Lecture 7: Convex Optimizations


 Amberlynn Richardson
 1 years ago
 Views:
Transcription
1 Lecture 7: Convex Optimizations Radu Balan, David Levermore March 29, 2018
2 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 t)y, 0 t 1} is included in S, [x, y] S. A function f : S R is called convex if for any x, y S and 0 t 1, f (t x + (1 t)y) t f (x) + (1 t)f (y). Here S is supposed to be a convex set in R n. Equivalently, f is convex if its epigraph is a convex set in R n+1. Epigraph: {(x, u) ; x S, u f (x)}. A function f : S R is called strictly convex if for any x y S and 0 < t < 1, f (t x + (1 t)y) < t f (x) + (1 t)f (y).
3 Convex Optimization Problems The general form of a convex optimization problem: min f (x) x S where S is a closed convex set, and f is a convex function on S. Properties: 1 Any local minimum is a global minimum. The set of minimizers is a convex subset of S. 2 If f is strictly convex, then the minimizer is unique: there is only one local minimizer. In general S is defined by equality and inequality constraints: S = {g i (x) 0, 1 i p} {h j (x) = 0, 1 j m}. Typically h j are required to be affine: h j (x) = a T x + b.
4 Convex Programs The hiarchy of convex optimization problems: 1 Linear Programs: Linear criterion with linear constraints 2 Quadratic Problems: Quadratic Programs (QP); Quadratically Constrained Quadratic Problems (QCQP); SecondOrder Cone Program (SOCP) 3 SemiDefinite Programs(SDP) Popular approach/solution: Primaldual interiorpoint using Newton s method (second order)
5 Linear Programs LP and standard LPs Linear Program: minimize subject to where G R m n, A R p n. c T x + d Gx h Ax = b
6 Linear Programs LP and standard LPs Linear Program: minimize subject to where G R m n, A R p n. c T x + d Gx h Ax = b Standard form LP: minimize subject to c T x Ax = b x 0 Inequality form LP: minimize subject to c T x Ax b
7 Linear Program: Example Basis Pursuit Cosnider a system of linear equations: Ax = b with more columns (unknowns) than rows (equations), i.e. A is fat. We want to find the sparsest solution minimize x 0 subject to Ax = b where x 0 denotes the number of nonzero entries in x (i.e. the support size). This is a nonconvex, NPhard problem. Instead we solve its socalled convexification : minimize x 1 subject to Ax = b where x 1 = n k=1 x k. It is shown that, under some sonditions (RIP) the solutions of the two problems coincide.
8 Linear Program: Example Basis Pursuit How to turn: into a LP? minimize x 1 subject to Ax = b
9 Linear Program: Example Basis Pursuit How to turn: minimize x 1 subject to Ax = b into a LP? Method 1. Use the following auxiliary variables: y = (y k ) 1 k n so that x k y k, or x k y k 0, x k y k 0: minimize subject to 1 T y Ax = b x y 0 x y 0 y 0
10 Linear Program: Example Basis Pursuit How to turn: into a LP? minimize x 1 subject to Ax = b
11 Linear Program: Example Basis Pursuit How to turn: minimize x 1 subject to Ax = b into a LP? Method 2. Use the following substitutions (positive and negative parts): x k = u k v k, x k = u k + v k, with u k, v k 0: minimize subject to 1 T u + 1 T v A(u v) = b u 0 v 0
12 Quadratic Problems QP: Quadratic Programs 1 minimize 2 x T PX + q T x + r subject to Gx h Ax = b where P = P T 0, P R n n, G R m n, A R p n.
13 Quadratic Problems QP: Quadratic Programs 1 minimize 2 x T PX + q T x + r subject to Gx h Ax = b where P = P T 0, P R n n, G R m n, A R p n. Example: Constrained Regression (constrained leastsquares). Typical LS problem: min Ax b 2 2 = min x T AX 2b T Ax + b T b has solution: x = A b = (A T A) 1 A T b. Constrained leastsquares: minimize Ax b 2 2 subject to l i x i u i, i = 1,, n
14 Quadratic Problems QCQP: Quadratically Constrained Quadratic Programs 1 minimize 2 x T PX + q T x + r 1 subject to 2 x T P i x + qi T x + r i 0, i = 1,, m Ax = b where P = P T 0, P i = P T i 0, i = 1,, m, P, P i R n n, A R p n.
