convex Optimization. Dani Yogatama. School of Computer Science, Carnegie Mellon University, Pittsburgh, PA, USA. February 12, 2014 Convex Optimization Dani Yogatama School of Computer Science, Carnegie Mellon University, Pittsburgh, PA, USA February 12, 2014 Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12,

More information

Nonlinear Optimization: What s important?

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

More information

I.3. LMI DUALITY. Didier HENRION EECI Graduate School on Control Supélec - Spring 2010

I.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 information

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

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

More information

CS-E4830 Kernel Methods in Machine Learning

CS-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 information

Lecture 6: Conic Optimization September 8

Lecture 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 information

EE/AA 578, Univ of Washington, Fall Duality

EE/AA 578, Univ of Washington, Fall Duality 7. Duality EE/AA 578, Univ of Washington, Fall 2016 Lagrange dual problem weak and strong duality geometric interpretation optimality conditions perturbation and sensitivity analysis examples generalized

More information

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

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

More information

Introduction to Machine Learning Prof. Sudeshna Sarkar Department of Computer Science and Engineering Indian Institute of Technology, Kharagpur

Introduction to Machine Learning Prof. Sudeshna Sarkar Department of Computer Science and Engineering Indian Institute of Technology, Kharagpur Introduction to Machine Learning Prof. Sudeshna Sarkar Department of Computer Science and Engineering Indian Institute of Technology, Kharagpur Module - 5 Lecture - 22 SVM: The Dual Formulation Good morning.

More information

4TE3/6TE3. Algorithms for. Continuous Optimization

4TE3/6TE3. Algorithms for. Continuous Optimization 4TE3/6TE3 Algorithms for Continuous Optimization (Duality in Nonlinear Optimization ) Tamás TERLAKY Computing and Software McMaster University Hamilton, January 2004 terlaky@mcmaster.ca Tel: 27780 Optimality

More information

CS711008Z Algorithm Design and Analysis

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

More information

CE 191: Civil & Environmental Engineering Systems Analysis. LEC 17 : Final Review

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

More information

Convex Optimization and Support Vector Machine

Convex Optimization and Support Vector Machine Convex Optimization and Support Vector Machine Problem 0. Consider a two-class classification problem. The training data is L n = {(x 1, t 1 ),..., (x n, t n )}, where each t i { 1, 1} and x i R p. We

More information

Constrained Optimization and Lagrangian Duality

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

More information

Optimality Conditions for Constrained Optimization

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

More information

The Karush-Kuhn-Tucker conditions

The Karush-Kuhn-Tucker conditions Chapter 6 The Karush-Kuhn-Tucker conditions 6.1 Introduction In this chapter we derive the first order necessary condition known as Karush-Kuhn-Tucker (KKT) conditions. To this aim we introduce the alternative

More information

Constrained Optimization

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

More information

Convex Optimization and SVM

Convex Optimization and SVM Convex Optimization and SVM Problem 0. Cf lecture notes pages 12 to 18. Problem 1. (i) A slab is an intersection of two half spaces, hence convex. (ii) A wedge is an intersection of two half spaces, hence

More information

Tutorial on Convex Optimization: Part II

Tutorial on Convex Optimization: Part II Tutorial on Convex Optimization: Part II Dr. Khaled Ardah Communications Research Laboratory TU Ilmenau Dec. 18, 2018 Outline Convex Optimization Review Lagrangian Duality Applications Optimal Power Allocation

More information

Lagrange Duality. Daniel P. Palomar. Hong Kong University of Science and Technology (HKUST)

Lagrange Duality. Daniel P. Palomar. Hong Kong University of Science and Technology (HKUST) Lagrange Duality Daniel P. Palomar Hong Kong University of Science and Technology (HKUST) ELEC5470 - Convex Optimization Fall 2017-18, HKUST, Hong Kong Outline of Lecture Lagrangian Dual function Dual

More information

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

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

More information

Introduction to Mathematical Programming IE406. Lecture 10. Dr. Ted Ralphs

Introduction 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 information

Extreme Abridgment of Boyd and Vandenberghe s Convex Optimization

Extreme 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 information

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

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

More information

Lecture 2: Linear SVM in the Dual

Lecture 2: Linear SVM in the Dual Lecture 2: Linear SVM in the Dual Stéphane Canu stephane.canu@litislab.eu São Paulo 2015 July 22, 2015 Road map 1 Linear SVM Optimization in 10 slides Equality constraints Inequality constraints Dual formulation

