Lecture 4: September 12
|
|
- Eustace Mathews
- 5 years ago
- Views:
Transcription
1 10-725/36-725: Conve Optimization Fall 2016 Lecture 4: September 12 Lecturer: Ryan Tibshirani Scribes: Jay Hennig, Yifeng Tao, Sriram Vasudevan Note: LaTeX template courtesy of UC Berkeley EECS dept. Disclaimer: These notes have not been subjected to the usual scrutiny reserved for formal publications. They may be distributed outside this class only with the permission of the Instructor. 4.1 Previous Lecture Eliating Equality Constraints If the problem is of the form f() s.t g i () 0, i = 1,..., m A = b then can be epressed as My + 0 (where A 0 = b and col(m) = null(a)). Doing so allows us to rewrite the above problem as: y f(my + 0 ) s.t g i (My + 0 ) 0, i = 1,..., m Introducing Slack Variables The concept of slack variables is opposite to that of eliating equality constraints. Thus the first formulation in the previous section can be written as:,s f() s.t s i 0, i = 1,..., m g i () + s i = 0, i = 1,..., m A = b This problem however is not conve unless the g i are all affine Relaing Nonaffine Equality Constraints Given an optimization problem f() such that C, we can consider an enlarged set C C and solve f() such that C instead. This is known as relaation, and its optimal value is always lesser than or equal to that of the original problem. 4-1
2 4-2 Lecture 4: September 12 An important special case is that of replacing conve nonaffine equality constraints h j () = 0, j = 1,..., r with h j () 0, j = 1,..., r Eamples 1. Maimum Utility Problem: This problem models investment/consumption. It can be formulated as: ma,b T α t u( t ) t=1 s.t. b t+1 = b t + f(b t ) t, t = 1,..., T 0 t b t, t = 1,..., T with b t being the budget and t being the amount consumed at time t. f is the investment return function, u is the utility function, and both are concave and increasing. The equality constraint is nonaffine, but if we rela it to an inequality, the problem doesn t change (relaation is tight), and the problem is now conve. 2. Principal Component Analysis: Given X R n p, consider the low rank approimation problem R X R 2 F such that rank(r) = k. Here A 2 F = n p i=1 j=1 A2 ij, the entrywise squared l 2 norm. This is equivalent to the PCA problem where R = U k D k Vk T with U k and V k being the first k columns of U and V, and D k being the first k diagonal elements of D (X = UDV T, the SVD decomposition of X). This is not a conve problem. To see this, suppose we take a matri A in the set C = {R : Rank(R) = k}, then A C, but 0.5A + 0.5( A) / C. This problem can be recast in a conve form by first rewriting the problem as X XZ 2 Z S P F subject to rank(r) = k ma tr(sz) subject to rank(r) = k Z S P where Z is a projection and S = X T X. Hence the constraint set is the nonconve set C = { Z S P : λ i (Z) {0, 1}, i = 1,..., p, tr(z) = k } where λ i (Z) are the n eigenvalues of Z. For this formulation, the solution becomes Z = V k Vk T V k gives first k columns of V. where If we rela the constraint set to F = conv(c), its conve hull, we have a linear maimiation over the fantope of order k, which is conve: ma Z F tr(sz). This is equivalent to the nonconve PCA problem, i.e., it admits the same solution. Note: The fantope of order k is given by: F = {Z S P : λ i (Z) [0, 1], i = 1,..., p, tr(z) = k} = {Z S P : 0 Z I, tr(z) = k}
3 Lecture 4: September Linear Programs Definition A linear program (LP) is an optimization problem of the form: c T s.t. D d A = b Note that this is always conve. A fundamental problem in conve optimization, it has many diverse applications and a rich history. Dantzig s simple algorithm gives a direct solver Eamples Some common LP problems are given below: 1. Diet Problem: The problem deals with finding the cheapest combination of food items that satisfies some nutritional requirements. It can be formulated as shown below: c T s.t. D d 0 where c j is the per-unit cost of item j, d i is the imum intake of nutrient i required, D ij is the amount of nutrient i contained in food j and j is the units of food j in the diet. 2. Transportation Problem: This problem deals with imizing the costs of shipping the commodities from given sources to destinations. It can be formulated as shown below: s.t. m i=1 j=1 n c ij ij n ij s i, i = 1,..., m j=1 m ij d ij, j = 1,..., n, 0 i=1 where s i is the supply at source i, d j is the demand at destination j, c ij is the per-unit shipping cost from source i to destination j and ij is number of units shipped from i to j. 3. Basis Pursuit: Given y R n and X R n p (with p > n), the aim is to detere the sparsest solution to the underdetered linear system Xβ = y. It can be formulated as below: β β 0 s.t. Xβ = y
4 4-4 Lecture 4: September 12 where β 0 = p j=1 1{β j 0}. This is a nonconve problem, which can be recast as a linear program through an l 1 approimation known as basis pursuit. This formulation is given below: The above problem can be reformulated as: β β 1 s.t. Xβ = y β,z 1T z s.t. z β z β Xβ = y 4. Dantzig Selector: The Dantzig selector is a modification of basis pursuit where strict equality is not enforced, i.e., Xβ y. Then the formulation becomes: β β 1 s.t. X T (y Xβ) λ where λ 0 is a tuning parameter. This too can be reformulated as a linear program if the constraint is written as: λ X T j (y Xβ) λ j = 1,..., p Standard Form A linear program is said to be in standard form when it is written as: Any LP can be written in standard form. c T s.t. A = b Quadratic Programs Definition Conve quadratic program (QP) is a kind of optimization problem of the form: c T T Q s.t. D d A = b We only discuss the case whose Q 0, since the problem is conve iff Q 0.