15 Quadratic Problems QCQP: Quadratically Constrained Quadratic Programs 1 minimize 2 x T PX + q T x + r 1 subject to 2 x T P i x + qi T x + r i 0, i = 1,, m Ax = b where P = P T 0, P i = P T i 0, i = 1,, m, P, P i R n n, A R p n. Remark 1. QP can be solved by QCQP: set P i = Criterion can always be recast in a linear form with unknown [x 0 ; x]: minimize x 0 1 subject to 2 x T Px + q T x x 0 + r x T P i x + qi T x + r i 0, i = 1,, m Ax = b
16 Problem: For p = 2 recast it as a SOCP. For p = 1 it is a LP. Quadratic Problems SOCP: SecondOrder Cone Programs minimize f T x subject to A i x + b i 2 ci T x + d i, i = 1,, m Fx = g where A i R n i n, F R p n. SOCP is the most general form of a quadratic problem. QCQP: c i = 0 except for i = 0. Example The placement problem: Given a set of weights {w ij } and of fixed points {x 1,, x L }, find the set of points {x L+1,, x N } that minimize forp 1: min 1 i<j N w ij x i x j p.
17 SemiDefinite Programs SDP and standard SDP SemiDefinite Program with unknown x R n : minimize c T x subject to x 1 F x n F n + G 0 Ax = b where G, F 1,, F n are k k symmetric matrices in S k, A R p n.
18 SemiDefinite Programs SDP and standard SDP SemiDefinite Program with unknown x R n : minimize c T x subject to x 1 F x n F n + G 0 Ax = b where G, F 1,, F n are k k symmetric matrices in S k, A R p n. Standard form SDP: minimize subject to trace(cx) trace(a i X) = b i X = X T 0 Inequality form SDP: minimize c T x subject to x 1 A 1 + x n A n B
19 CVX Matlab package Downloadable from: Follows Disciplined Convex Programming à la Boyd [2]. m = 20; n = 10; p = 4; A = randn(m,n); b = randn(m,1); C = randn(p,n); d = randn(p,1); e = rand; cvx_begin variable x(n) minimize( norm( A * x  b, 2 ) ) subject to C * x == d norm( x, Inf ) <= e cvx_end min Cx = d x e Ax b
20 CVX SDP Example cvx_begin sdp cvx_end variable X(n,n) semidefinite; minimize trace(x); subject to abs(trace(e1*x)d1)<=epsx; abs(trace(e2*x)d2)<=epsx; minimize subject to trace(x) trace(e 1 X) d 1 ε trace(e 2 X) d 2 ε X = X T 0
21 Dual Problem Lagrangian Primal Problem: p = minimize f 0 (x) subject to f i (x) 0, i = 1,, m h i (x) = 0, i = 1,, p Define the Lagrangian, L : R n R m R p R: m p L(x, λ, ν) = f 0 (x) + λ i f i (x) + ν i h i (x) i=1 i=1 Variables λ R m and ν R p are called dual variables, or Lagrange multipliers.
22 Dual Problem Lagrange Dual Function The Lagrange dual function (or the dual function) is given by: g(λ, ν) = inf L(x, λ, ν) = inf x D x D ( ) m p f 0 (x) + λ i f i (x) + ν i h i (x) i=1 i=1 where D R n is the domain of definition of all functions. Remark 1. Key estimate: For any λ 0, and any ν, g(λ, ν) p because g(λ, ν) = L(x, λ, ν) = f 0 (x ) + λ T f (x ) + ν T h(x ) f 0 (x ). 2. g(λ, ν) is a concave function regardless of problem convexity.
