Semidefinite Programming


 Emery Dean
 2 years ago
 Views:
Transcription
1 Semidefinite Programming Basics and SOS Fernando Mário de Oliveira Filho Campos do Jordão, 2 November 23 Available at: under talks
2 Conic programming V is a real vector space h, i is an inner product Convex cone: set C V such that x + y 2 C 8 x, y 2 C and, Given: c, a,...,a m 2 V and b,...,b m 2 R max hc, xi ha i, xi = b i x 2 C for i =,..., m Complexity: depends on the cone!
3 Examples V = R n and C = R n +: linear programming V = S n and C = cone of PSD matrices: SDP A, B 2 S n ha, Bi =tr(a T B)= nx i,j= A ij B ij = trace inner product A is PSD (also A ): x T Ax 8 x 2 R n () all eigenvalues are nonnegative () A = Q T Q for some Q 2 R n n () A = P n i= iu i u T i,with i
4 Semidefinite programming Given: C, A,...,A m 2 S n and b,...,b m 2 R max hc, Xi ha i, Xi = b i X for i =,..., m x z z y x, y and xy z 2 () Generalizes LP: make all matrices diagonal! Duality theory: comes from conic programming Solvable in polytime: under some conditions (later) Numerical methods: efficient interiorpoint methods
5 Application: polynomial optimization Is the polynomial p 2 R[x,...,x n ] nonnegative? Sure, if it is a sumofsquares: p = q qk 2 Exercise: A univariate poly is nonnegative iff it is SOS P n,d = nonnegative polys, n variables, degree d n,d = SOS polys, n variables, degree d Theorem. n,d P n,d with equality i bivariate polys (n =2) quadratics (d =2) n =3, d =4 David Hilbert 888
6 SOS is SDP p 2 R[x,...,x n ] degree d v(x) = vector with all monomials degree apple d/2 Theorem. p is SOS i p(x) =hq, v(x)v(x) T i for some Q. n = 2, d =6 v(x) =, x,x 2,x 3,y,xy,x 2 y, y 2,xy 2,y 3 x x 2 x 3 y xy x 2 y y 2 xy 2 y 3 x x 2 x 3 x 4 xy x 2 y x 3 y xy 2 x 2 y 2 xy 3 x 2 x 3 x 4 x 5 x 2 y x 3 y x 4 y x 2 y 2 x 3 y 2 x 2 y 3 x 3 x 4 x 5 x 6 x 3 y x 4 y x 5 y x 3 y 2 x 4 y 2 x 3 y 3 y xy x 2 y x 3 y y 2 xy 2 x 2 y 2 y 3 xy 3 y 4 xy x 2 y x 3 y x 4 y xy 2 x 2 y 2 x 3 y 2 xy 3 x 2 y 3 xy 4 x 2 y x 3 y x 4 y x 5 y x 2 y 2 x 3 y 2 x 4 y 2 x 2 y 3 x 3 y 3 x 2 y 4 y 2 xy 2 x 2 y 2 x 3 y 2 y 3 xy 3 x 2 y 3 y 4 xy 4 y 5 xy 2 x 2 y 2 x 3 y 2 x 4 y 2 xy 3 x 2 y 3 x 3 y 3 xy 4 x 2 y 4 xy 5 y 3 xy 3 x 2 y 3 x 3 y 3 y 4 xy 4 x 2 y 4 y 5 xy 5 y 6 C A
7 A small example p(x, y) =x 4 +2x 3 y + x 2 y 2 6x 2 y 6xy 2 + x 2 + xy +9y 2 3y + 25 v(x) =(,x,x 2,y,y 2,xy) Q x x 2 y y 2 xy q q 2 q 3 q 4 q 5 q 6 x q 2 q 22 q 23 q 24 q 25 q 26 x 2 q 3 q 32 q 33 q 34 q 35 q 36 y q 4 q 42 q 43 q 44 q 45 q 46 y 2 q 5 q 52 q 53 q 54 q 55 q 56 xy q 6 q 62 q 63 q 64 q 65 q 66 C A v(x)v(x) T x x 2 y y 2 xy x x 2 y y 2 xy x x x 2 x 3 xy xy 2 x 2 y x 2 x 2 x 3 x 4 x 2 y x 2 y 2 x 3 y y y xy x 2 y y 2 y 3 xy 2 y 2 y 2 xy 2 x 2 y 2 y 3 y 4 xy 3 xy xy x 2 y x 3 y xy 2 xy 3 x 2 y 2 C A q = 25 q 23 + q 32 = q 25 + q 52 + q 46 + q 64 = 6 Q = X u i u T i =) p = X (u T i v(x)) 2
8 Sparse SDPA format: Setup maximize hf,yi hf i,yi = c i Y. for i =,..., m 2 Matrices have a block structure 3 Block 4, a diagonal block
9 Sparse SDPA format maximize Text file problem.sdpa : Number of constraint matrices: Number of blocks: hf,yi hf i,yi = c i Y. m N for i =,..., m Block sizes: Righthand side: Matrix entries: s s 2 s N c c 2 c m h matrix number i h block number i h i i h j i h entry i Remarks: Indices always start at Write the size of a block as a negative number to indicate a diagonal block Only upperdiagonal entries need to be given!