5 Lecture 4: September Eamples Here are some common QP problems: 1. Portfolio optimization We can use the QP: µ T + γ 2 T Q s.t. 1 T = 1 0 to trade off performance and risk in a financial portfolio. Here µ is epected assets returns, Q is covariance matri of assets returns, γ is risk aversion, is portfolio holdings (sum is normalized to be 1). 2. Support vector machine Given y { 1, 1} n, X R n p with rows 1, 2,..., n. SVM problem is: 3. Lasso 1 β,β 0,ξ 2 β C n i=1 ξ i s.t. ξ i 0, i = 1,..., n y i ( T i β + β 0 ) 1 ξ i, i = 1,..., n. Given y R n, X R n p, recall the lasso problem: β R p y Xβ 2 2 s.t. β 1 s. Here s 0 is a tuning parameter. This can be rewritten as a quadratic program. An alternative way to parametrize the lasso problem is in the penalized / Lagrange form: β R p y Xβ λ β 1 Here λ is the tuning parameter. The can also be rewritten in a quadratic form Standard Form Any QP can be rewritten in the standard form: c T T Q s.t. A = b 0.
6 4-6 Lecture 4: September Semidefinite programs (SDPs) Motivation Recall that Linear programs (LPs) have the following form: c T subject to D d A = b (4.1) Here, is a vector. But we can generalize this problem to optimize over matrices, X, by changing to. This defines a partial ordering over matrices. (More on this later.) Background Recall: S n is the space of n n symmetric matrices. S n + is the space of positive semidefinite matrices: S n + = {X S n u T Xu 0 for all u R n } S n ++ is the space of positive definite matrices. S n ++ = {X S n u T Xu > 0 for all u R n \{0}} If X is a matri in one of the above sets, this constrains its eigenvalues. Letting λ(x) denote the eigenvalues of a matri X: X S n λ(x) R n X S n + λ(x) R n + = { R n 0} X S n ++ λ(x) R n ++ = { R n > 0} We can define an inner product between two symmetric matrices X, Y S n using the trace operator: X Y = tr(xy ) = i,j X i,j Y i,j We can also partially order S n by defining as follows: X Y X Y S n + If we consider diagonal matrices, then this ordering for matrices becomes the same as our ordering for vectors. Below, diag() denotes a matri X S n which has the vector R n as its diagonal elements, and 0 elsewhere. Now let, y R n. Then:
7 Lecture 4: September diag() diag(y) y Semidefinite programs (SDPs) An SDP is an optimization problem of the form: c T subject to 1 F n F n F 0 A = b (4.2) Here F j S d for j = 0, 1,, n, and A R m n, c R n, b R m. Recall that in an LP we have the constraint D d. If we let D i be the i th column of D, then this is the same as i id i d. So here, the SDP simply generalizes the vectors D i and d to be symmetric matrices, F i. This problem is always conve because linear matri inequalities are conve. An SDP is said to be in standard form if it is written as: C X X S n subject to A i X = b i, i = 1,, m X 0 (4.3) With C S n, A i S n, and b i R. Any SDP can be written in this form, though a proof of this fact will require some effort! Finally, any linear program is also a semidefinite program. To see this, consider the SDP where X = diag(). Eample: trace norm imization Let A 1,, A p R m n. Then the following is a linear mapping from R m n R p : A(X) = A 1 X (4.4) A p X (Note that because A i is not necessarily symmetric, we use the standard defintion of trace: A i X = tr(a T i X).) Finding a matri X that satisfies A(X) = b for some b such that X has the lowest rank is a nonconve problem, because calculating rank is not a conve function. But we can use the trace norm as a surrogate objective, resulting in the following trace norm approimation: X tr X subject to A(X) = b (4.5)