23 Dual Problem The dual problem The dual problem is given by: d = maximize g(λ, ν) subject to λ 0
24 Dual Problem The dual problem The dual problem is given by: d = maximize g(λ, ν) subject to λ 0 Remark 1. The dual problem is always a convex optimization problem (maximization of a concave function, with convex constraints). 2. We always have weak duality: d p
25 Dual Problem The duality gap. Strong duality The duality gap is p d. If the primal is a convex optimization problem: p = minimize f 0 (x) subject to f i (x) 0, i = 1,, m Ax = b and the Slater s constraint qualification condition holds: there is a feasible x relint(d) so that f i (x) < 0, i = 1,, m, then the strong duality holds: d = p. (Slater s condition is a sufficient, not a necessary condition.)
26 The KarushKuhnTucker (KKT) Conditions Necessary Conditions Assume f 0, f 1,, f m, h 1,, h p are differentiable with open domains. Assume x and (λ, ν ) be any primal and dual optimal points with zero duality gap. It follows that L(x, λ, ν ) x=x = 0 and g(λ, ν ) = L(x, λ, ν ) = f 0 (x ). We obtain the following set of equations called the KKT conditions: f i (x ) 0, i = 1,, m h i (x ) = 0, i = 1,, p λ i 0, i = 1,, m λ i f i (x ) = 0, i = 1,, m m p f 0 (x ) + λ i f i (x ) + νi h i (x ) = 0 i=1 i=1
27 The KarushKuhnTucker (KKT) Conditions Necessary Conditions Assume f 0, f 1,, f m, h 1,, h p are differentiable with open domains. Assume x and (λ, ν ) be any primal and dual optimal points with zero duality gap. It follows that L(x, λ, ν ) x=x = 0 and g(λ, ν ) = L(x, λ, ν ) = f 0 (x ). We obtain the following set of equations called the KKT conditions: f i (x ) 0, i = 1,, m h i (x ) = 0, i = 1,, p λ i 0, i = 1,, m λ i f i (x ) = 0, i = 1,, m m p f 0 (x ) + λ i f i (x ) + νi h i (x ) = 0 i=1 i=1 Remark: λ i f i(x ) = 0 for each i are called the complementary slackness conditions.
28 The KarushKuhnTucker (KKT) Conditions Sufficient Conditions Assume the primal problem is convex with h(x) = Ax b and f 0,, f m differentiable. Assume x, ( λ, ν) satisfy the KKT conditions: f 0 ( x) + m λ i f i ( x) + A T ν = 0 i=1 f i ( x) 0, i = 1,, m A x = b, i = 1,, p λ i 0, i = 1,, m λ i f i ( x) = 0, i = 1,, m Then x and ( λ, ν) are primal and dual optimal with zero duality gap.
29 The KarushKuhnTucker (KKT) Conditions Sufficient Conditions Assume the primal problem is convex with h(x) = Ax b and f 0,, f m differentiable. Assume x, ( λ, ν) satisfy the KKT conditions: f 0 ( x) + m λ i f i ( x) + A T ν = 0 i=1 f i ( x) 0, i = 1,, m A x = b, i = 1,, p λ i 0, i = 1,, m λ i f i ( x) = 0, i = 1,, m Then x and ( λ, ν) are primal and dual optimal with zero duality gap. A primaldual interiorpoint algorithm is an iterative algorithm that approaches a solution of the KKT system of conditions.
30 PrimalDual InteriorPoint Method Idea: Solve the nonlinear system r t (x, λ, ν) = 0 with r t (x, λ, ν) = f 0 (x) + Df (x) T λ + A T ν diag(λ)f (x) (1/t)1 Ax b, f (x) = f 1 (x). f m (x) and t > 0, using Newton s method (second order). The search direction is obtained by solving the linear system: 2 f 0 (x) + m i=1 λ i 2 f i (x) Df (x) T A T x diag(λ)df (x) diag(f (x)) 0 λ = r t (x, λ, ν). A 0 0 ν At each iteration t increases by the reciprocal of the (surogate) duality gap 1 f (x) T λ.