More information

A Brief Review on Convex Optimization

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

More information

Support Vector Machines

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

More information

Introduction 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 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 information

Motivation. Lecture 2 Topics from Optimization and Duality. network utility maximization (NUM) problem:

Motivation. Lecture 2 Topics from Optimization and Duality. network utility maximization (NUM) problem: CDS270 Maryam Fazel Lecture 2 Topics from Optimization and Duality Motivation network utility maximization (NUM) problem: consider a network with S sources (users), each sending one flow at rate x s, through

More information

Lecture Notes on Support Vector Machine

Lecture 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

Lecture 7: Convex Optimizations

Lecture 7: Convex Optimizations Lecture 7: Convex Optimizations Radu Balan, David Levermore March 29, 2018 Convex Sets. Convex Functions A set S R n is called a convex set if for any points x, y S the line segment [x, y] := {tx + (1

More information

Linear and Combinatorial Optimization

Linear 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 information

Support vector machines

Support vector machines Support vector machines Guillaume Obozinski Ecole des Ponts - ParisTech SOCN course 2014 SVM, kernel methods and multiclass 1/23 Outline 1 Constrained optimization, Lagrangian duality and KKT 2 Support

More information

More on Lagrange multipliers

More on Lagrange multipliers More on Lagrange multipliers CE 377K April 21, 2015 REVIEW The standard form for a nonlinear optimization problem is min x f (x) s.t. g 1 (x) 0. g l (x) 0 h 1 (x) = 0. h m (x) = 0 The objective function

More information

Linear Support Vector Machine. Classification. Linear SVM. Huiping Cao. Huiping Cao, Slide 1/26

Linear Support Vector Machine. Classification. Linear SVM. Huiping Cao. Huiping Cao, Slide 1/26 Huiping Cao, Slide 1/26 Classification Linear SVM Huiping Cao linear hyperplane (decision boundary) that will separate the data Huiping Cao, Slide 2/26 Support Vector Machines rt Vector Find a linear Machines

More information

ISM206 Lecture Optimization of Nonlinear Objective with Linear Constraints

ISM206 Lecture Optimization of Nonlinear Objective with Linear Constraints ISM206 Lecture Optimization of Nonlinear Objective with Linear Constraints Instructor: Prof. Kevin Ross Scribe: Nitish John October 18, 2011 1 The Basic Goal The main idea is to transform a given constrained

More information

Machine Learning A Geometric Approach

Machine Learning A Geometric Approach Machine Learning A Geometric Approach CIML book Chap 7.7 Linear Classification: Support Vector Machines (SVM) Professor Liang Huang some slides from Alex Smola (CMU) Linear Separator Ham Spam From Perceptron

More information

10701 Recitation 5 Duality and SVM. Ahmed Hefny

10701 Recitation 5 Duality and SVM. Ahmed Hefny 10701 Recitation 5 Duality and SVM Ahmed Hefny Outline Langrangian and Duality The Lagrangian Duality Eamples Support Vector Machines Primal Formulation Dual Formulation Soft Margin and Hinge Loss Lagrangian

More information

Scientific Computing: Optimization

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

More information

Optimization Problems with Constraints - introduction to theory, numerical Methods and applications

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

More information

Chap 2. Optimality conditions

Chap 2. Optimality conditions Chap 2. Optimality conditions Version: 29-09-2012 2.1 Optimality conditions in unconstrained optimization Recall the definitions of global, local minimizer. Geometry of minimization Consider for f C 1

More information

Generalization to inequality constrained problem. Maximize

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

More information

Interior Point Algorithms for Constrained Convex Optimization

Interior Point Algorithms for Constrained Convex Optimization Interior Point Algorithms for Constrained Convex Optimization Chee Wei Tan CS 8292 : Advanced Topics in Convex Optimization and its Applications Fall 2010 Outline Inequality constrained minimization problems

More information

Machine Learning. Support Vector Machines. Manfred Huber

Machine 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 information

Optimality, Duality, Complementarity for Constrained Optimization

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

More information

10 Numerical methods for constrained problems

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

More information

Machine Learning. Lecture 6: Support Vector Machine. Feng Li.

Machine Learning. Lecture 6: Support Vector Machine. Feng Li. Machine Learning Lecture 6: Support Vector Machine Feng Li fli@sdu.edu.cn https://funglee.github.io School of Computer Science and Technology Shandong University Fall 2018 Warm Up 2 / 80 Warm Up (Contd.)