8 4-8 Lecture 4: September 12 This is an SDP, though this is not a trivial fact. 4.5 Conic Programs Definition A conic program is an optimization problem of the form: c T s.t. A = b D() + b K Here, c, R n, A R m n, b R m. D : R n Y is a linear mapping, d Y for Euclidean space Y, K Y is a closed conve cone. This is very similar to LP, the only distinction is the set of linear inequalities are replaced with conic inequalities, i.e., D() + d K 0. Notice that if K = S n +, we recover SDP. Thus, this is a very broad class of problems Eamples Second-order cone program. A second-order cone program (SOCP) is an optimization problem of the form: c T s.t. D i + d 2 e T i + f i, i = 1, 2,..., n A = b This is a conic program with specific choice of K. In particular, it is a combination of second-order cones that are defined as: Q = {(, t) : 2 t}. From the definition, it is easy to see D i + d 2 e T i + f i (D i + d, e T i + f i ) Q i for appropriate dimensions, then taking K = Q 1 Q 2... Q p will lead to the conic program form. It is easy to see every LP is SOCP. In addition, to see every SOCP is and SDP, recall the Schur complement theorem: For A, C sysmmetric and C 0. Apply this the theorem to the following matri, [ ] A B B T 0 A BC C 1 B T 0.
9 Lecture 4: September [ ti ] T t 0 ti T t 0 2 t. Thus, we can convert the second-order cone constraint to PSD constraint. 4.6 Relationship between Programs The relationship between Linear Program (LP), Quadratic Program (QP), Second-Order Cone Program (SOCP), Semidefinite Program (SDP) and Conic Program (CP) is shown in the following figure. While the relationship between conve problems and non-conve problems is shown in the following figure. Conve problems just contain the amount of a bubble compared with non-conve ones in this figure. Acknowledgements The slides and scribe notes in the former years were referred to while making this scribe note.
Canonical Problem Forms. Ryan Tibshirani Convex Optimization
Canonical Problem Forms Ryan Tibshirani Convex Optimization 10-725 Last time: optimization basics Optimization terology (e.g., criterion, constraints, feasible points, solutions) Properties and first-order
More informationConvexity II: Optimization Basics
Conveity II: Optimization Basics Lecturer: Ryan Tibshirani Conve Optimization 10-725/36-725 See supplements for reviews of basic multivariate calculus basic linear algebra Last time: conve sets and functions
More informationLecture 4: January 26
10-725/36-725: Conve Optimization Spring 2015 Lecturer: Javier Pena Lecture 4: January 26 Scribes: Vipul Singh, Shinjini Kundu, Chia-Yin Tsai Note: LaTeX template courtesy of UC Berkeley EECS dept. Disclaimer:
More informationLecture 26: April 22nd
10-725/36-725: Conve Optimization Spring 2015 Lecture 26: April 22nd Lecturer: Ryan Tibshirani Scribes: Eric Wong, Jerzy Wieczorek, Pengcheng Zhou Note: LaTeX template courtesy of UC Berkeley EECS dept.
More informationLecture 23: November 19
10-725/36-725: Conve Optimization Fall 2018 Lecturer: Ryan Tibshirani Lecture 23: November 19 Scribes: Charvi Rastogi, George Stoica, Shuo Li Charvi Rastogi: 23.1-23.4.2, George Stoica: 23.4.3-23.8, Shuo
More informationLecture 20: November 1st
10-725: Optimization Fall 2012 Lecture 20: November 1st Lecturer: Geoff Gordon Scribes: Xiaolong Shen, Alex Beutel Note: LaTeX template courtesy of UC Berkeley EECS dept. Disclaimer: These notes have not
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 informationLecture 14: Optimality Conditions for Conic Problems
EE 227A: Conve Optimization and Applications March 6, 2012 Lecture 14: Optimality Conditions for Conic Problems Lecturer: Laurent El Ghaoui Reading assignment: 5.5 of BV. 14.1 Optimality for Conic Problems
More informationDuality Uses and Correspondences. Ryan Tibshirani Convex Optimization
Duality Uses and Correspondences Ryan Tibshirani Conve Optimization 10-725 Recall that for the problem Last time: KKT conditions subject to f() h i () 0, i = 1,... m l j () = 0, j = 1,... r the KKT conditions
More informationLecture 5: September 15
10-725/36-725: Convex Optimization Fall 2015 Lecture 5: September 15 Lecturer: Lecturer: Ryan Tibshirani Scribes: Scribes: Di Jin, Mengdi Wang, Bin Deng Note: LaTeX template courtesy of UC Berkeley EECS
More informationLecture 7: Weak Duality
EE 227A: Conve Optimization and Applications February 7, 2012 Lecture 7: Weak Duality Lecturer: Laurent El Ghaoui 7.1 Lagrange Dual problem 7.1.1 Primal problem In this section, we consider a possibly
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 informationLECTURE 7. Least Squares and Variants. Optimization Models EE 127 / EE 227AT. Outline. Least Squares. Notes. Notes. Notes. Notes.