10 A feasibility problem Is the polynomial a sum of squares? p(x) =x 4 +6x 3 + 5x 2 x + 7 v(x) =(,x,x 2 ) F = F 3 A F A F 4 A F 2 A F 5 A c = 7, c 2 =, c 3 = 5, c 4 = 6, c 5 = maximize hf,yi hf i,yi = c i for i =,..., 5 Y.
11 The SDPA file F = F 3 A F A F 4 A F 2 A F 5 A c = 7, c 2 =, c 3 = 5, c 4 = 6, c 5 = File sos.sdpa : 5 Number of constraint matrices! We have only one block! 3 Which is 3x3! The righthand side. Matrix entries! 2 2.! 3 3.! ! !
12 CSDP output..... Dual solution, only numbers! e Dual solution, matrix! e! e! 2.7e+ Primal solution, matrix! e+! e+! e+! e+! e+ Y A p (x) = x x p 2 (x) = x x p 3 (x) = x 2 p 2 + p p 2 3 = x 4 +6x 3 + 5x 2 x + 7
13 Other uses of SOS Finding the minimum of a polynomial minimum of p is () p is nonnegative () p is SOS Formulate as SDP: max{ : p is SOS } Nonnegativity in a domain p 2 R[x] is nonnegative in [a, b] () 9 q, q 2 SOS s.t. p =(x a)(b x)q + q 2
14 Duality in conic programming C V a cone Dual cone: C = { x 2 V : hx, yi for all y 2 C } Primal standard form: max hc, xi ha i, xi = b i for i =,..., m x 2 C strictly feasible: x in interior of C Dual standard form: min y b + + y m b m y a + + y m a m strictly feasible: y a + + y m a m c 2 C c in interior of C
15 Duality theorems Weak duality: x is primal feasible, y is dual feasible =) hc, xi appley T b Strong duality: If: Then: If: Then: dual is bounded and strictly feasible primal attains optimal and values coincide primal is bounded and strictly feasible dual attains optimal and values coincide
16 Applied to SDP Exercise: PSD cone is selfdual Primal standard form: max hc, Xi ha i, Xi = b i X for i =,..., m Dual standard form: strictly feasible: X is positive definite min y b + + y m b m y A + + y m A m C strictly feasible: y A + + y m A m C is positive definite
17 SDP and duality gone wrong Optimal not attained: min max X 2 X 2 X =,X 22 = Positive duality gap: max X X 22 X = 2X 3 + X 22 = X min y 2 y + y y 2 + A y 2 Optimal:  Optimal:
18 Summary and coming soon Semidefinite programming Generalizes LP Fits framework of conic programming Solvable in poly time (under some mild assumptions) Has a duality theory (inherits from conic programming) One useful application: SOS polynomials Tomorrow: Approximation algorithms Maximumcut problem Grothendieck inequalities and applications
Global Optimization of Polynomials
Semidefinite Programming Lecture 9 OR 637 Spring 2008 April 9, 2008 Scribe: Dennis Leventhal Global Optimization of Polynomials Recall we were considering the problem min z R n p(z) where p(z) is a degree
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 informationIntroduction to Semidefinite Programming I: Basic properties a
Introduction to Semidefinite Programming I: Basic properties and variations on the GoemansWilliamson approximation algorithm for maxcut MFO seminar on Semidefinite Programming May 30, 2010 Semidefinite
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 informationApproximation Algorithms
Approximation Algorithms Chapter 26 Semidefinite Programming Zacharias Pitouras 1 Introduction LP place a good lower bound on OPT for NPhard problems Are there other ways of doing this? Vector programs
More informationLMI MODELLING 4. CONVEX LMI MODELLING. Didier HENRION. LAASCNRS Toulouse, FR Czech Tech Univ Prague, CZ. Universidad de Valladolid, SP March 2009
LMI MODELLING 4. CONVEX LMI MODELLING Didier HENRION LAASCNRS 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: 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 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 informationContinuous Optimisation, Chpt 9: Semidefinite Optimisation
