l p -Norm Constrained Quadratic Programming: Conic Approximation Methods
|
|
- Prosper Casey
- 5 years ago
- Views:
Transcription
1 OUTLINE l p -Norm Constrained Quadratic Programming: Conic Approximation Methods Wenxun Xing Department of Mathematical Sciences Tsinghua University, Beijing wxing@math.tsinghua.edu.cn
2 OUTLINE OUTLINE 1 l p -Norm Constrained Quadratic Programming
3 Data fitting l 2 -norm: least-square data fitting min Ax b 2 s.t. x R n. When A is full rank in column, then x = A T A 1 A T b. A 2nd-order conic programming formulation min s.t. t Ax b 2 t x R n. Experts in numerical analysis prefer the direct calculation much more than the optimal solution method.
4 l 1 - norm problem l 1 -norm. min x 1 s.t. Ax = b x R n. A linear programming formulation min s.t. n i=1 t i t i x i t i, i = 1, 2,..., n Ax = b t, x R n.
5 Heuristic method for finding a sparse solution Regressor selection problem: A potential regressors, b to be fit by a linear combination of A min Ax b 2 s.t. cardx k x Z n +. It is NP-hard. Let m = 1, A = a 1, a 2,..., a n, b = 1 n 2 i=1 a i, k n 2. It is a partition problem. Heuristic method. min Ax b 2 + γ x 1 s.t. x R n. Ref. S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004.
6 Regularized approximation min Ax b 2 + γ x 1 s.t. x R n. l 1 -norm and l 2 -norm constrained programming min t 1 + γt 2 s.t. Ax b 2 t 1 x 1 t 2 x R n, t 1, t 2 R. The objective function is linear, the first constraint is a 2nd-order cone and the 2nd is a 1st-order cone. It is a convex optimization problem of polynomially solvable.
7 p-norm domain Black: 1-norm. Red: 2-norm. Green: 3-norm. Yellow: 8-norm.
8 Convex l p -norm problems p-norm domain is convex p 1. For set {x x p 1}, the smallest one is the domain with p = 1, which is the smallest convex set containing integer points { 1, 1} n. For p 1, the l p -norm problems with linear objective or linear constraints are polynomially solvable. Variants of l p -norm problems should be considered.
9 Variants of l p -norm problems l 2 -norm constrained quadratic problem min x T Qx + q T x s.t. Ax b 2 c T x c T x = d 0 x R n. l 1 -norm constrained quadratic problem min s.t. x T Qx + q T x x 1 k x R n, where Q is a general symmetric matrix.
10 l p -Norm Constrained Quadratic Programming min where p xt Qx + q T x s.t. 1 2 xt Q i x + q T i x + c i 0, i = 1, 2,..., m Ax b p c T x x R n,
11 QCQP reformulation where D R n. min 1 2 xt Q 0 x + q T 0 x + c 0 s.t. 1 2 xt Q i x + q T i x + c i 0, i = 1, 2,..., m x D,
12 p-norm form l 1 -norm problem min x T Qx + q T x s.t. x 1 k x R n. Denote D = {x R n x 1 k}. QCQP form min x T Qx + q T x s.t. x D.
13 2-norm form 2-norm problem min x T Qx + q T x s.t. Ax b 2 c T x c T x = d 0 x R n. Denote D = { x R n Ax b 2 c T x } QCQP form min s.t. x T Qx + q T x c T x = d x D.
14 Lifting reformulation min f x = 1 2 xt Q 0 x + q T 0 x + c 0 s.t. g i x = 1 2 xt Q i x + q T i x + c i 0, i = 1, 2,..., m QCQP x D. Denote: F = {x D g i x 0, i = 1, 2,..., m}. Lifting min 1 2c 0 q0 T 2 X q 0 Q 0 1 2c s.t. i qi T 2 X 0, i = 1, 2,..., m q i Q i T 1 1 X =, x F. x x
15 Convex reformulation min 1 2c 0 q0 T 2 X q 0 Q 0 1 2c s.t. i qi T 2 X 0, i = 1, 2,..., m q i Q i 1 0 X = T 1 1 X clconv x F x x.