Optimization Models EE 127 / EE 227AT Laurent El Ghaoui EECS department UC Berkeley Spring 2015 Sp 15 1 / 23 LECTURE 7 Least Squares and Variants If others would but reflect on mathematical truths as deeply
More informationLecture 16: October 22
0-725/36-725: Conve Optimization Fall 208 Lecturer: Ryan Tibshirani Lecture 6: October 22 Scribes: Nic Dalmasso, Alan Mishler, Benja LeRoy Note: LaTeX template courtesy of UC Berkeley EECS dept. Disclaimer:
More informationLecture 5: September 12
10-725/36-725: Convex Optimization Fall 2015 Lecture 5: September 12 Lecturer: Lecturer: Ryan Tibshirani Scribes: Scribes: Barun Patra and Tyler Vuong Note: LaTeX template courtesy of UC Berkeley EECS
More informationLecture 9: SVD, Low Rank Approximation
CSE 521: Design and Analysis of Algorithms I Spring 2016 Lecture 9: SVD, Low Rank Approimation Lecturer: Shayan Oveis Gharan April 25th Scribe: Koosha Khalvati Disclaimer: hese notes have not been subjected
More informationSparse Optimization Lecture: Basic Sparse Optimization Models
Sparse Optimization Lecture: Basic Sparse Optimization Models Instructor: Wotao Yin July 2013 online discussions on piazza.com Those who complete this lecture will know basic l 1, l 2,1, and nuclear-norm
More informationLecture 23: Conditional Gradient Method
10-725/36-725: Conve Optimization Spring 2015 Lecture 23: Conditional Gradient Method Lecturer: Ryan Tibshirani Scribes: Shichao Yang,Diyi Yang,Zhanpeng Fang Note: LaTeX template courtesy of UC Berkeley
More informationLecture 9: September 28
0-725/36-725: Convex Optimization Fall 206 Lecturer: Ryan Tibshirani Lecture 9: September 28 Scribes: Yiming Wu, Ye Yuan, Zhihao Li Note: LaTeX template courtesy of UC Berkeley EECS dept. Disclaimer: These
More informationLecture 6: September 12
10-725: Optimization Fall 2013 Lecture 6: September 12 Lecturer: Ryan Tibshirani Scribes: Micol Marchetti-Bowick Note: LaTeX template courtesy of UC Berkeley EECS dept. Disclaimer: These notes have not
More informationConvex Optimization. 4. Convex Optimization Problems. Prof. Ying Cui. Department of Electrical Engineering Shanghai Jiao Tong University
Conve Optimization 4. Conve Optimization Problems Prof. Ying Cui Department of Electrical Engineering Shanghai Jiao Tong University 2017 Autumn Semester SJTU Ying Cui 1 / 58 Outline Optimization problems
More informationLecture 6: September 19
36-755: Advanced Statistical Theory I Fall 2016 Lecture 6: September 19 Lecturer: Alessandro Rinaldo Scribe: YJ Choe Note: LaTeX template courtesy of UC Berkeley EECS dept. Disclaimer: These notes have
More information10-725/36-725: Convex Optimization Spring Lecture 21: April 6
10-725/36-725: Conve Optimization Spring 2015 Lecturer: Ryan Tibshirani Lecture 21: April 6 Scribes: Chiqun Zhang, Hanqi Cheng, Waleed Ammar Note: LaTeX template courtesy of UC Berkeley EECS dept. Disclaimer:
More informationConvex Optimization Problems. Prof. Daniel P. Palomar
Conve Optimization Problems Prof. Daniel P. Palomar The Hong Kong University of Science and Technology (HKUST) MAFS6010R- Portfolio Optimization with R MSc in Financial Mathematics Fall 2018-19, HKUST,
More informationLecture 13: Duality Uses and Correspondences
10-725/36-725: Conve Optimization Fall 2016 Lectre 13: Dality Uses and Correspondences Lectrer: Ryan Tibshirani Scribes: Yichong X, Yany Liang, Yanning Li Note: LaTeX template cortesy of UC Berkeley EECS
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 2017-18, HKUST, Hong Kong Outline of Lecture Lagrangian Dual function Dual
More informationLecture: Examples of LP, SOCP and SDP
1/34 Lecture: Examples of LP, SOCP and SDP Zaiwen Wen Beijing International Center For Mathematical Research Peking University http://bicmr.pku.edu.cn/~wenzw/bigdata2018.html wenzw@pku.edu.cn Acknowledgement:
More informationORF 523 Lecture 9 Spring 2016, Princeton University Instructor: A.A. Ahmadi Scribe: G. Hall Thursday, March 10, 2016
ORF 523 Lecture 9 Spring 2016, Princeton University Instructor: A.A. Ahmadi Scribe: G. Hall Thursday, March 10, 2016 When in doubt on the accuracy of these notes, please cross check with the instructor
More informationLecture 1: January 12