Continuous Optimisation, Chpt 9: Semidefinite Optimisation Peter J.C. Dickinson DMMP, University of Twente p.j.c.dickinson@utwente.nl http://dickinson.website/teaching/2017co.html version: 28/11/17 Monday
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 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 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 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 informationSDP Relaxations for MAXCUT
SDP Relaxations for MAXCUT from Random Hyperplanes to SumofSquares Certificates CATS @ UMD March 3, 2017 Ahmed Abdelkader MAXCUT SDP SOS March 3, 2017 1 / 27 Overview 1 MAXCUT, Hardness and UGC 2 LP
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 informationLecture 3: Semidefinite Programming
Lecture 3: Semidefinite Programming Lecture Outline Part I: Semidefinite programming, examples, canonical form, and duality Part II: Strong Duality Failure Examples Part III: Conditions for strong duality
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 informationCOURSE ON LMI PART I.2 GEOMETRY OF LMI SETS. Didier HENRION henrion
COURSE ON LMI PART I.2 GEOMETRY OF LMI SETS Didier HENRION www.laas.fr/ henrion October 2006 Geometry of LMI sets Given symmetric matrices F i we want to characterize the shape in R n of the LMI set F
More informationIII. Applications in convex optimization
III. Applications in convex optimization nonsymmetric interiorpoint methods partial separability and decomposition partial separability first order methods interiorpoint methods Conic linear optimization
More informationSemidefinite and Second Order Cone Programming Seminar Fall 2001 Lecture 2
Semidefinite and Second Order Cone Programming Seminar Fall 2001 Lecture 2 Instructor: Farid Alizadeh Scribe: Xuan Li 9/17/2001 1 Overview We survey the basic notions of cones and conelp and give several
More informationContinuous Optimisation, Chpt 9: Semidefinite Problems
Continuous Optimisation, Chpt 9: Semidefinite Problems Peter J.C. Dickinson DMMP, University of Twente p.j.c.dickinson@utwente.nl http://dickinson.website/teaching/2016co.html version: 21/11/16 Monday
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 information61 The Positivstellensatz P. Parrilo and S. Lall, ECC
61 The Positivstellensatz P. Parrilo and S. Lall, ECC 2003 2003.09.02.10 6. The Positivstellensatz Basic semialgebraic sets Semialgebraic sets TarskiSeidenberg and quantifier elimination Feasibility
More informationAgenda. 1 Duality for LP. 2 Theorem of alternatives. 3 Conic Duality. 4 Dual cones. 5 Geometric view of cone programs. 6 Conic duality theorem
Agenda 1 Duality for LP 2 Theorem of alternatives 3 Conic Duality 4 Dual cones 5 Geometric view of cone programs 6 Conic duality theorem 7 Examples Lower bounds on LPs By eliminating variables (if needed)
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 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 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 information15. Conic optimization
L. Vandenberghe EE236C (Spring 216) 15. Conic optimization conic linear program examples modeling duality 151 Generalized (conic) inequalities Conic inequality: a constraint x K where K is a convex cone
More informationPOLYNOMIAL OPTIMIZATION WITH SUMSOFSQUARES INTERPOLANTS