16 Linear conic programming reformulation min 1 2c 0 q0 T 2 X q 0 Q 0 1 2c s.t. i qi T 2 X 0, i = 1, 2,..., m q i Q i 1 0 X = T 1 1 X clcone x F x x. It is a linear conic programming and has the same optimal value with QCQP.
17 Quadratic-Function Conic Programming PRIMAL min 1 2c 0 q0 T 2 V q 0 Q 0 1 2c s.t. i qi T 2 V 0, i = 1, 2,..., m QFCP q i Q i 1 0 V = T V DF = cl 1 1 cone, x F. x x F R n, A B = traceab T,
18 Quadratic-Function Conic Programming DUAL s.t. max σ 2σ + 2c m i=1 λ ic i q 0 + m i=1 λ iq i T q 0 + m i=1 λ iq i Q 0 + m i=1 λ iq i σ R, λ R m +, D F F R n, D F = U Sn+1 1 x T U 1 x 0, x F.
19 Properties of the Quadratic-Function Cone Cone of nonnegative quadratic functions Sturm and Zhang, MOR 28, T D F = U 1 1 Sn+1 U 0, x F x x. If F, then D F is the dual cone of D F and vice versa. If F is a bounded nonempty set, then DF 1 1 = cone x x T, x F. If intf, then D F and D F are proper.
20 Properties The complexity of checking whether V D F or U D F depends on F. When F = R n, D F = Sn+1 +. When F = R n +, DF is the copositive cone! Ref: recent survey papers I. M. Bomze, EJOR, ; Mirjam Dür, Recent Advances in Optimization and its Applications in Engineering, 2010; J.-B. Hiriart-Urruty and A. Seeger, SIAM Review 524, Relaxation or restriction D F S n + D F. Approximation: Computable cover of F.
21 Checking U D F is an optimization problem! D F = U Sn+1 1 x T U 1 x 0, x F. Theorem U D F if and only if the optimal value of the following problem is not negative min 1 x s.t. x F. T U 1 x If F is a p-norm constraint, then it is a p-norm constrained quadratic programming.
22 Easy cases min T 1 U x s.t. x F. 1 x If F is a p-norm constraint, then it is a p-norm constrained quadratic programming. When F = {x R n 1 2 xt Px + p T x + d 0}, P 0, intf, it is computable. { When F = Socn = x R n } x T Px c T x, P 0, intf, it is computable.
23 A special case of p-norm constrained quadratic programming min s.t. 1 2 xt Qx + q T x x p k x R n, where p 1. Equivalent formulation min s.t. 1 2 xt Qx + 1 k tqt x x p t t = k x R n.
24 Homogenous quadratic constrained model min s.t. 1 2 xt Qx + 1 k tqt x x p t t = k x R n. Homogenous quadratic form T min 1 t 2 x s.t. 0 1 k qt 1 k q Q t = k t { t, x R R n x p t } x t x
25 of the problem Homogenous: It is polynomially computable when p = 2. min x T Qx s.t. x Socn = { x R n } x T Px c T x, where Q is a general symmetric matrix, P is positive definite and Socn + 1 has an interior Ref: Ye Tian et. al., JIMO 93, Variant? min s.t. x T Qx + q T x Ax b 2 c T x c T x = d 0 x R n.
26 of the problem Homogeneous QP over the 1st-order cone is NP-hard min s.t. x 0 x x 0 x T Q x 0 x Focn + 1, where Focn + 1 = {x 0, x R R n x 1 x 0 }, and Q is a general symmetric matrix. It is NP-hard.
27 of the problem A cross section problem T 1 min Q x s.t. x 1 1 x R n 1 x Guo et. al. conjectured NP-hard Ref: Xiaoling Guo et. al., JIMO 103, It is NP-hard Ref: Yong Hsia, Optimization Letters 8, 2014.
28 of the problem A general case p 1. min s.t. t x t x T Q t x { t, x R R n x p t }. Zhou et. al. conjectured NP-hard Ref: Jing Zhou et. al., PJO to appear, Provided with many solvable subcases.