10-725/36-725: Convex Optimization Fall 2015 Lecturer: Ryan Tibshirani Lecture 1: January 12 Scribes: Seo-Jin Bang, Prabhat KC, Josue Orellana 1.1 Review We begin by going through some examples and key
More informationLecture 1: September 25
0-725: Optimization Fall 202 Lecture : September 25 Lecturer: Geoff Gordon/Ryan Tibshirani Scribes: Subhodeep Moitra Note: LaTeX template courtesy of UC Berkeley EECS dept. Disclaimer: These notes have
More information9. Interpretations, Lifting, SOS and Moments
9-1 Interpretations, Lifting, SOS and Moments P. Parrilo and S. Lall, CDC 2003 2003.12.07.04 9. Interpretations, Lifting, SOS and Moments Polynomial nonnegativity Sum of squares (SOS) decomposition Eample
More informationLecture 1. 1 Conic programming. MA 796S: Convex Optimization and Interior Point Methods October 8, Consider the conic program. min.
MA 796S: Convex Optimization and Interior Point Methods October 8, 2007 Lecture 1 Lecturer: Kartik Sivaramakrishnan Scribe: Kartik Sivaramakrishnan 1 Conic programming Consider the conic program min s.t.
More informationMidterm 1 Solutions. 1. (2 points) Show that the Frobenius norm of a matrix A depends only on its singular values. Precisely, show that
EE127A L. El Ghaoui YOUR NAME HERE: SOLUTIONS YOUR SID HERE: 42 3/27/9 Midterm 1 Solutions The eam is open notes, but access to the Internet is not allowed. The maimum grade is 2. When asked to prove something,
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 information1 Kernel methods & optimization
Machine Learning Class Notes 9-26-13 Prof. David Sontag 1 Kernel methods & optimization One eample of a kernel that is frequently used in practice and which allows for highly non-linear discriminant functions
More informationLecture 6: September 17
10-725/36-725: Convex Optimization Fall 2015 Lecturer: Ryan Tibshirani Lecture 6: September 17 Scribes: Scribes: Wenjun Wang, Satwik Kottur, Zhiding Yu Note: LaTeX template courtesy of UC Berkeley EECS
More informationLecture 24: August 28
10-725: Optimization Fall 2012 Lecture 24: August 28 Lecturer: Geoff Gordon/Ryan Tibshirani Scribes: Jiaji Zhou,Tinghui Zhou,Kawa Cheung Note: LaTeX template courtesy of UC Berkeley EECS dept. Disclaimer:
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 informationLecture 17: Primal-dual interior-point methods part II
10-725/36-725: Convex Optimization Spring 2015 Lecture 17: Primal-dual interior-point methods part II Lecturer: Javier Pena Scribes: Pinchao Zhang, Wei Ma Note: LaTeX template courtesy of UC Berkeley EECS
More informationU.C. Berkeley CS294: Beyond Worst-Case Analysis Handout 12 Luca Trevisan October 3, 2017
U.C. Berkeley CS94: Beyond Worst-Case Analysis Handout 1 Luca Trevisan October 3, 017 Scribed by Maxim Rabinovich Lecture 1 In which we begin to prove that the SDP relaxation exactly recovers communities
More informationConvex Optimization M2
Convex Optimization M2 Lecture 8 A. d Aspremont. Convex Optimization M2. 1/57 Applications A. d Aspremont. Convex Optimization M2. 2/57 Outline Geometrical problems Approximation problems Combinatorial
More informationLecture 7: September 17
10-725: Optimization Fall 2013 Lecture 7: September 17 Lecturer: Ryan Tibshirani Scribes: Serim Park,Yiming Gu 7.1 Recap. The drawbacks of Gradient Methods are: (1) requires f is differentiable; (2) relatively
More informationSparse and Robust Optimization and Applications
Sparse and and Statistical Learning Workshop Les Houches, 2013 Robust Laurent El Ghaoui with Mert Pilanci, Anh Pham EECS Dept., UC Berkeley January 7, 2013 1 / 36 Outline Sparse Sparse Sparse Probability
More informationIntro to Nonlinear Optimization
Intro to Nonlinear Optimization We now rela the proportionality and additivity assumptions of LP What are the challenges of nonlinear programs NLP s? Objectives and constraints can use any function: ma
More informationMidterm Solutions. EE127A L. El Ghaoui 3/19/11
EE27A L. El Ghaoui 3/9/ Midterm Solutions. (6 points Find the projection z of the vector = (2, on the line that passes through 0 = (, 2 and with direction given by the vector u = (,. Solution: The line