More information

On the Method of Lagrange Multipliers

On the Method of Lagrange Multipliers On the Method of Lagrange Multipliers Reza Nasiri Mahalati November 6, 2016 Most of what is in this note is taken from the Convex Optimization book by Stephen Boyd and Lieven Vandenberghe. This should

More information

Numerical Optimization. Review: Unconstrained Optimization

Numerical 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 information

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

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

More information

Numerical Optimization

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

More information

Lecture 13: Constrained optimization

Lecture 13: Constrained optimization 2010-12-03 Basic ideas A nonlinearly constrained problem must somehow be converted relaxed into a problem which we can solve (a linear/quadratic or unconstrained problem) We solve a sequence of such problems

More information

EE364a Review Session 5

EE364a 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 information

12. Interior-point methods

12. 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 information

Support Vector Machine (SVM) and Kernel Methods

Support Vector Machine (SVM) and Kernel Methods Support Vector Machine (SVM) and Kernel Methods CE-717: Machine Learning Sharif University of Technology Fall 2016 Soleymani Outline Margin concept Hard-Margin SVM Soft-Margin SVM Dual Problems of Hard-Margin

More information

Part 4: Active-set methods for linearly constrained optimization. Nick Gould (RAL)

Part 4: Active-set methods for linearly constrained optimization. Nick Gould (RAL) Part 4: Active-set methods for linearly constrained optimization Nick Gould RAL fx subject to Ax b Part C course on continuoue optimization LINEARLY CONSTRAINED MINIMIZATION fx subject to Ax { } b where

More information

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

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

More information

Support Vector Machines for Regression

Support Vector Machines for Regression COMP-566 Rohan Shah (1) Support Vector Machines for Regression Provided with n training data points {(x 1, y 1 ), (x 2, y 2 ),, (x n, y n )} R s R we seek a function f for a fixed ɛ > 0 such that: f(x

More information

Tutorial on Convex Optimization for Engineers Part II

Tutorial on Convex Optimization for Engineers Part II Tutorial on Convex Optimization for Engineers Part II M.Sc. Jens Steinwandt Communications Research Laboratory Ilmenau University of Technology PO Box 100565 D-98684 Ilmenau, Germany jens.steinwandt@tu-ilmenau.de

More information

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

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

More information

Nonlinear Programming (Hillier, Lieberman Chapter 13) CHEM-E7155 Production Planning and Control

Nonlinear 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 information

Primal/Dual Decomposition Methods

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

More information

Part IB Optimisation

Part IB Optimisation Part IB Optimisation Theorems Based on lectures by F. A. Fischer Notes taken by Dexter Chua Easter 2015 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after

More information

The Lagrangian L : R d R m R r R is an (easier to optimize) lower bound on the original problem:

The Lagrangian L : R d R m R r R is an (easier to optimize) lower bound on the original problem: HT05: SC4 Statistical Data Mining and Machine Learning Dino Sejdinovic Department of Statistics Oxford Convex Optimization and slides based on Arthur Gretton s Advanced Topics in Machine Learning course

More information

Computational Finance

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

More information

Lecture 3: Lagrangian duality and algorithms for the Lagrangian dual problem

Lecture 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 information

LECTURE 7 Support vector machines

LECTURE 7 Support vector machines LECTURE 7 Support vector machines SVMs have been used in a multitude of applications and are one of the most popular machine learning algorithms. We will derive the SVM algorithm from two perspectives:

More information

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

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

More information

IE 5531 Midterm #2 Solutions

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

More information

CS711008Z Algorithm Design and Analysis

CS711008Z Algorithm Design and Analysis .. CS711008Z Algorithm Design and Analysis Lecture 9. Algorithm design technique: Linear programming and duality Dongbo Bu Institute of Computing Technology Chinese Academy of Sciences, Beijing, China

More information

LECTURE 10 LECTURE OUTLINE

LECTURE 10 LECTURE OUTLINE LECTURE 10 LECTURE OUTLINE Min Common/Max Crossing Th. III Nonlinear Farkas Lemma/Linear Constraints Linear Programming Duality Convex Programming Duality Optimality Conditions Reading: Sections 4.5, 5.1,5.2,

More information

Optimisation in Higher Dimensions

Optimisation in Higher Dimensions CHAPTER 6 Optimisation in Higher Dimensions Beyond optimisation in 1D, we will study two directions. First, the equivalent in nth dimension, x R n such that f(x ) f(x) for all x R n. Second, constrained

