Agenda. Applications of semidefinite programming. 1 Control and system theory. 2 Combinatorial and nonconvex optimization


 Felicia Brown
 2 years ago
 Views:
Transcription
1 Agenda Applications of semidefinite programming 1 Control and system theory 2 Combinatorial and nonconvex optimization 3 Spectral estimation & superresolution
2 Control and system theory SDP in wide use in control theory Example: differential inclusion (ẋ(t) is time derivative) ẋ(t) = Ax(t) + Bu(t) (S) y(t) = Cx(t) u i (t) y i (t) x(t) R n, y(t), u(t) R p Problem: Find ellipsoid E such that for any x, u obeying (S) x(0) E x(t) E t 0 Implication: if such E exists, then all solutions of differential inclusion are bounded
3 Quadratic Lyapunov function Ellipsoid E = { x : x T P x 1 } P 0 Quadratic Lyapunov function V (t) = x(t) T P x(t) Claim E invariant V (t) nonincreasing Proof: obvious V (0) 0 (otherwise leave E ) V (t) > 0 and x(t) λe. Then starting at x(0) = λ 1 x(t) E, we would leave E Hence, existence of a Lyapunov function proves stability of (S)
4 (i) V (t) 0 [ x(t) u(t) ] T [ A T P + P A P B B T P 0 ] [ x(t) u(t) ] 0 (ii) u i (t) y i (t) u 2 i (t) y 2 i (t) 0 With y i (t) = c i (t) T x(t), this can be expressed as [ ] T [ ] [ ] x(t) c T i c i 0 x(t) 0 u(t) 0 E ii u(t) where E ii is matrix with all zero entries except (i, i)th equal to 1 E invariant (ii) (i) That is, constraint quadratic holds V (t) 0 Formally, we want z R n+p z T T i z 0 i = 1,..., p z T T 0 z 0 where T 0 = [ A T P + P A ] P B B T P 0 [ ] c T T i = i c i 0 0 E ii
5 Obvious sufficient condition: λ 1,..., λ p 0 such that T 0 λ 1 T λ p T p called Sprocedure in control (analogy with later cvx relaxations) Λ = diagλ i T 0 [ ] c λ i T i = T Λc 0 0 Λ [ ] A T P + P A + c T Λc P B B T 0 P Λ By solving an SDP feasibilty problem, we can certify stability (find an invariant E)
6 Applications of semidefinite programming 1 Control and system theory 2 Combinatorial and nonconvex optimization 3 Spectral estimation & superresolution
7 Combinatorial and nonconvex optimization min f 0(x) s.t. f i (x) 0 i = 1,..., m f i (x) = x T A i x + 2b T i x + c i A i 0 cvx prob A i S n indefinite non cvx, very hard
8 Examples: A. Boolean leastsquares min Ax b 2 s.t. x i { 1, 1} i = 1,..., n Basic problem in digital communications: MLE for digital signals Boolean least squares can be cast as nonconvex QCQP min x T A T Ax 2b T x + b T b s.t. x 2 i 1 = 0 i = 1,..., n x 2 i 1 = 0 { x 2 i 1 0 x 2 i 1 0
9 Examples: B. minimum cardinality problems min card(x) s.t. Ax b card(x) = x l0 = {i : x i 0} Many applications in signal processing, statistics, finance; e.g. optimization with fixed transaction costs portfolio z i = 1{x i 0} (1 z i )x i = 0 z i {0, 1} Min cardinality problem can be cast as nonconvex QCQP in (x, z) min zi s.t. Ax b (1 z i )x i = 0 zi 2 z i = 0
10 Examples: C. partitioning problems min xt Qx s.t. x 2 i = 1 i = 1,..., n Q S n, x R n Feasible gives a partition {1,..., n} = {i : x i = 1} {i : x i = 1} Interpretation Q ij is cost of having i and j in same partition Q ij is cost of having i and j in different partitions x T Qx is total cost Problem: Find partition with least total cost noncvx QCQP
11 Examples: D. MAXCUT Graph G = (V, E) with weighted edges { w ij (i, j) E 0 otherwise MAXCUT cut of G with largest possible weight: partition (V 1, V 2 ) s.t. sum of weights of edges between V 1 and V 2 is maximized classical problem in network optimization special case of partitioning probem
12 Weight of a particular cut f 0 (x) = 1 2 i,j:x ix j= 1 w ij = 1 w ij (1 x i x j ) 4 i,j Set W ij = { w ij i j 0 i = j D ij = { 0 i j j i w ij i = j MAXCUT max x T (D W )x := x T Ax s.t. x 2 i = 1 i = 1,..., n
13 Examples: E. polynomial problems min p 0 (x) s.t. p i (x) 0 i = 1,..., m more complex than QCQP? No! All polynomial problems can be cast as QCQPs e.g. min x3 2xyz + y + z s.t. x 2 + y 2 + z new variables u = x 2, v = yz. min xu 2xv + y + z s.t. x 2 + y 2 + z u x 2 = 0 v yz = 0
14 Two tricks: (i) can reduce max degree of an equation via { y 2n u + (...) n + (...) 0 u = y 2 (ii) can eliminate product terms { ux + (...) 0 xyz + (...) 0 u = yz Apply tricks iteratively reduction to QCQP (noncvx)