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 information10701 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 informationLecture 25: November 27
10-725: Optimization Fall 2012 Lecture 25: November 27 Lecturer: Ryan Tibshirani Scribes: Matt Wytock, Supreeth Achar Note: LaTeX template courtesy of UC Berkeley EECS dept. Disclaimer: These notes have
More informationIntroduction to Machine Learning Spring 2018 Note Duality. 1.1 Primal and Dual Problem
CS 189 Introduction to Machine Learning Spring 2018 Note 22 1 Duality As we have seen in our discussion of kernels, ridge regression can be viewed in two ways: (1) an optimization problem over the weights
More informationConvex optimization problems. Optimization problem in standard form
Convex optimization problems optimization problem in standard form convex optimization problems linear optimization quadratic optimization geometric programming quasiconvex optimization generalized inequality
More informationCSCI 1951-G Optimization Methods in Finance Part 10: Conic Optimization
CSCI 1951-G Optimization Methods in Finance Part 10: Conic Optimization April 6, 2018 1 / 34 This material is covered in the textbook, Chapters 9 and 10. Some of the materials are taken from it. Some of
More informationChapter 2: Linear Programming Basics. (Bertsimas & Tsitsiklis, Chapter 1)
Chapter 2: Linear Programming Basics (Bertsimas & Tsitsiklis, Chapter 1) 33 Example of a Linear Program Remarks. minimize 2x 1 x 2 + 4x 3 subject to x 1 + x 2 + x 4 2 3x 2 x 3 = 5 x 3 + x 4 3 x 1 0 x 3
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 informationLMI MODELLING 4. CONVEX LMI MODELLING. Didier HENRION. LAAS-CNRS Toulouse, FR Czech Tech Univ Prague, CZ. Universidad de Valladolid, SP March 2009
LMI MODELLING 4. CONVEX LMI MODELLING Didier HENRION LAAS-CNRS Toulouse, FR Czech Tech Univ Prague, CZ Universidad de Valladolid, SP March 2009 Minors A minor of a matrix F is the determinant of a submatrix
More informationLecture 3: Review of Linear Algebra
ECE 83 Fall 2 Statistical Signal Processing instructor: R Nowak, scribe: R Nowak Lecture 3: Review of Linear Algebra Very often in this course we will represent signals as vectors and operators (eg, filters,
More informationLecture 9: Numerical Linear Algebra Primer (February 11st)
10-725/36-725: Convex Optimization Spring 2015 Lecture 9: Numerical Linear Algebra Primer (February 11st) Lecturer: Ryan Tibshirani Scribes: Avinash Siravuru, Guofan Wu, Maosheng Liu Note: LaTeX template
More informationLecture 14: October 17
1-725/36-725: Convex Optimization Fall 218 Lecture 14: October 17 Lecturer: Lecturer: Ryan Tibshirani Scribes: Pengsheng Guo, Xian Zhou Note: LaTeX template courtesy of UC Berkeley EECS dept. Disclaimer:
More 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 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: first-order condition: f(y) f(x) + f x,y x, second-order
More informationLecture 14: October 11
10-725: Optimization Fall 2012 Lecture 14: October 11 Lecturer: Geoff Gordon/Ryan Tibshirani Scribes: Zitao Liu Note: LaTeX template courtesy of UC Berkeley EECS dept. Disclaimer: These notes have not
More informationCSC Linear Programming and Combinatorial Optimization Lecture 10: Semidefinite Programming
CSC2411 - Linear Programming and Combinatorial Optimization Lecture 10: Semidefinite Programming Notes taken by Mike Jamieson March 28, 2005 Summary: In this lecture, we introduce semidefinite programming
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 informationCourse Notes for EE227C (Spring 2018): Convex Optimization and Approximation
Course otes for EE7C (Spring 018): Conve Optimization and Approimation Instructor: Moritz Hardt Email: hardt+ee7c@berkeley.edu Graduate Instructor: Ma Simchowitz Email: msimchow+ee7c@berkeley.edu October
More informationMA 575 Linear Models: Cedric E. Ginestet, Boston University Regularization: Ridge Regression and Lasso Week 14, Lecture 2
MA 575 Linear Models: Cedric E. Ginestet, Boston University Regularization: Ridge Regression and Lasso Week 14, Lecture 2 1 Ridge Regression Ridge regression and the Lasso are two forms of regularized