29 Quadratic-Function Conic Programming PRIMAL min 1 2c 0 q0 T 2 V q 0 Q 0 1 2c s.t. i qi T 2 V 0, i = 1, 2,..., m q i Q i 1 0 V = V D F. QFCP F R n, A B = traceab T, T DF 1 1 = cl cone, x F. x x
30 Quadratically Constrained Quadratic Programming QCQP Theorem If F, then the QFCP primal, its dual and the QCQP have the same optimal objective value. Theorem Suppose F, G 1 and G 2 be nonempty sets. Denote vf, vg 1 and vg 2 be the optimal objective value of the QFCP with F selecting different sets respectively. i If G 1 G 2, then D G1 D G2 and D G 1 D G 2. ii If F G 1 G 2, then vf vg 1 vg 2.
31 Relaxation Relaxation C DF and computable. min 1 2c 0 q0 T 2 q 0 Q 0 s.t. v 11 = 1 The worst one: C = S n+1 +. V 1 2 H i V 0, i = 1, 2,..., s V = v ij C,
32 Ellipsoid Cover of Bounded Feasible Set Easy case: Quadratic-function cone over one ellipsoid constraint. Theorem Let F = {x R n gx 0}, where gx = 1 2 xt Qx + q T x + c, intf and Q S++. n For an n + 1 n + 1 real symmetric matrix V, V DF if and only if 1 2c q T 2 V 0 q Q V S+ n+1.
33 Ellipsoid Cover of Bounded Feasible Set Ellipsoid cover Lu et al, 2011 Theorem Let G = G 1 G 2 G s, where G i = {x R n 1 2 xt B i x + b T i x + d i 0}, 1 i s, are ellipsoids with an interior, then D G = D G 1 + D G D G s. And V DG if and only if the following system is feasible V = V 1 + V V s 2d i bi T V i 0, i = 1, 2,..., s 1 2 b i B i V i S n+1 +, i = 1, 2,..., s.
34 Ellipsoid Cover of Bounded Feasible Set min H 0 V s.t. V 11 = 1 H i V 0, i = 1, 2..., m V = V V s [ ] d i bi T V i 0, V i 0, i = 1, 2,..., s. b i B i EC It is a SDP, computable!
35 Ellipsoid Cover: Decomposition Theorem Under some assumptions, if V = V V s is an optimal solution of EC, then for each j, j = 1,.., s, there exists a decomposition of V j = n j [ ] [ 1 1 µ ji i=1 x ji x ji ] T for some n j > 0, x ji G j, µ ji > 0 and n j i=1 µ ji = [Y j ] 11. Moreover, V can be decomposed in the form of V = n s j [ ] [ 1 1 µ ji j=1 i=1 with x ji G j, µ ji > 0 and s nj j=1 i=1 µ ji = V11 = 1. x ji x ji ] T
36 Ellipsoid Cover: Step 1 Cover the feasible set F with some ellipsoids. Step 2 Solve EC. Step 3 Decompose the optimal solution of EC and find a x ji with the smallest objective value sensitive point. Step 4 Check if the sensitive point x ji F. If it is, then it is a global optimum of QCQP. Otherwise, cover G j with two smaller ellipsoids. Repeat above procedure. Step 5 The approximation objective values converge to the optimal value of QCQP. Applications: QP Lu et al, to appear in OPT, 2014, 0-1 knapsack Zhou et al, JIMO 93, 2013, to detect copositve cone Deng et al, EJOR 229, 2013 etc.
37 Adaptive ellipsoid covering
38 Applications to p-norm problems: bounded feasible sets p-norm problem min x T Qx + q T x s.t. x p k x R n. F = D = { x R n x p k }. 2-norm problem min x T Qx + q T x s.t. Ax b 2 c T x c T x = d, x R n. D = { x R n Ax b 2 c T x }, F = { x D c T x = d }.
39 lp -Norm Constrained Quadratic Programming Second-order Cone Cover W. Xing Sept. 2-4, 2014, Peking University
40 For the least square problem, why the 2nd-order conic model is not used generally? Can we have more efficient algorithms than the interior point method for SDP?