POLYNOMIAL OPTIMIZATION WITH SUMSOFSQUARES INTERPOLANTS Sercan Yıldız syildiz@samsi.info in collaboration with Dávid Papp (NCSU) OPT Transition Workshop May 02, 2017 OUTLINE Polynomial optimization and
More informationFast ADMM for Sum of Squares Programs Using Partial Orthogonality
Fast ADMM for Sum of Squares Programs Using Partial Orthogonality Antonis Papachristodoulou Department of Engineering Science University of Oxford www.eng.ox.ac.uk/control/sysos antonis@eng.ox.ac.uk with
More informationReal Symmetric Matrices and Semidefinite Programming
Real Symmetric Matrices and Semidefinite Programming Tatsiana Maskalevich Abstract Symmetric real matrices attain an important property stating that all their eigenvalues are real. This gives rise to many
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 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 informationm i=1 c ix i i=1 F ix i F 0, X O.
What is SDP? for a beginner of SDP Copyright C 2005 SDPA Project 1 Introduction This note is a short course for SemiDefinite Programming s SDP beginners. SDP has various applications in, for example, control
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 FermatWeber Problem. Next we will
More informationSummer School: Semidefinite Optimization
Summer School: Semidefinite Optimization Christine Bachoc Université Bordeaux I, IMB Research Training Group Experimental and Constructive Algebra Haus Karrenberg, Sept. 3  Sept. 7, 2012 Duality Theory
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 informationConic Linear Optimization and its Dual. yyye
Conic Linear Optimization and Appl. MS&E314 Lecture Note #04 1 Conic Linear Optimization and its Dual Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A.
More informationAn E cient A nescaling Algorithm for Hyperbolic Programming
An E cient A nescaling Algorithm for Hyperbolic Programming Jim Renegar joint work with Mutiara Sondjaja 1 Euclidean space A homogeneous polynomial p : E!R is hyperbolic if there is a vector e 2E such
More informationLecture 5. 1 GoermansWilliamson Algorithm for the maxcut problem
Math 280 Geometric and Algebraic Ideas in Optimization April 26, 2010 Lecture 5 Lecturer: Jesús A De Loera Scribe: HuyDung Han, Fabio Lapiccirella 1 GoermansWilliamson Algorithm for the maxcut problem
More informationSemidefinite Programming, Combinatorial Optimization and Real Algebraic Geometry
Semidefinite Programming, Combinatorial Optimization and Real Algebraic Geometry assoc. prof., Ph.D. 1 1 UNM  Faculty of information studies Edinburgh, 16. September 2014 Outline Introduction Definition
More informationConvex Optimization Theory. Chapter 5 Exercises and Solutions: Extended Version
Convex Optimization Theory Chapter 5 Exercises and Solutions: Extended Version Dimitri P. Bertsekas Massachusetts Institute of Technology Athena Scientific, Belmont, Massachusetts http://www.athenasc.com
More informationSparse Matrix Theory and Semidefinite Optimization
Sparse Matrix Theory and Semidefinite Optimization Lieven Vandenberghe Department of Electrical Engineering University of California, Los Angeles Joint work with Martin S. Andersen and Yifan Sun Third
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 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 informationSF2822 Applied nonlinear optimization, final exam Wednesday June
SF2822 Applied nonlinear optimization, final exam Wednesday June 3 205 4.00 9.00 Examiner: Anders Forsgren, tel. 08790 7 27. Allowed tools: Pen/pencil, ruler and eraser. Note! Calculator is not allowed.