31 References B. Bollobás, Graph Theory. An Introductory Course, SpringerVerlag (25), (2002). S. Boyd, L. Vandenberghe, Convex Optimization, available online at: boyd/cvxbook/ F. Chung, Spectral Graph Theory, AMS F. Chung, L. Lu, The average distances in random graphs with given expected degrees, Proc. Nat.Acad.Sci F. Chung, L. Lu, V. Vu, The spectra of random graphs with Given Expected Degrees, Internet Math. 1(3), (2004). R. Diestel, Graph Theory, 3rd Edition, SpringerVerlag P. Erdös, A. Rényi, On The Evolution of Random Graphs
32 G. Grimmett, Probability on Graphs. Random Processes on Graphs and Lattices, Cambridge Press C. Hoffman, M. Kahle, E. Paquette, Spectral Gap of Random Graphs and Applications to Random Topology, arxiv: [math.co] 17 Sept A. Javanmard, A. Montanari, Localization from Incomplete Noisy Distance Measurements, arxiv: , Nov. 2012; also ISIT J. Leskovec, J. Kleinberg, C. Faloutsos, Graph Evolution: Densification and Shrinking Diameters, ACM Trans. on Knowl.Disc.Data, 1(1) 2007.
Lecture 9: Laplacian Eigenmaps
Lecture 9: Radu Balan Department of Mathematics, AMSC, CSCAMM and NWC University of Maryland, College Park, MD April 18, 2017 Optimization Criteria Assume G = (V, W ) is a undirected weighted graph with
More informationLecture: 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 informationDuality. 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 informationE5295/5B5749 Convex optimization with engineering applications. Lecture 5. Convex programming and semidefinite programming
E5295/5B5749 Convex optimization with engineering applications Lecture 5 Convex programming and semidefinite programming A. Forsgren, KTH 1 Lecture 5 Convex optimization 2006/2007 Convex quadratic program
More information5. 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 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 informationICSE4030 Kernel Methods in Machine Learning
ICSE4030 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 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 informationConvex 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 informationLecture: Duality.
Lecture: Duality http://bicmr.pku.edu.cn/~wenzw/opt2016fall.html Acknowledgement: this slides is based on Prof. Lieven Vandenberghe s lecture notes Introduction 2/35 Lagrange dual problem weak and strong
More informationLecture 1: Graphs, Adjacency Matrices, Graph Laplacian
Lecture 1: Graphs, Adjacency Matrices, Graph Laplacian Radu Balan January 31, 2017 G = (V, E) An undirected graph G is given by two pieces of information: a set of vertices V and a set of edges E, G =
More informationConvex 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 informationTutorial 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 D98684 Ilmenau, Germany jens.steinwandt@tuilmenau.de
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 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 informationLagrange 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 201718, HKUST, Hong Kong Outline of Lecture Lagrangian Dual function Dual
More informationLecture 18: Optimization Programming
Fall, 2016 Outline Unconstrained Optimization 1 Unconstrained Optimization 2 Equalityconstrained Optimization Inequalityconstrained Optimization Mixtureconstrained Optimization 3 Quadratic Programming
More informationCSE4830 Kernel Methods in Machine Learning
CSE4830 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 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 wellwritten and a pleasure to read. The
More informationSemidefinite Programming Basics and Applications
Semidefinite Programming Basics and Applications Ray Pörn, principal lecturer Åbo Akademi University Novia University of Applied Sciences Content What is semidefinite programming (SDP)? How to represent
More information14. 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 information12. Interiorpoint methods
12. Interiorpoint methods Convex Optimization Boyd & Vandenberghe inequality constrained minimization logarithmic barrier function and central path barrier method feasibility and phase I methods complexity
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 informationMotivation. 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 informationLagrangian Duality and Convex Optimization
Lagrangian Duality and Convex Optimization David Rosenberg New York University February 11, 2015 David Rosenberg (New York University) DSGA 1003 February 11, 2015 1 / 24 Introduction Why Convex Optimization?
