Canonical Problem Forms. Ryan Tibshirani Convex Optimization
|
|
- Bryce McDaniel
- 5 years ago
- Views:
Transcription
1 Canonical Problem Forms Ryan Tibshirani Convex Optimization
2 Last time: optimization basics Optimization terology (e.g., criterion, constraints, feasible points, solutions) Properties and first-order optimality Equivalent transformations (e.g., partial optimization, change of variables, eliating equality constraints) 2
3 Outline Today: Linear programs Quadratic programs Semidefinite programs Cone programs 3
4 4
5 Linear program A linear program or LP is an optimization problem of the form x c T x Dx d Ax = b Observe that this is always a convex optimization problem First introduced by Kantorovich in the late 1930s and Dantzig in the 1940s Dantzig s simplex algorithm gives a direct (noniterative) solver for LPs (later in the course we ll see interior point methods) Fundamental problem in convex optimization. Many diverse applications, rich history 5
6 Example: diet problem Find cheapest combination of foods that satisfies some nutritional requirements (useful for graduate students!) x c T x Dx d x 0 Interpretation: c j : per-unit cost of food j d i : imum required intake of nutrient i D ij : content of nutrient i per unit of food j x j : units of food j in the diet 6
7 Example: transportation problem Ship commodities from given sources to destinations at cost x m n c ij x ij i=1 j=1 n x ij s i, i = 1,..., m j=1 m x ij d j, j = 1,..., n, x 0 i=1 Interpretation: s i : supply at source i d j : demand at destination j c ij : per-unit shipping cost from i to j x ij : units shipped from i to j 7
8 Example: basis pursuit Given y R n and X R n p, where p > n. Suppose that we seek the sparsest solution to underdetered linear system Xβ = y Nonconvex formulation: β β 0 Xβ = y where recall β 0 = p j=1 1{β j 0}, the l 0 norm The l 1 approximation, often called basis pursuit: β β 1 Xβ = y 8
9 Basis pursuit is a linear program. Reformulation: β β 1 Xβ = y β,z 1 T z z β z β Xβ = y (Check that this makes sense to you) 9
10 Example: Dantzig selector Modification of previous problem, where we allow for Xβ y (we don t require exact equality), the Dantzig selector: 1 β β 1 Here λ 0 is a tuning parameter X T (y Xβ) λ Again, this can be reformulated as a linear program (check this!) 1 Candes and Tao (2007), The Dantzig selector: statistical estimation when p is much larger than n 10
11 11 Standard form A linear program is said to be in standard form when it is written as x c T x Ax = b x 0 Any linear program can be rewritten in standard form (check this!)
12 12 Convex quadratic program A convex quadratic program or QP is an optimization problem of the form x c T x xt Qx Dx d Ax = b where Q 0, i.e., positive semidefinite Note that this problem is not convex when Q 0 From now on, when we say quadratic program or QP, we implicitly assume that Q 0 (so the problem is convex)
13 13 Example: portfolio optimization Construct a financial portfolio, trading off performance and risk: Interpretation: µ : expected assets returns max µ T x γ x 2 xt Qx 1 T x = 1 x 0 Q : covariance matrix of assets returns γ : risk aversion x : portfolio holdings (percentages)
14 14 Example: support vector machines Given y { 1, 1} n, X R n p having rows x 1,... x n, recall the support vector machine or SVM problem: β,β 0,ξ 1 2 β C n i=1 ξ i ξ i 0, i = 1,... n y i (x T i β + β 0 ) 1 ξ i, i = 1,... n This is a quadratic program
15 15 Example: lasso Given y R n, X R n p, recall the lasso problem: β y Xβ 2 2 β 1 s Here s 0 is a tuning parameter. Indeed, this can be reformulated as a quadratic program (check this!) Alternative parametrization (called Lagrange, or penalized form): 1 β 2 y Xβ λ β 1 Now λ 0 is a tuning parameter. And again, this can be rewritten as a quadratic program (check this!)