More informationLecture 5. The Dual Cone and Dual Problem
IE 8534 1 Lecture 5. The Dual Cone and Dual Problem IE 8534 2 For a convex cone K, its dual cone is defined as K = {y x, y 0, x K}. The innerproduct can be replaced by x T y if the coordinates of the
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 informationThe Simplest Semidefinite Programs are Trivial
The Simplest Semidefinite Programs are Trivial Robert J. Vanderbei Bing Yang Program in Statistics & Operations Research Princeton University Princeton, NJ 08544 January 10, 1994 Technical Report SOR9312
More informationCSCI 1951G Optimization Methods in Finance Part 01: Linear Programming
CSCI 1951G Optimization Methods in Finance Part 01: Linear Programming January 26, 2018 1 / 38 Liability/asset cashflow matching problem Recall the formulation of the problem: max w c 1 + p 1 e 1 = 150
More informationAssignment 1: From the Definition of Convexity to Helley Theorem
Assignment 1: From the Definition of Convexity to Helley Theorem Exercise 1 Mark in the following list the sets which are convex: 1. {x R 2 : x 1 + i 2 x 2 1, i = 1,..., 10} 2. {x R 2 : x 2 1 + 2ix 1x
More informationResearch overview. Seminar September 4, Lehigh University Department of Industrial & Systems Engineering. Research overview.
Research overview Lehigh University Department of Industrial & Systems Engineering COR@L Seminar September 4, 2008 1 Duality without regularity condition Duality in nonexact arithmetic 2 interior point
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 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 informationPositive semidefinite rank
1/15 Positive semidefinite rank Hamza Fawzi (MIT, LIDS) Joint work with João Gouveia (Coimbra), Pablo Parrilo (MIT), Richard Robinson (Microsoft), James Saunderson (Monash), Rekha Thomas (UW) DIMACS Workshop
More informationLECTURE 13 LECTURE OUTLINE
LECTURE 13 LECTURE OUTLINE Problem Structures Separable problems Integer/discrete problems Branchandbound Large sum problems Problems with many constraints Conic Programming Second Order Cone Programming
More informationFacial Reduction and Geometry on Conic Programming
Facial Reduction and Geometry on Conic Programming Masakazu Muramatsu UEC Bruno F. Lourenço, and Takashi Tsuchiya Seikei University GRIPS Based on the papers of LMT: 1. A structural geometrical analysis
More informationInterior Point Methods: SecondOrder 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: SecondOrder Cone Programming and Semidefinite Programming Jacek Gondzio Email: J.Gondzio@ed.ac.uk URL: http://www.maths.ed.ac.uk/~gondzio
More information6.854J / J Advanced Algorithms Fall 2008
MIT OpenCourseWare http://ocw.mit.edu 6.85J / 8.5J Advanced Algorithms Fall 008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 8.5/6.85 Advanced Algorithms
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 informationminimize x x2 2 x 1x 2 x 1 subject to x 1 +2x 2 u 1 x 1 4x 2 u 2, 5x 1 +76x 2 1,
4 Duality 4.1 Numerical perturbation analysis example. Consider the quadratic program with variables x 1, x 2, and parameters u 1, u 2. minimize x 2 1 +2x2 2 x 1x 2 x 1 subject to x 1 +2x 2 u 1 x 1 4x
More informationToday: Linear Programming (con t.)
Today: Linear Programming (con t.) COSC 581, Algorithms April 10, 2014 Many of these slides are adapted from several online sources Reading Assignments Today s class: Chapter 29.4 Reading assignment for
More informationWhat can be expressed via Conic Quadratic and Semidefinite Programming?