More informationKarushKuhnTucker Conditions. Lecturer: Ryan Tibshirani Convex Optimization /36725
KarushKuhnTucker Conditions Lecturer: Ryan Tibshirani Convex Optimization 10725/36725 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 informationCSCI : Optimization and Control of Networks. Review on Convex Optimization
CSCI7000016: 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 informationA 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 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 informationTutorial 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 informationConvex 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 informationLecture 4: Phase Transition in Random Graphs
Lecture 4: Phase Transition in Random Graphs Radu Balan February 21, 2017 Distributions Today we discuss about phase transition in random graphs. Recall on the ErdödRényi class G n,p of random graphs,
More informationOptimization for Machine Learning
Optimization for Machine Learning (Problems; Algorithms  A) SUVRIT SRA Massachusetts Institute of Technology PKU Summer School on Data Science (July 2017) Course materials http://suvrit.de/teaching.html
More information12. Interiorpoint methods
12. Interiorpoint methods Convex Optimization Boyd & Vandenberghe inequality constrained minimization logarithmic barrier function and central path barrier method feasibility and phase I methods complexity
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 00 The Relaxation Theorem 1 Problem: find f := infimum f(x), x subject to x S, (1a) (1b) where f : R n R
More informationEE/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 informationConvex Optimization and Modeling
Convex Optimization and Modeling Duality Theory and Optimality Conditions 5th lecture, 12.05.2010 Jun.Prof. Matthias Hein Program of today/next lecture Lagrangian and duality: the Lagrangian the dual
More informationApplications of Linear Programming
Applications of Linear Programming lecturer: András London University of Szeged Institute of Informatics Department of Computational Optimization Lecture 9 Nonlinear programming In case of LP, the goal
More informationOn 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 informationLecture 1: From Data to Graphs, Weighted Graphs and Graph Laplacian
Lecture 1: From Data to Graphs, Weighted Graphs and Graph Laplacian Radu Balan February 5, 2018 Datasets diversity: Social Networks: Set of individuals ( agents, actors ) interacting with each other (e.g.,
More informationHomework Set #6  Solutions
EE 15  Applications of Convex Optimization in Signal Processing and Communications Dr Andre Tkacenko JPL Third Term 111 Homework Set #6  Solutions 1 a The feasible set is the interval [ 4] The unique
More informationDuality. Geoff Gordon & Ryan Tibshirani Optimization /
Duality Geoff Gordon & Ryan Tibshirani Optimization 10725 / 36725 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 informationSupport 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 FuJie Huang 1 Outline What is behind
More informationSupport 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 informationThe 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 information4TE3/6TE3. 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 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 ndimensional space, a hyper plane is defined by ω T x + b = 0 (1) where ω R n is
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 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 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 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 informationConstrained Optimization
1 / 22 Constrained Optimization ME598/494 Lecture Max Yi Ren Department of Mechanical Engineering, Arizona State University March 30, 2015 2 / 22 1. Equality constraints only 1.1 Reduced gradient 1.2 Lagrange
More informationLectures 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 informationLinear and nonlinear programming
Linear and nonlinear programming Benjamin Recht March 11, 2005 The Gameplan Constrained Optimization Convexity Duality Applications/Taxonomy 1 Constrained Optimization minimize f(x) subject to g j (x)
More informationOptimization. YuhJye Lee. March 28, Data Science and Machine Intelligence Lab National Chiao Tung University 1 / 40
Optimization YuhJye 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 informationOptimality, Duality, Complementarity for Constrained Optimization
Optimality, Duality, Complementarity for Constrained Optimization Stephen Wright University of WisconsinMadison May 2014 Wright (UWMadison) Optimality, Duality, Complementarity May 2014 1 / 41 Linear
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 informationLagrangian Duality Theory
Lagrangian Duality Theory Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A. http://www.stanford.edu/ yyye Chapter 14.14 1 Recall Primal and Dual
More informationELE539A: Optimization of Communication Systems Lecture 15: Semidefinite Programming, Detection and Estimation Applications
ELE539A: Optimization of Communication Systems Lecture 15: Semidefinite Programming, Detection and Estimation Applications Professor M. Chiang Electrical Engineering Department, Princeton University March
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 informationConvex Optimization and l 1 minimization
Convex Optimization and l 1 minimization Sangwoon Yun Computational Sciences Korea Institute for Advanced Study December 11, 2009 2009 NIMS Thematic Winter School Outline I. Convex Optimization II. l
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 information4. Algebra and Duality
41 Algebra and Duality P. Parrilo and S. Lall, CDC 2003 2003.12.07.01 4. Algebra and Duality Example: nonconvex polynomial optimization Weak duality and duality gap The dual is not intrinsic The cone
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 informationHW1 solutions. 1. α Ef(x) β, where Ef(x) is the expected value of f(x), i.e., Ef(x) = n. i=1 p if(a i ). (The function f : R R is given.