41 Thank You!
Semidefinite 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 informationAn improved characterisation of the interior of the completely positive cone
Electronic Journal of Linear Algebra Volume 2 Volume 2 (2) Article 5 2 An improved characterisation of the interior of the completely positive cone Peter J.C. Dickinson p.j.c.dickinson@rug.nl Follow this
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 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 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 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 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 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 informationLecture 18: Optimization Programming
Fall, 2016 Outline Unconstrained Optimization 1 Unconstrained Optimization 2 Equality-constrained Optimization Inequality-constrained Optimization Mixture-constrained Optimization 3 Quadratic 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 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 inner-product can be replaced by x T y if the coordinates of the
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 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 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 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 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 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 informationConvexification of Mixed-Integer Quadratically Constrained Quadratic Programs
Convexification of Mixed-Integer Quadratically Constrained Quadratic Programs Laura Galli 1 Adam N. Letchford 2 Lancaster, April 2011 1 DEIS, University of Bologna, Italy 2 Department of Management Science,
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 informationCS711008Z Algorithm Design and Analysis
CS711008Z Algorithm Design and Analysis Lecture 8 Linear programming: interior point method Dongbo Bu Institute of Computing Technology Chinese Academy of Sciences, Beijing, China 1 / 31 Outline Brief
More informationELE539A: Optimization of Communication Systems Lecture 6: Quadratic Programming, Geometric Programming, and Applications
ELE539A: Optimization of Communication Systems Lecture 6: Quadratic Programming, Geometric Programming, and Applications Professor M. Chiang Electrical Engineering Department, Princeton University February
More 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 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: 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 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 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 informationarxiv: v1 [math.oc] 14 Oct 2014
arxiv:110.3571v1 [math.oc] 1 Oct 01 An Improved Analysis of Semidefinite Approximation Bound for Nonconvex Nonhomogeneous Quadratic Optimization with Ellipsoid Constraints Yong Hsia a, Shu Wang a, Zi Xu
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 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 information8. Geometric problems
8. Geometric problems Convex Optimization Boyd & Vandenberghe extremal volume ellipsoids centering classification placement and facility location 8 Minimum volume ellipsoid around a set Löwner-John ellipsoid
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 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 information8. Geometric problems
8. Geometric problems Convex Optimization Boyd & Vandenberghe extremal volume ellipsoids centering classification placement and facility location 8 1 Minimum volume ellipsoid around a set Löwner-John ellipsoid
More informationRelations between Semidefinite, Copositive, Semi-infinite and Integer Programming
Relations between Semidefinite, Copositive, Semi-infinite and Integer Programming Author: Faizan Ahmed Supervisor: Dr. Georg Still Master Thesis University of Twente the Netherlands May 2010 Relations
More informationA semidefinite relaxation scheme for quadratically constrained quadratic problems with an additional linear constraint
Iranian Journal of Operations Research Vol. 2, No. 2, 20, pp. 29-34 A semidefinite relaxation scheme for quadratically constrained quadratic problems with an additional linear constraint M. Salahi Semidefinite
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 informationMatrix Rank-One Decomposition. and Applications