What can be expressed via Conic Quadratic and Semidefinite Programming? A. Nemirovski Faculty of Industrial Engineering and Management Technion Israel Institute of Technology Abstract Tremendous recent
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 informationA solution approach for linear optimization with completely positive matrices
A solution approach for linear optimization with completely positive matrices Franz Rendl http://www.math.uniklu.ac.at AlpenAdriaUniversität Klagenfurt Austria joint work with M. Bomze (Wien) and F.
More information1 Quantum states and von Neumann entropy
Lecture 9: Quantum entropy maximization CSE 599S: Entropy optimality, Winter 2016 Instructor: James R. Lee Last updated: February 15, 2016 1 Quantum states and von Neumann entropy Recall that S sym n n
More informationAdvanced SDPs Lecture 6: March 16, 2017
Advanced SDPs Lecture 6: March 16, 2017 Lecturers: Nikhil Bansal and Daniel Dadush Scribe: Daniel Dadush 6.1 Notation Let N = {0, 1,... } denote the set of nonnegative integers. For α N n, define the
More informationLecture 20: November 1st
10725: 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 informationOptimization for Communications and Networks. Poompat Saengudomlert. Session 4 Duality and Lagrange Multipliers
Optimization for Communications and Networks Poompat Saengudomlert Session 4 Duality and Lagrange Multipliers P Saengudomlert (2015) Optimization Session 4 1 / 14 24 Dual Problems Consider a primal convex
More 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 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: 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 5. Theorems of Alternatives and SelfDual Embedding
IE 8534 1 Lecture 5. Theorems of Alternatives and SelfDual Embedding IE 8534 2 A system of linear equations may not have a solution. It is well known that either Ax = c has a solution, or A T y = 0, c
More informationThe Ongoing Development of CSDP
The Ongoing Development of CSDP Brian Borchers Department of Mathematics New Mexico Tech Socorro, NM 87801 borchers@nmt.edu Joseph Young Department of Mathematics New Mexico Tech (Now at Rice University)
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 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 nonconvex problem that you want to solve, and replacing it with a convex problem which you can actually solve the solution
More informationQuadratic reformulation techniques for 01 quadratic programs
OSE SEMINAR 2014 Quadratic reformulation techniques for 01 quadratic programs Ray Pörn CENTER OF EXCELLENCE IN OPTIMIZATION AND SYSTEMS ENGINEERING ÅBO AKADEMI UNIVERSITY ÅBO NOVEMBER 14th 2014 2 Structure
More informationHilbert s 17th Problem to Semidefinite Programming & Convex Algebraic Geometry
Hilbert s 17th Problem to Semidefinite Programming & Convex Algebraic Geometry Rekha R. Thomas University of Washington, Seattle References Monique Laurent, Sums of squares, moment matrices and optimization
More informationOptimization Methods. Lecture 23: Semidenite Optimization
5.93 Optimization Methods Lecture 23: Semidenite Optimization Outline. Preliminaries Slide 2. SDO 3. Duality 4. SDO Modeling Power 5. Barrier lgorithm for SDO 2 Preliminaries symmetric matrix is positive
More informationCopositive Programming and Combinatorial Optimization
Copositive Programming and Combinatorial Optimization Franz Rendl http://www.math.uniklu.ac.at AlpenAdriaUniversität Klagenfurt Austria joint work with M. Bomze (Wien) and F. Jarre (Düsseldorf) and
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 informationRobust and Optimal Control, Spring 2015
Robust and Optimal Control, Spring 2015 Instructor: Prof. Masayuki Fujita (S5303B) G. Sum of Squares (SOS) G.1 SOS Program: SOS/PSD and SDP G.2 Duality, valid ineqalities and Cone G.3 Feasibility/Optimization
More information16.1 L.P. Duality Applied to the Minimax Theorem