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 informationConvex 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 informationLecture 8: Linear Algebra Background
CSE 521: Design and Analysis of Algorithms I Winter 2017 Lecture 8: Linear Algebra Background Lecturer: Shayan Oveis Gharan 2/1/2017 Scribe: Swati Padmanabhan Disclaimer: These notes have not been subjected
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 informationQuadratic reformulation techniques for 0-1 quadratic programs
OSE SEMINAR 2014 Quadratic reformulation techniques for 0-1 quadratic programs Ray Pörn CENTER OF EXCELLENCE IN OPTIMIZATION AND SYSTEMS ENGINEERING ÅBO AKADEMI UNIVERSITY ÅBO NOVEMBER 14th 2014 2 Structure
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 informationKarush-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 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 informationLecture Semidefinite Programming and Graph Partitioning
Approximation Algorithms and Hardness of Approximation April 16, 013 Lecture 14 Lecturer: Alantha Newman Scribes: Marwa El Halabi 1 Semidefinite Programming and Graph Partitioning In previous lectures,
More informationLecture 3: Review of Linear Algebra
ECE 83 Fall 2 Statistical Signal Processing instructor: R Nowak Lecture 3: Review of Linear Algebra Very often in this course we will represent signals as vectors and operators (eg, filters, transforms,
More informationLecture 23: November 21
10-725/36-725: Convex Optimization Fall 2016 Lecturer: Ryan Tibshirani Lecture 23: November 21 Scribes: Yifan Sun, Ananya Kumar, Xin Lu Note: LaTeX template courtesy of UC Berkeley EECS dept. Disclaimer:
More information2. Linear algebra. matrices and vectors. linear equations. range and nullspace of matrices. function of vectors, gradient and Hessian
FE661 - Statistical Methods for Financial Engineering 2. Linear algebra Jitkomut Songsiri matrices and vectors linear equations range and nullspace of matrices function of vectors, gradient and Hessian
More informationLecture 10: Duality in Linear Programs
10-725/36-725: Convex Optimization Spring 2015 Lecture 10: Duality in Linear Programs Lecturer: Ryan Tibshirani Scribes: Jingkun Gao and Ying Zhang Disclaimer: These notes have not been subjected to the
More informationDimensionality Reduction: PCA. Nicholas Ruozzi University of Texas at Dallas
Dimensionality Reduction: PCA Nicholas Ruozzi University of Texas at Dallas Eigenvalues λ is an eigenvalue of a matrix A R n n if the linear system Ax = λx has at least one non-zero solution If Ax = λx
More informationSemidefinite and Second Order Cone Programming Seminar Fall 2001 Lecture 5
Semidefinite and Second Order Cone Programming Seminar Fall 2001 Lecture 5 Instructor: Farid Alizadeh Scribe: Anton Riabov 10/08/2001 1 Overview We continue studying the maximum eigenvalue SDP, and generalize
More informationMathematical Optimization Models and Applications
Mathematical Optimization Models and Applications Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A. http://www.stanford.edu/ yyye Chapters 1, 2.1-2,
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 informationAgenda. 1 Cone programming. 2 Convex cones. 3 Generalized inequalities. 4 Linear programming (LP) 5 Second-order cone programming (SOCP)
Agenda 1 Cone programming 2 Convex cones 3 Generalized inequalities 4 Linear programming (LP) 5 Second-order cone programming (SOCP) 6 Semidefinite programming (SDP) 7 Examples Optimization problem in
More informationSparse PCA with applications in finance
Sparse PCA with applications in finance A. d Aspremont, L. El Ghaoui, M. Jordan, G. Lanckriet ORFE, Princeton University & EECS, U.C. Berkeley Available online at www.princeton.edu/~aspremon 1 Introduction
More informationEE Applications of Convex Optimization in Signal Processing and Communications Dr. Andre Tkacenko, JPL Third Term
EE 150 - Applications of Convex Optimization in Signal Processing and Communications Dr. Andre Tkacenko JPL Third Term 2011-2012 Due on Thursday May 3 in class. Homework Set #4 1. (10 points) (Adapted
More informationReview of Linear Algebra