Matrix rank-one decomposition and applications 1 Matrix Rank-One Decomposition and Applications Shuzhong Zhang Department of Systems Engineering and Engineering Management The Chinese University of Hong
More informationAcyclic Semidefinite Approximations of Quadratically Constrained Quadratic Programs
Acyclic Semidefinite Approximations of Quadratically Constrained Quadratic Programs Raphael Louca & Eilyan Bitar School of Electrical and Computer Engineering American Control Conference (ACC) Chicago,
More informationLecture: Duality of LP, SOCP and SDP
1/33 Lecture: Duality of LP, SOCP and SDP Zaiwen Wen Beijing International Center For Mathematical Research Peking University http://bicmr.pku.edu.cn/~wenzw/bigdata2017.html wenzw@pku.edu.cn Acknowledgement:
More information1 Introduction Semidenite programming (SDP) has been an active research area following the seminal work of Nesterov and Nemirovski [9] see also Alizad
Quadratic Maximization and Semidenite Relaxation Shuzhong Zhang Econometric Institute Erasmus University P.O. Box 1738 3000 DR Rotterdam The Netherlands email: zhang@few.eur.nl fax: +31-10-408916 August,
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 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
Semidefinite Programming Basics and SOS Fernando Mário de Oliveira Filho Campos do Jordão, 2 November 23 Available at: www.ime.usp.br/~fmario under talks Conic programming V is a real vector space h, i
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 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 informationLecture 8. Strong Duality Results. September 22, 2008
Strong Duality Results September 22, 2008 Outline Lecture 8 Slater Condition and its Variations Convex Objective with Linear Inequality Constraints Quadratic Objective over Quadratic Constraints Representation
More informationAcyclic Semidefinite Approximations of Quadratically Constrained Quadratic Programs
2015 American Control Conference Palmer House Hilton July 1-3, 2015. Chicago, IL, USA Acyclic Semidefinite Approximations of Quadratically Constrained Quadratic Programs Raphael Louca and Eilyan Bitar
More informationarxiv: v1 [math.oc] 20 Nov 2012
Manuscript submitted to AIMS Journals Volume X, Number 0X, XX 201X Website: http://aimsciences.org pp. X XX arxiv:1211.4670v1 math.oc 20 Nov 2012 DOUBLY NONNEGATIVE RELAXATION METHOD FOR SOLVING MULTIPLE
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 informationExercises. Exercises. Basic terminology and optimality conditions. 4.2 Consider the optimization problem
Exercises Basic terminology and optimality conditions 4.1 Consider the optimization problem f 0(x 1, x 2) 2x 1 + x 2 1 x 1 + 3x 2 1 x 1 0, x 2 0. Make a sketch of the feasible set. For each of the following
More informationarxiv: v1 [math.oc] 23 Nov 2012
arxiv:1211.5406v1 [math.oc] 23 Nov 2012 The equivalence between doubly nonnegative relaxation and semidefinite relaxation for binary quadratic programming problems Abstract Chuan-Hao Guo a,, Yan-Qin Bai
More informationInterior points of the completely positive cone
Electronic Journal of Linear Algebra Volume 17 Volume 17 (2008) Article 5 2008 Interior points of the completely positive cone Mirjam Duer duer@mathematik.tu-darmstadt.de Georg Still Follow this and additional
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 non-exact arithmetic 2 interior point
More informationLECTURE 13 LECTURE OUTLINE
LECTURE 13 LECTURE OUTLINE Problem Structures Separable problems Integer/discrete problems Branch-and-bound Large sum problems Problems with many constraints Conic Programming Second Order Cone Programming
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 informationApproximation Algorithms
Approximation Algorithms Chapter 26 Semidefinite Programming Zacharias Pitouras 1 Introduction LP place a good lower bound on OPT for NP-hard problems Are there other ways of doing this? Vector programs
More informationModule 04 Optimization Problems KKT Conditions & Solvers
Module 04 Optimization Problems KKT Conditions & Solvers Ahmad F. Taha EE 5243: Introduction to Cyber-Physical Systems Email: ahmad.taha@utsa.edu Webpage: http://engineering.utsa.edu/ taha/index.html September
More informationA Note on KKT Points of Homogeneous Programs 1
A Note on KKT Points of Homogeneous Programs 1 Y. B. Zhao 2 and D. Li 3 Abstract. Homogeneous programming is an important class of optimization problems. The purpose of this note is to give a truly equivalent