CS787: Advanced Algorithms Scribe: David Malec and Xiaoyong Chai Lecturer: Shuchi Chawla Topic: Minimax Theorem and SemiDefinite Programming Date: October 22 2007 In this lecture, we first conclude our
More informationCSCI 1951G Optimization Methods in Finance Part 10: Conic Optimization
CSCI 1951G 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 informationCopositive Programming and Combinatorial Optimization
Copositive Programming and Combinatorial Optimization Franz Rendl http://www.math.uniklu.ac.at AlpenAdriaUniversität Klagenfurt Austria joint work with I.M. Bomze (Wien) and F. Jarre (Düsseldorf) IMA
More informationMore FirstOrder Optimization Algorithms
More FirstOrder Optimization Algorithms Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A. http://www.stanford.edu/ yyye Chapters 3, 8, 3 The SDM
More informationConvex Optimization. (EE227A: UC Berkeley) Lecture 28. Suvrit Sra. (Algebra + Optimization) 02 May, 2013
Convex Optimization (EE227A: UC Berkeley) Lecture 28 (Algebra + Optimization) 02 May, 2013 Suvrit Sra Admin Poster presentation on 10th May mandatory HW, Midterm, Quiz to be reweighted Project final report
More informationarxiv: v1 [math.oc] 31 Jan 2017
CONVEX CONSTRAINED SEMIALGEBRAIC VOLUME OPTIMIZATION: APPLICATION IN SYSTEMS AND CONTROL 1 Ashkan Jasour, Constantino Lagoa School of Electrical Engineering and Computer Science, Pennsylvania State University
More informationLecture #21. c T x Ax b. maximize subject to
COMPSCI 330: Design and Analysis of Algorithms 11/11/2014 Lecture #21 Lecturer: Debmalya Panigrahi Scribe: Samuel Haney 1 Overview In this lecture, we discuss linear programming. We first show that the
More 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 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 4: January 26
10725/36725: Conve Optimization Spring 2015 Lecturer: Javier Pena Lecture 4: January 26 Scribes: Vipul Singh, Shinjini Kundu, ChiaYin Tsai Note: LaTeX template courtesy of UC Berkeley EECS dept. Disclaimer:
More informationA Review of Linear Programming
A Review of Linear Programming Instructor: Farid Alizadeh IEOR 4600y Spring 2001 February 14, 2001 1 Overview In this note we review the basic properties of linear programming including the primal simplex
More informationDuality revisited. Javier Peña Convex Optimization /36725
Duality revisited Javier Peña Conve Optimization 10725/36725 1 Last time: barrier method Main idea: approimate the problem f() + I C () with the barrier problem f() + 1 t φ() tf() + φ() where t > 0 and
More informationPrimalDual Geometry of Level Sets and their Explanatory Value of the Practical Performance of InteriorPoint Methods for Conic Optimization
PrimalDual Geometry of Level Sets and their Explanatory Value of the Practical Performance of InteriorPoint Methods for Conic Optimization Robert M. Freund M.I.T. June, 2010 from papers in SIOPT, Mathematics
More informationSolving large Semidefinite Programs  Part 1 and 2
Solving large Semidefinite Programs  Part 1 and 2 Franz Rendl http://www.math.uniklu.ac.at AlpenAdriaUniversität Klagenfurt Austria F. Rendl, Singapore workshop 2006 p.1/34 Overview Limits of Interior
More informationU.C. Berkeley CS294: Beyond WorstCase Analysis Handout 12 Luca Trevisan October 3, 2017
U.C. Berkeley CS94: Beyond WorstCase 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 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 information9. Interpretations, Lifting, SOS and Moments
91 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 informationIterative LP and SOCPbased. approximations to. sum of squares programs. Georgina Hall Princeton University. Joint work with:
Iterative LP and SOCPbased approximations to sum of squares programs Georgina Hall Princeton University Joint work with: Amir Ali Ahmadi (Princeton University) Sanjeeb Dash (IBM) Sum of squares programs
More information