16 16 Standard form A quadratic program is in standard form if it is written as x c T x xt Qx Ax = b x 0 Any quadratic program can be rewritten in standard form
17 17 Motivation for semidefinite programs Consider linear programg again: x c T x Dx d Ax = b Can generalize by changing to different (partial) order. Recall: S n is space of n n symmetric matrices S n + is the space of positive semidefinite matrices, i.e., S n + = {X S n : u T Xu 0 for all u R n } S n ++ is the space of positive definite matrices, i.e., S n ++ = { X S n : u T Xu > 0 for all u R n \ {0} }
18 18 Facts about S n, S n +, S n ++ Basic linear algebra facts, here λ(x) = (λ 1 (X),..., λ n (X)): X S n = λ(x) R n X S n + λ(x) R n + X S n ++ λ(x) R n ++ We can define an inner product over S n : given X, Y S n, X Y = tr(xy ) We can define a partial ordering over S n : given X, Y S n, X Y X Y S n + Note: for x, y R n, diag(x) diag(y) x y (recall, the latter is interpreted elementwise)
19 19 Semidefinite program A semidefinite program or SDP is an optimization problem of the form x c T x x 1 F x n F n F 0 Ax = b Here F j S d, for j = 0, 1,... n, and A R m n, c R n, b R m. Observe that this is always a convex optimization problem Also, any linear program is a semidefinite program (check this!)
20 20 Standard form A semidefinite program is in standard form if it is written as X C X A i X = b i, i = 1,... m X 0 Any semidefinite program can be written in standard form (for a challenge, check this!)
21 Example: theta function Let G = (N, E) be an undirected graph, N = {1,..., n}, and ω(g) : clique number of G χ(g) : chromatic number of G The Lovasz theta function: 2 ϑ(g) = max X 11 T X I X = 1 X ij = 0, (i, j) / E X 0 The Lovasz sandwich theorem: ω(g) ϑ(ḡ) χ(g), where Ḡ is the complement graph of G 2 Lovasz (1979), On the Shannon capacity of a graph 21
22 22 Example: trace norm imization Let A : R m n R p be a linear map, A 1 X A(X) =... A p X for A 1,... A p R m n (and where A i X = tr(a T i X)). Finding lowest-rank solution to an underdetered system, nonconvex: Trace norm approximation: X X rank(x) A(X) = b X tr A(X) = b This is indeed an SDP (but harder to show, requires duality...)
23 23 Conic program A conic program is an optimization problem of the form: x c T x Ax = b D(x) + d K Here: c, x R n, and A R m n, b R m D : R n Y is a linear map, d Y, for Euclidean space Y K Y is a closed convex cone Both LPs and SDPs are special cases of conic programg. For LPs, K = R n +; for SDPs, K = S n +
24 24 Second-order cone program A second-order cone program or SOCP is an optimization problem of the form: x c T x D i x + d i 2 e T i x + f i, i = 1,... p Ax = b This is indeed a cone program. Why? Recall the second-order cone So we have Q = {(x, t) : x 2 t} D i x + d i 2 e T i x + f i (D i x + d i, e T i x + f i ) Q i for second-order cone Q i of appropriate dimensions. Now take K = Q 1... Q p
25 25 Observe that every LP is an SOCP. Further, every SOCP is an SDP Why? Turns out that x 2 t [ ti x x T t ] 0 Hence we can write any SOCP constraint as an SDP constraint The above is a special case of the Schur complement theorem: [ ] A B B T 0 A BC 1 B T 0 C for A, C symmetric and C 0
26 26 Hey, what about QPs? Finally, our old friend QPs sneak into the hierarchy. Turns out QPs are SOCPs, which we can see by rewriting a QP as x,t c T x + t Dx d, 1 2 xt Qx t Ax = b Now write 1 2 xt Qx t ( 1 2 Q 1/2 x, 1 2 (1 t)) (1 + t) Take a breath (phew!). Thus we have established the hierachy LPs QPs SOCPs SDPs Conic programs completing the picture we saw at the start
27 27 References and further reading D. Bertsimas and J. Tsitsiklis (1997), Introduction to linear optimization, Chapters 1, 2 A. Nemirovski and A. Ben-Tal (2001), Lectures on modern convex optimization, Chapters 1 4 S. Boyd and L. Vandenberghe (2004), Convex optimization, Chapter 4
Lecture 4: September 12