More informationPOLYNOMIAL OPTIMIZATION WITH SUMS-OF-SQUARES INTERPOLANTS
POLYNOMIAL OPTIMIZATION WITH SUMS-OF-SQUARES INTERPOLANTS Sercan Yıldız syildiz@samsi.info in collaboration with Dávid Papp (NCSU) OPT Transition Workshop May 02, 2017 OUTLINE Polynomial optimization and
More informationL. Vandenberghe EE236C (Spring 2016) 18. Symmetric cones. definition. spectral decomposition. quadratic representation. log-det barrier 18-1
L. Vandenberghe EE236C (Spring 2016) 18. Symmetric cones definition spectral decomposition quadratic representation log-det barrier 18-1 Introduction This lecture: theoretical properties of the following
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 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 informationApproximate Farkas Lemmas in Convex Optimization
Approximate Farkas Lemmas in Convex Optimization Imre McMaster University Advanced Optimization Lab AdvOL Graduate Student Seminar October 25, 2004 1 Exact Farkas Lemma Motivation 2 3 Future plans The
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 information6.3 Partial Fractions
6.3 Partial Fractions Mark Woodard Furman U Fall 2009 Mark Woodard (Furman U) 6.3 Partial Fractions Fall 2009 1 / 11 Outline 1 The method illustrated 2 Terminology 3 Factoring Polynomials 4 Partial fraction
More informationEE5138R: Problem Set 5 Assigned: 16/02/15 Due: 06/03/15
EE538R: Problem Set 5 Assigned: 6/0/5 Due: 06/03/5. BV Problem 4. The feasible set is the convex hull of (0, ), (0, ), (/5, /5), (, 0) and (, 0). (a) x = (/5, /5) (b) Unbounded below (c) X opt = {(0, x
More information6-1 The Positivstellensatz P. Parrilo and S. Lall, ECC
6-1 The Positivstellensatz P. Parrilo and S. Lall, ECC 2003 2003.09.02.10 6. The Positivstellensatz Basic semialgebraic sets Semialgebraic sets Tarski-Seidenberg and quantifier elimination Feasibility
More informationAn E cient A ne-scaling Algorithm for Hyperbolic Programming
An E cient A ne-scaling 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 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 informationConvex Optimization for Signal Processing and Communications: From Fundamentals to Applications
Convex Optimization for Signal Processing and Communications: From Fundamentals to Applications Chong-Yung Chi Institute of Communications Engineering & Department of Electrical Engineering National Tsing
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 informationNew Rank-One Matrix Decomposition Techniques and Applications to Signal Processing
New Rank-One Matrix Decomposition Techniques and Applications to Signal Processing Yongwei Huang Hong Kong Baptist University SPOC 2012 Hefei China July 1, 2012 Outline Trust-region subproblems in nonlinear
More informationTRUST REGION SUBPROBLEM WITH A FIXED NUMBER OF ADDITIONAL LINEAR INEQUALITY CONSTRAINTS HAS POLYNOMIAL COMPLEXITY
TRUST REGION SUBPROBLEM WITH A FIXED NUMBER OF ADDITIONAL LINEAR INEQUALITY CONSTRAINTS HAS POLYNOMIAL COMPLEXITY YONG HSIA AND RUEY-LIN SHEU Abstract. The trust region subproblem with a fixed number m
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 informationIE 521 Convex Optimization Homework #1 Solution
IE 521 Convex Optimization Homework #1 Solution your NAME here your NetID here February 13, 2019 Instructions. Homework is due Wednesday, February 6, at 1:00pm; no late homework accepted. Please use the
More informationQuadratic Optimization over a Second-Order Cone with Linear Equality Constraints
JORC (2014) 2:17 38 DOI 10.1007/s40305-013-0035-6 Quadratic Optimization over a Second-Order Cone with Linear Equality Constraints Xiao-ling Guo Zhi-bin Deng Shu-Cherng Fang Zhen-bo Wang Wen-xun Xing Received:
More informationEE/ACM Applications of Convex Optimization in Signal Processing and Communications Lecture 18
EE/ACM 150 - Applications of Convex Optimization in Signal Processing and Communications Lecture 18 Andre Tkacenko Signal Processing Research Group Jet Propulsion Laboratory May 31, 2012 Andre Tkacenko
More informationDeterminant maximization with linear. S. Boyd, L. Vandenberghe, S.-P. Wu. Information Systems Laboratory. Stanford University
Determinant maximization with linear matrix inequality constraints S. Boyd, L. Vandenberghe, S.-P. Wu Information Systems Laboratory Stanford University SCCM Seminar 5 February 1996 1 MAXDET problem denition
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 informationContinuous Optimisation, Chpt 7: Proper Cones