HW1 solutions Exercise 1 (Some sets of probability distributions.) Let x be a realvalued random variable with Prob(x = a i ) = p i, i = 1,..., n, where a 1 < a 2 < < a n. Of course p R n lies in the standard
More informationConstrained optimization
Constrained optimization DSGA 1013 / MATHGA 2824 Optimizationbased Data Analysis http://www.cims.nyu.edu/~cfgranda/pages/obda_fall17/index.html Carlos FernandezGranda Compressed sensing Convex constrained
More informationISM206 Lecture Optimization of Nonlinear Objective with Linear Constraints
ISM206 Lecture Optimization of Nonlinear Objective with Linear Constraints Instructor: Prof. Kevin Ross Scribe: Nitish John October 18, 2011 1 The Basic Goal The main idea is to transform a given constrained
More informationPart 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 informationCS711008Z 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 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 informationAdditional Homework Problems
Additional Homework Problems Robert M. Freund April, 2004 2004 Massachusetts Institute of Technology. 1 2 1 Exercises 1. Let IR n + denote the nonnegative orthant, namely IR + n = {x IR n x j ( ) 0,j =1,...,n}.
More informationInterior 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 informationAgenda. 1 Cone programming. 2 Convex cones. 3 Generalized inequalities. 4 Linear programming (LP) 5 Secondorder cone programming (SOCP)
Agenda 1 Cone programming 2 Convex cones 3 Generalized inequalities 4 Linear programming (LP) 5 Secondorder cone programming (SOCP) 6 Semidefinite programming (SDP) 7 Examples Optimization problem in
More informationLecture: Introduction to LP, SDP and SOCP
Lecture: Introduction to LP, SDP and SOCP Zaiwen Wen Beijing International Center For Mathematical Research Peking University http://bicmr.pku.edu.cn/~wenzw/bigdata2015.html wenzw@pku.edu.cn Acknowledgement:
More informationUC Berkeley Department of Electrical Engineering and Computer Science. EECS 227A Nonlinear and Convex Optimization. Solutions 6 Fall 2009
UC Berkele Department of Electrical Engineering and Computer Science EECS 227A Nonlinear and Convex Optimization Solutions 6 Fall 2009 Solution 6.1 (a) p = 1 (b) The Lagrangian is L(x,, λ) = e x + λx 2
More informationSemidefinite Programming
Semidefinite Programming Notes by Bernd Sturmfels for the lecture on June 26, 208, in the IMPRS Ringvorlesung Introduction to Nonlinear Algebra The transition from linear algebra to nonlinear algebra has
More informationA Tutorial on Convex Optimization II: Duality and Interior Point Methods
A Tutorial on Convex Optimization II: Duality and Interior Point Methods Haitham Hindi Palo Alto Research Center (PARC), Palo Alto, California 94304 email: hhindi@parc.com Abstract In recent years, convex
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 informationConvex Optimization Overview (cnt d)
Conve Optimization Overview (cnt d) Chuong B. Do November 29, 2009 During last week s section, we began our study of conve optimization, the study of mathematical optimization problems of the form, minimize
More informationConvex 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 informationInterior Point Methods. We ll discuss linear programming first, followed by three nonlinear problems. Algorithms for Linear Programming Problems
AMSC 607 / CMSC 764 Advanced Numerical Optimization Fall 2008 UNIT 3: Constrained Optimization PART 4: Introduction to Interior Point Methods Dianne P. O Leary c 2008 Interior Point Methods We ll discuss
More informationminimize 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 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 informationAlgorithms for nonlinear programming problems II
Algorithms for nonlinear programming problems II Martin Branda Charles University in Prague Faculty of Mathematics and Physics Department of Probability and Mathematical Statistics Computational Aspects
More informationNumerical 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 informationCSCI 1951G Optimization Methods in Finance Part 09: Interior Point Methods
CSCI 1951G Optimization Methods in Finance Part 09: Interior Point Methods March 23, 2018 1 / 35 This material is covered in S. Boyd, L. Vandenberge s book Convex Optimization https://web.stanford.edu/~boyd/cvxbook/.