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:
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 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 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 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 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 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 informationIE 521 Convex Optimization
Lecture 14: and Applications 11th March 2019 Outline LP SOCP SDP LP SOCP SDP 1 / 21 Conic LP SOCP SDP Primal Conic Program: min c T x s.t. Ax K b (CP) : b T y s.t. A T y = c (CD) y K 0 Theorem. (Strong
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 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 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 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 information15. Conic optimization
L. Vandenberghe EE236C (Spring 216) 15. Conic optimization conic linear program examples modeling duality 15-1 Generalized (conic) inequalities Conic inequality: a constraint x K where K is a convex cone
More 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 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: 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 informationRelaxations and Randomized Methods for Nonconvex QCQPs
Relaxations and Randomized Methods for Nonconvex QCQPs Alexandre d Aspremont, Stephen Boyd EE392o, Stanford University Autumn, 2003 Introduction While some special classes of nonconvex problems can be
More informationMathematics of Data: From Theory to Computation
Mathematics of Data: From Theory to Computation Prof. Volkan Cevher volkan.cevher@epfl.ch Lecture 13: Disciplined convex optimization Laboratory for Information and Inference Systems (LIONS) École Polytechnique
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 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 informationIntroduction to Convex Optimization
Introduction to Convex Optimization Daniel P. Palomar Hong Kong University of Science and Technology (HKUST) ELEC5470 - Convex Optimization Fall 2018-19, HKUST, Hong Kong Outline of Lecture Optimization
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 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 informationModeling with semidefinite and copositive matrices
Modeling with semidefinite and copositive matrices Franz Rendl http://www.math.uni-klu.ac.at Alpen-Adria-Universität Klagenfurt Austria F. Rendl, Singapore workshop 2006 p.1/24 Overview Node and Edge relaxations
More informationThe maximal stable set problem : Copositive programming and Semidefinite Relaxations
The maximal stable set problem : Copositive programming and Semidefinite Relaxations Kartik Krishnan Department of Mathematical Sciences Rensselaer Polytechnic Institute Troy, NY 12180 USA kartis@rpi.edu
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 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 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 informationCopositive Programming and Combinatorial Optimization
Copositive Programming and Combinatorial Optimization Franz Rendl http://www.math.uni-klu.ac.at Alpen-Adria-Universität Klagenfurt Austria joint work with M. Bomze (Wien) and F. Jarre (Düsseldorf) and
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 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 informationPrimal-Dual Interior-Point Methods. Ryan Tibshirani Convex Optimization /36-725
Primal-Dual Interior-Point Methods Ryan Tibshirani Convex Optimization 10-725/36-725 Given the problem Last time: barrier method min x subject to f(x) h i (x) 0, i = 1,... m Ax = b where f, h i, i = 1,...
More informationIE 521 Convex Optimization
Lecture 1: 16th January 2019 Outline 1 / 20 Which set is different from others? Figure: Four sets 2 / 20 Which set is different from others? Figure: Four sets 3 / 20 Interior, Closure, Boundary Definition.