Review of Linear Algebra Dr Gerhard Roth COMP 40A Winter 05 Version Linear algebra Is an important area of mathematics It is the basis of computer vision Is very widely taught, and there are many resources
More informationFantope Regularization in Metric Learning
Fantope Regularization in Metric Learning CVPR 2014 Marc T. Law (LIP6, UPMC), Nicolas Thome (LIP6 - UPMC Sorbonne Universités), Matthieu Cord (LIP6 - UPMC Sorbonne Universités), Paris, France Introduction
More informationOperations Research Letters
Operations Research Letters 37 (2009) 1 6 Contents lists available at ScienceDirect Operations Research Letters journal homepage: www.elsevier.com/locate/orl Duality in robust optimization: Primal worst
More information10-725/36-725: Convex Optimization Prerequisite Topics
10-725/36-725: Convex Optimization Prerequisite Topics February 3, 2015 This is meant to be a brief, informal refresher of some topics that will form building blocks in this course. The content of the
More informationHomework 4. Convex Optimization /36-725
Homework 4 Convex Optimization 10-725/36-725 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 informationLecture 14: Newton s Method
10-725/36-725: Conve Optimization Fall 2016 Lecturer: Javier Pena Lecture 14: Newton s ethod Scribes: Varun Joshi, Xuan Li Note: LaTeX template courtesy of UC Berkeley EECS dept. Disclaimer: These notes
More informationInterior Point Methods: Second-Order Cone Programming and Semidefinite Programming
School of Mathematics T H E U N I V E R S I T Y O H F E D I N B U R G Interior Point Methods: Second-Order Cone Programming and Semidefinite Programming Jacek Gondzio Email: J.Gondzio@ed.ac.uk URL: http://www.maths.ed.ac.uk/~gondzio
More informationCS295: Convex Optimization. Xiaohui Xie Department of Computer Science University of California, Irvine
CS295: Convex Optimization Xiaohui Xie Department of Computer Science University of California, Irvine Course information Prerequisites: multivariate calculus and linear algebra Textbook: Convex Optimization
More informationSemidefinite and Second Order Cone Programming Seminar Fall 2001 Lecture 4
Semidefinite and Second Order Cone Programming Seminar Fall 2001 Lecture 4 Instructor: Farid Alizadeh Scribe: Haengju Lee 10/1/2001 1 Overview We examine the dual of the Fermat-Weber Problem. Next we will
More information1 Robust optimization
ORF 523 Lecture 16 Princeton University Instructor: A.A. Ahmadi Scribe: G. Hall Any typos should be emailed to a a a@princeton.edu. In this lecture, we give a brief introduction to robust optimization
More informationNonconvex? NP! (No Problem!) Ryan Tibshirani Convex Optimization
Nonconvex? NP! (No Problem!) Ryan Tibshirani Convex Optimization 10-725 1 Outline Today: Convex versus nonconvex? Classical nonconvex problems Eigen problems Graph problems Nonconvex proximal operators
More informationLearning the Kernel Matrix with Semi-Definite Programming
Learning the Kernel Matrix with Semi-Definite Programg Gert R.G. Lanckriet gert@cs.berkeley.edu Department of Electrical Engineering and Computer Science University of California, Berkeley, CA 94720, USA
More informationSelected Methods for Modern Optimization in Data Analysis Department of Statistics and Operations Research UNC-Chapel Hill Fall 2018
Selected Methods for Modern Optimization in Data Analysis Department of Statistics and Operations Research UNC-Chapel Hill Fall 208 Instructor: Quoc Tran-Dinh Scriber: Quoc Tran-Dinh Lecture : Introduction
More informationPreliminaries Overview OPF and Extensions. Convex Optimization. Lecture 8 - Applications in Smart Grids. Instructor: Yuanzhang Xiao
Convex Optimization Lecture 8 - Applications in Smart Grids Instructor: Yuanzhang Xiao University of Hawaii at Manoa Fall 2017 1 / 32 Today s Lecture 1 Generalized Inequalities and Semidefinite Programming
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 informationLecture 12: Introduction to Spectral Graph Theory, Cheeger s inequality
CSE 521: Design and Analysis of Algorithms I Spring 2016 Lecture 12: Introduction to Spectral Graph Theory, Cheeger s inequality Lecturer: Shayan Oveis Gharan May 4th Scribe: Gabriel Cadamuro Disclaimer:
More information