More information

Support Vector Machine (SVM) and Kernel Methods

Support Vector Machine (SVM) and Kernel Methods Support Vector Machine (SVM) and Kernel Methods CE-717: Machine Learning Sharif University of Technology Fall 2015 Soleymani Outline Margin concept Hard-Margin SVM Soft-Margin SVM Dual Problems of Hard-Margin

More information

1. f(β) 0 (that is, β is a feasible point for the constraints)

1. f(β) 0 (that is, β is a feasible point for the constraints) xvi 2. The lasso for linear models 2.10 Bibliographic notes Appendix Convex optimization with constraints In this Appendix we present an overview of convex optimization concepts that are particularly useful

More information

Cheng Soon Ong & Christian Walder. Canberra February June 2018

Cheng Soon Ong & Christian Walder. Canberra February June 2018 Cheng Soon Ong & Christian Walder Research Group and College of Engineering and Computer Science Canberra February June 2018 Outlines Overview Introduction Linear Algebra Probability Linear Regression

More information

Homework Set #6 - Solutions

Homework Set #6 - Solutions EE 15 - Applications of Convex Optimization in Signal Processing and Communications Dr Andre Tkacenko JPL Third Term 11-1 Homework Set #6 - Solutions 1 a The feasible set is the interval [ 4] The unique

More information

ELE539A: 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 ELE539A: Optimization of Communication Systems Lecture 6: Quadratic Programming, Geometric Programming, and Applications Professor M. Chiang Electrical Engineering Department, Princeton University February

More information

Primal-dual Subgradient Method for Convex Problems with Functional Constraints

Primal-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 information

Quiz Discussion. IE417: Nonlinear Programming: Lecture 12. Motivation. Why do we care? Jeff Linderoth. 16th March 2006

Quiz 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 information

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

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

More information

Algorithms for constrained local optimization

Algorithms for constrained local optimization Algorithms for constrained local optimization Fabio Schoen 2008 http://gol.dsi.unifi.it/users/schoen Algorithms for constrained local optimization p. Feasible direction methods Algorithms for constrained

More information

Pattern Classification, and Quadratic Problems

Pattern Classification, and Quadratic Problems Pattern Classification, and Quadratic Problems (Robert M. Freund) March 3, 24 c 24 Massachusetts Institute of Technology. 1 1 Overview Pattern Classification, Linear Classifiers, and Quadratic Optimization

More information

Support Vector Machine (SVM) & Kernel CE-717: Machine Learning Sharif University of Technology. M. Soleymani Fall 2012

Support Vector Machine (SVM) & Kernel CE-717: Machine Learning Sharif University of Technology. M. Soleymani Fall 2012 Support Vector Machine (SVM) & Kernel CE-717: Machine Learning Sharif University of Technology M. Soleymani Fall 2012 Linear classifier Which classifier? x 2 x 1 2 Linear classifier Margin concept x 2

More information

Machine Learning and Data Mining. Support Vector Machines. Kalev Kask

Machine Learning and Data Mining. Support Vector Machines. Kalev Kask Machine Learning and Data Mining Support Vector Machines Kalev Kask Linear classifiers Which decision boundary is better? Both have zero training error (perfect training accuracy) But, one of them seems

More information

What s New in Active-Set Methods for Nonlinear Optimization?

What s New in Active-Set Methods for Nonlinear Optimization? What s New in Active-Set Methods for Nonlinear Optimization? Philip E. Gill Advances in Numerical Computation, Manchester University, July 5, 2011 A Workshop in Honor of Sven Hammarling UCSD Center for

More information

Linear classifiers selecting hyperplane maximizing separation margin between classes (large margin classifiers)

Linear classifiers selecting hyperplane maximizing separation margin between classes (large margin classifiers) Support vector machines In a nutshell Linear classifiers selecting hyperplane maximizing separation margin between classes (large margin classifiers) Solution only depends on a small subset of training

More information

Duality. Geoff Gordon & Ryan Tibshirani Optimization /

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

More information

Linear classifiers selecting hyperplane maximizing separation margin between classes (large margin classifiers)

Linear classifiers selecting hyperplane maximizing separation margin between classes (large margin classifiers) Support vector machines In a nutshell Linear classifiers selecting hyperplane maximizing separation margin between classes (large margin classifiers) Solution only depends on a small subset of training

More information