More informationA CONIC DANTZIG-WOLFE DECOMPOSITION APPROACH FOR LARGE SCALE SEMIDEFINITE PROGRAMMING
A CONIC DANTZIG-WOLFE DECOMPOSITION APPROACH FOR LARGE SCALE SEMIDEFINITE PROGRAMMING Kartik Krishnan Advanced Optimization Laboratory McMaster University Joint work with Gema Plaza Martinez and Tamás
More informationLMI Methods in Optimal and Robust Control
LMI Methods in Optimal and Robust Control Matthew M. Peet Arizona State University Lecture 02: Optimization (Convex and Otherwise) What is Optimization? An Optimization Problem has 3 parts. x F f(x) :
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 informationGradient Descent. Ryan Tibshirani Convex Optimization /36-725
Gradient Descent Ryan Tibshirani Convex Optimization 10-725/36-725 Last time: canonical convex programs Linear program (LP): takes the form min x subject to c T x Gx h Ax = b Quadratic program (QP): like
More informationBCOL RESEARCH REPORT 07.04
BCOL RESEARCH REPORT 07.04 Industrial Engineering & Operations Research University of California, Berkeley, CA 94720-1777 LIFTING FOR CONIC MIXED-INTEGER PROGRAMMING ALPER ATAMTÜRK AND VISHNU NARAYANAN
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 information4. Convex optimization problems
Convex Optimization Boyd & Vandenberghe 4. Convex optimization problems optimization problem in standard form convex optimization problems quasiconvex optimization linear optimization quadratic optimization
More informationThe Q-parametrization (Youla) Lecture 13: Synthesis by Convex Optimization. Lecture 13: Synthesis by Convex Optimization. Example: Spring-mass System
The Q-parametrization (Youla) Lecture 3: Synthesis by Convex Optimization controlled variables z Plant distubances w Example: Spring-mass system measurements y Controller control inputs u Idea for lecture
More informationMIT Algebraic techniques and semidefinite optimization February 14, Lecture 3
MI 6.97 Algebraic techniques and semidefinite optimization February 4, 6 Lecture 3 Lecturer: Pablo A. Parrilo Scribe: Pablo A. Parrilo In this lecture, we will discuss one of the most important applications
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 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 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 informationFour new upper bounds for the stability number of a graph
Four new upper bounds for the stability number of a graph Miklós Ujvári Abstract. In 1979, L. Lovász defined the theta number, a spectral/semidefinite upper bound on the stability number of a graph, which
More informationDuality in Linear Programs. Lecturer: Ryan Tibshirani Convex Optimization /36-725
Duality in Linear Programs Lecturer: Ryan Tibshirani Convex Optimization 10-725/36-725 1 Last time: proximal gradient descent Consider the problem x g(x) + h(x) with g, h convex, g differentiable, and
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 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 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 informationConvex relaxation. In example below, we have N = 6, and the cut we are considering
Convex relaxation The art and science of convex relaxation revolves around taking a non-convex problem that you want to solve, and replacing it with a convex problem which you can actually solve the solution
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 informationCopositive Programming and Combinatorial Optimization
Copositive Programming and Combinatorial Optimization Franz Rendl http://www.math.uni-klu.ac.at Alpen-Adria-Universität Klagenfurt Austria joint work with I.M. Bomze (Wien) and F. Jarre (Düsseldorf) IMA
More informationPrimal-Dual Interior-Point Methods. Ryan Tibshirani Convex Optimization
Primal-Dual Interior-Point Methods Ryan Tibshirani Convex Optimization 10-725 Given the problem Last time: barrier method min x subject to f(x) h i (x) 0, i = 1,... m Ax = b where f, h i, i = 1,... m are
More informationCourse Outline. FRTN10 Multivariable Control, Lecture 13. General idea for Lectures Lecture 13 Outline. Example 1 (Doyle Stein, 1979)
Course Outline FRTN Multivariable Control, Lecture Automatic Control LTH, 6 L-L Specifications, models and loop-shaping by hand L6-L8 Limitations on achievable performance L9-L Controller optimization:
More informationFRTN10 Multivariable Control, Lecture 13. Course outline. The Q-parametrization (Youla) Example: Spring-mass System
FRTN Multivariable Control, Lecture 3 Anders Robertsson Automatic Control LTH, Lund University Course outline The Q-parametrization (Youla) L-L5 Purpose, models and loop-shaping by hand L6-L8 Limitations
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 informationOn the Sandwich Theorem and a approximation algorithm for MAX CUT
On the Sandwich Theorem and a 0.878-approximation algorithm for MAX CUT Kees Roos Technische Universiteit Delft Faculteit Electrotechniek. Wiskunde en Informatica e-mail: C.Roos@its.tudelft.nl URL: http://ssor.twi.tudelft.nl/
More informationConvex relaxation. In example below, we have N = 6, and the cut we are considering
Convex relaxation The art and science of convex relaxation revolves around taking a non-convex problem that you want to solve, and replacing it with a convex problem which you can actually solve the solution
More informationSemidefinite and Second Order Cone Programming Seminar Fall 2012 Project: Robust Optimization and its Application of Robust Portfolio Optimization
Semidefinite and Second Order Cone Programming Seminar Fall 2012 Project: Robust Optimization and its Application of Robust Portfolio Optimization Instructor: Farid Alizadeh Author: Ai Kagawa 12/12/2012
More informationSparsity Matters. Robert J. Vanderbei September 20. IDA: Center for Communications Research Princeton NJ.
Sparsity Matters Robert J. Vanderbei 2017 September 20 http://www.princeton.edu/ rvdb IDA: Center for Communications Research Princeton NJ The simplex method is 200 times faster... The simplex method is
More informationDuality. 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 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 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 informationIntroduction to Semidefinite Programming I: Basic properties a
Introduction to Semidefinite Programming I: Basic properties and variations on the Goemans-Williamson approximation algorithm for max-cut MFO seminar on Semidefinite Programming May 30, 2010 Semidefinite
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 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 information1 Strict local optimality in unconstrained optimization
ORF 53 Lecture 14 Spring 016, Princeton University Instructor: A.A. Ahmadi Scribe: G. Hall Thursday, April 14, 016 When in doubt on the accuracy of these notes, please cross check with the instructor s
More 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 informationMS-E2140. Lecture 1. (course book chapters )
Linear Programming MS-E2140 Motivations and background Lecture 1 (course book chapters 1.1-1.4) Linear programming problems and examples Problem manipulations and standard form Graphical representation
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 informationLearning the Kernel Matrix with Semidefinite Programming
Journal of Machine Learning Research 5 (2004) 27-72 Submitted 10/02; Revised 8/03; Published 1/04 Learning the Kernel Matrix with Semidefinite Programg Gert R.G. Lanckriet Department of Electrical Engineering
More informationDistributionally Robust Convex Optimization
Submitted to Operations Research manuscript OPRE-2013-02-060 Authors are encouraged to submit new papers to INFORMS journals by means of a style file template, which includes the journal title. However,
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 informationSymmetries in Experimental Design and Group Lasso Kentaro Tanaka and Masami Miyakawa
Symmetries in Experimental Design and Group Lasso Kentaro Tanaka and Masami Miyakawa Workshop on computational and algebraic methods in statistics March 3-5, Sanjo Conference Hall, Hongo Campus, University
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 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 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 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 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 informationLifting for conic mixed-integer programming
Math. Program., Ser. A DOI 1.17/s117-9-282-9 FULL LENGTH PAPER Lifting for conic mixed-integer programming Alper Atamtürk Vishnu Narayanan Received: 13 March 28 / Accepted: 28 January 29 The Author(s)
More informationHandout 6: Some Applications of Conic Linear Programming