More informationConvex Optimization and Modeling
Convex Optimization and Modeling Convex Optimization Fourth lecture, 05.05.2010 Jun.Prof. Matthias Hein Reminder from last time Convex functions: firstorder condition: f(y) f(x) + f x,y x, secondorder
More informationSparse Optimization Lecture: Dual Certificate in l 1 Minimization
Sparse Optimization Lecture: Dual Certificate in l 1 Minimization Instructor: Wotao Yin July 2013 Note scriber: Zheng Sun Those who complete this lecture will know what is a dual certificate for l 1 minimization
More informationAlgorithms for nonlinear programming problems II
Algorithms for nonlinear programming problems II Martin Branda Charles University Faculty of Mathematics and Physics Department of Probability and Mathematical Statistics Computational Aspects of Optimization
More informationSupport 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 informationHomework 4. Convex Optimization /36725
Homework 4 Convex Optimization 10725/36725 Due Friday November 4 at 5:30pm submitted to Christoph Dann in Gates 8013 (Remember to a submit separate writeup for each problem, with your name at the top)
More informationStructural and Multidisciplinary Optimization. P. Duysinx and P. Tossings
Structural and Multidisciplinary Optimization P. Duysinx and P. Tossings 20182019 CONTACTS Pierre Duysinx Institut de Mécanique et du Génie Civil (B52/3) Phone number: 04/366.91.94 Email: P.Duysinx@uliege.be
More informationModule 04 Optimization Problems KKT Conditions & Solvers
Module 04 Optimization Problems KKT Conditions & Solvers Ahmad F. Taha EE 5243: Introduction to CyberPhysical Systems Email: ahmad.taha@utsa.edu Webpage: http://engineering.utsa.edu/ taha/index.html September
More informationLecture 13: Constrained optimization
20101203 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 informationEE364b Homework 2. λ i f i ( x) = 0, i=1
EE364b Prof. S. Boyd EE364b Homework 2 1. Subgradient optimality conditions for nondifferentiable inequality constrained optimization. Consider the problem minimize f 0 (x) subject to f i (x) 0, i = 1,...,m,
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 informationConvex Optimization. (EE227A: UC Berkeley) Lecture 6. Suvrit Sra. (Conic optimization) 07 Feb, 2013
Convex Optimization (EE227A: UC Berkeley) Lecture 6 (Conic optimization) 07 Feb, 2013 Suvrit Sra Organizational Info Quiz coming up on 19th Feb. Project teams by 19th Feb Good if you can mix your research
More informationEE364 Review Session 4
EE364 Review Session 4 EE364 Review Outline: convex optimization examples solving quasiconvex problems by bisection exercise 4.47 1 Convex optimization problems we have seen linear programming (LP) quadratic
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 informationExample: feasibility. Interpretation as formal proof. Example: linear inequalities and Farkas lemma
41 Algebra and Duality P. Parrilo and S. Lall 2006.06.07.01 4. Algebra and Duality Example: nonconvex polynomial optimization Weak duality and duality gap The dual is not intrinsic The cone of valid
More information