Continuous Optimisation, Chpt 7: Proper Cones Peter J.C. Dickinson DMMP, University of Twente p.j.c.dickinson@utwente.nl http://dickinson.website/teaching/2017co.html version: 10/11/17 Monday 13th November
More informationTutorial on Convex Optimization for Engineers Part II
Tutorial on Convex Optimization for Engineers Part II M.Sc. Jens Steinwandt Communications Research Laboratory Ilmenau University of Technology PO Box 100565 D-98684 Ilmenau, Germany jens.steinwandt@tu-ilmenau.de
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 informationAdvances in Convex Optimization: Theory, Algorithms, and Applications
Advances in Convex Optimization: Theory, Algorithms, and Applications Stephen Boyd Electrical Engineering Department Stanford University (joint work with Lieven Vandenberghe, UCLA) ISIT 02 ISIT 02 Lausanne
More informationProblem 1 (Exercise 2.2, Monograph)
MS&E 314/CME 336 Assignment 2 Conic Linear Programming January 3, 215 Prof. Yinyu Ye 6 Pages ASSIGNMENT 2 SOLUTIONS Problem 1 (Exercise 2.2, Monograph) We prove the part ii) of Theorem 2.1 (Farkas Lemma
More informationConvex sets, conic matrix factorizations and conic rank lower bounds
Convex sets, conic matrix factorizations and conic rank lower bounds Pablo A. Parrilo Laboratory for Information and Decision Systems Electrical Engineering and Computer Science Massachusetts Institute
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 informationCutting Planes for First Level RLT Relaxations of Mixed 0-1 Programs
Cutting Planes for First Level RLT Relaxations of Mixed 0-1 Programs 1 Cambridge, July 2013 1 Joint work with Franklin Djeumou Fomeni and Adam N. Letchford Outline 1. Introduction 2. Literature Review
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 informationAgenda. Applications of semidefinite programming. 1 Control and system theory. 2 Combinatorial and nonconvex optimization
Agenda Applications of semidefinite programming 1 Control and system theory 2 Combinatorial and nonconvex optimization 3 Spectral estimation & super-resolution Control and system theory SDP in wide use
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 informationOn Conic QPCCs, Conic QCQPs and Completely Positive Programs
Noname manuscript No. (will be inserted by the editor) On Conic QPCCs, Conic QCQPs and Completely Positive Programs Lijie Bai John E.Mitchell Jong-Shi Pang July 28, 2015 Received: date / Accepted: date
More informationMotivating examples Introduction to algorithms Simplex algorithm. On a particular example General algorithm. Duality An application to game theory
Instructor: Shengyu Zhang 1 LP Motivating examples Introduction to algorithms Simplex algorithm On a particular example General algorithm Duality An application to game theory 2 Example 1: profit maximization
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 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 real-valued 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 informationCHAPTER 2: QUADRATIC PROGRAMMING
CHAPTER 2: QUADRATIC PROGRAMMING Overview Quadratic programming (QP) problems are characterized by objective functions that are quadratic in the design variables, and linear constraints. In this sense,
More information4y Springer NONLINEAR INTEGER PROGRAMMING
NONLINEAR INTEGER PROGRAMMING DUAN LI Department of Systems Engineering and Engineering Management The Chinese University of Hong Kong Shatin, N. T. Hong Kong XIAOLING SUN Department of Mathematics Shanghai
More informationAn Adaptive Linear Approximation Algorithm for Copositive Programs
1 An Adaptive Linear Approximation Algorithm for Copositive Programs Stefan Bundfuss and Mirjam Dür 1 Department of Mathematics, Technische Universität Darmstadt, Schloßgartenstr. 7, D 64289 Darmstadt,
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 informationA Simple Derivation of a Facial Reduction Algorithm and Extended Dual Systems
A Simple Derivation of a Facial Reduction Algorithm and Extended Dual Systems Gábor Pataki gabor@unc.edu Dept. of Statistics and OR University of North Carolina at Chapel Hill Abstract The Facial Reduction
More informationRanks of Real Symmetric Tensors
Ranks of Real Symmetric Tensors Greg Blekherman SIAM AG 2013 Algebraic Geometry of Tensor Decompositions Real Symmetric Tensor Decompositions Let f be a form of degree d in R[x 1,..., x n ]. We would like
More information