ENGG 550: Foundations of Optimization 08 9 First Term Handout 6: Some Applications of Conic Linear Programming Instructor: Anthony Man Cho So November, 08 Introduction Conic linear programming CLP, and
More informationHandout 8: Dealing with Data Uncertainty
MFE 5100: Optimization 2015 16 First Term Handout 8: Dealing with Data Uncertainty Instructor: Anthony Man Cho So December 1, 2015 1 Introduction Conic linear programming CLP, and in particular, semidefinite
More informationSEMIDEFINITE PROGRAM BASICS. Contents
SEMIDEFINITE PROGRAM BASICS BRIAN AXELROD Abstract. A introduction to the basics of Semidefinite programs. Contents 1. Definitions and Preliminaries 1 1.1. Linear Algebra 1 1.2. Convex Analysis (on R n
More informationIII. Applications in convex optimization
III. Applications in convex optimization nonsymmetric interior-point methods partial separability and decomposition partial separability first order methods interior-point methods Conic linear optimization
More informationSolution to EE 617 Mid-Term Exam, Fall November 2, 2017
Solution to EE 67 Mid-erm Exam, Fall 207 November 2, 207 EE 67 Solution to Mid-erm Exam - Page 2 of 2 November 2, 207 (4 points) Convex sets (a) (2 points) Consider the set { } a R k p(0) =, p(t) for t
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 informationCO350 Linear Programming Chapter 6: The Simplex Method
CO50 Linear Programming Chapter 6: The Simplex Method rd June 2005 Chapter 6: The Simplex Method 1 Recap Suppose A is an m-by-n matrix with rank m. max. c T x (P ) s.t. Ax = b x 0 On Wednesday, we learned
More informationChapter 3. Some Applications. 3.1 The Cone of Positive Semidefinite Matrices
Chapter 3 Some Applications Having developed the basic theory of cone programming, it is time to apply it to our actual subject, namely that of semidefinite programming. Indeed, any semidefinite program
More informationHomework 3. Convex Optimization /36-725
Homework 3 Convex Optimization 10-725/36-725 Due Friday October 14 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 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 informationMS-E2140. Lecture 1. (course book chapters )
Linear Programming MS-E2140 Motivations and background Lecture 1 (course book chapters 1.1-1.4) Linear programming problems and examples Problem manipulations and standard form problems Graphical representation
More information4. Algebra and Duality
4-1 Algebra and Duality P. Parrilo and S. Lall, CDC 2003 2003.12.07.01 4. Algebra and Duality Example: non-convex polynomial optimization Weak duality and duality gap The dual is not intrinsic The cone
More 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 informationLecture: Cone programming. Approximating the Lorentz cone.
Strong relaxations for discrete optimization problems 10/05/16 Lecture: Cone programming. Approximating the Lorentz cone. Lecturer: Yuri Faenza Scribes: Igor Malinović 1 Introduction Cone programming is
More information12. Interior-point methods
12. Interior-point methods Convex Optimization Boyd & Vandenberghe inequality constrained minimization logarithmic barrier function and central path barrier method feasibility and phase I methods complexity
More informationA direct formulation for sparse PCA using semidefinite programming
A direct formulation for sparse PCA using semidefinite programming A. d Aspremont, L. El Ghaoui, M. Jordan, G. Lanckriet ORFE, Princeton University & EECS, U.C. Berkeley A. d Aspremont, INFORMS, Denver,
More informationLecture 1: Introduction
EE 227A: Convex Optimization and Applications January 17 Lecture 1: Introduction Lecturer: Anh Pham Reading assignment: Chapter 1 of BV 1. Course outline and organization Course web page: http://www.eecs.berkeley.edu/~elghaoui/teaching/ee227a/
More informationBarrier Method. Javier Peña Convex Optimization /36-725
Barrier Method Javier Peña Convex Optimization 10-725/36-725 1 Last time: Newton s method For root-finding F (x) = 0 x + = x F (x) 1 F (x) For optimization x f(x) x + = x 2 f(x) 1 f(x) Assume f strongly
More informationRobust linear optimization under general norms
Operations Research Letters 3 (004) 50 56 Operations Research Letters www.elsevier.com/locate/dsw Robust linear optimization under general norms Dimitris Bertsimas a; ;, Dessislava Pachamanova b, Melvyn
More information