15 Convex relaxations Q: How to get lower bound on opt. value? (QCQP) min xt A 0 x + 2b T 0 x + c 0 s.t. x T A i x + 2b T i x + c i 0 Semidefinite relaxation (QCQP) x T A i x = trace(x T A i x) = trace(a i xx T ) min trace(a 0 X) + 2b T 0 x + c 0 s.t. trace(a i X) + 2b T i x + c i i = 1,..., m X = xx T Relax noncvx constraint X = xx T by considering X xx T. [ ] X xx T X x x T 0 (Schur complement) 1
16 Convex relaxations SDP relaxation min trace(xa 0 ) + 2b T 0 x + c 0 s.t. trace(xa [ ] i ) + 2b T i x + c i i = 1,..., m X x x T 0 1 Lagrangian relaxation min xt A 0 x + 2b T 0 + c 0 s.t. x T A i x + 2b T i x + c i i = 1,..., m Lagrangian: L(x, λ) = x T A(λ)x + 2b T (λ)x + c(λ) A(λ) = A 0 + λ i A i b(λ) = b 0 + λ i b i c(λ) = c 0 + λ i c i
17 min x T Ax + 2b T x + c = { c b T A b A 0, b R(A) otherwise Dual function: g(λ) = b(λ) T A b(λ) + c(λ) max γ + c(λ) Dual problem: s.t. λ i 0 A(λ) 0 b(λ) T A (λ)b(λ) γ This is an SDP! max γ + c(λ) s.t. λ [ i 0 ] A(λ) b(λ) b(λ) T 0 γ
18 Question: Which relaxation is better? Two problems are dual from each other If strictly feasible, bounds are the same Perfect duality: Sometimes cvx relaxation is exact, i.e. under some conditions OPT(D) = OPT(P )
19 Examples Boolean LS min Ax b 2 s.t. x 2 i = 1 min trace(a T Ax) 2b T Ax + b T b s. t. X ii = 1 [ ] X x X xx T or x T 0 1 Partitioning and max cut min xt W x s.t. x 2 i = 1 min trace(w X) s.t X ii = 1 X xx T or [ ] X x x T 0 1
20 Delicate issue Relaxations provide a lower bound on optimal value but provide no hints on how to compute a good feasible point Frequently discusssed approach: randomization Case study: MAXCUT max s.t. X 0 X ii = 1 i,j w ij(1 X ij ) is the SDP relaxation of MAXCUT (P ) max wij (1 x i x j ) = x T Ax s.t. x 2 i = 1
21 Goemans and Williamson (1996) Theorem If X feasible for SDP: 0.878SDP OPT SDP X 0 X ij = vi T v j (X = V T V ) X ii = 1 v i = 1 V can be obtained by Cholevsky factorization Pick v at random on unit sphere: cut V + = {i : v T v i > 0} V = {i : v T v i < 0} x i = sgn(v T v i )
22 What is the expected value of this cut? Expected weight of random cut E w ij (1 x i x j ) = 2w ij P(V separates i and j) 2 π = 2w ij P(sgn(vi T v) sgn (vi T v)) ( ) 2θ = 2w ij 2π = 2 π w ij cos 1 (v T i v j ) w ij cos 1 (X ij ) i,j
23 and so 2 π cos 1 (t) α(1 t) α = w ij cos 1 (X ij ) α w ij (1 X ij ) = αtrace(ax) π i,j i,j True for all feasible X true for optimal X Expected weight from random cut generated by X opt is at least α SDP This gives OPT α SDP Provides an algorithm (randomized) for finding a good cut which on average has weight at least 87.5 % of OPT
24 Expected weight of a random cut Suppose x i iid with P(x i = ±1) = 1/2 E x T Ax = ij w ij (1 E x i x j ) = ij w ij Expected weight of a random cut is at least 50% of total edge weight No polynomial approximation algorithm with constant better than exists unless P = NP [Hästaad 97]
25 Extension I: diagonal dominance max xt Ax s.t. x 2 i = 1 max trace(ax) s.t. X ii = 1 X 0 If A is diagonally dominant, then same result holds Diagonal dominance a ii a ij for all i j:j i
26 If A diag. dominant, then x T Ax is a sum of terms of the form x 2 i and (x i ± x j ) 2 with positive coefficients. In expectation 1 2 E(x i ± x j ) 2 = E(1 ± x i x j ) = 1 ± 2 π sin 1 (X ij ) 0.878(1 ± X ij ) Value of GW randomized cut obeys 0.878trace(AX) E x T Ax p trace(ax) For graph Laplacian A = D W x T Ax = 1 w ij (x i x j ) 2 2 ij
27 Extension: A 0 max xt Ax s.t. x 2 i = 1 max trace(ax) s.t. X ii = 1 X 0 Theorem (Nesterov s theorem) If A 0, then 2 π SDP E xt Ax SDP with the same randomized construction
28 X 0 = sin 1 (X) X (*) Hence, E x T Ax = 2 π trace(a sin 1 (X)) 2 π trace(ax) Proof of (*) relies on a fact: assume f : R R has a Taylor series with nonnegative coefficients and set Y = f(x) [Y ij = f(x ij )]. Then X 0 = Y 0 Apply with f(t) = sin 1 (t) t to get (*) Proof of fact is a direct consequence of this: A, B 0 = A B 0 Hadamard product: (A B) ij = A ij B ij
29 Extension: bipartite graphs [ ] 0 S T A = 1 2 S 0 max xt Ax s.t. x 2 i = 1 max v T Su s.t. u 2 i = 1 vi 2 = 1 First analyzed by Gothendieck κ G = sup A trace(ax) p Theorem (Krivine) κ G
30 Lemma f, g : R R s.t. f + g and f g have nonnegative Taylor coefficients. Let [ ] [ ] X11 X X = 12 Y11 Y X12 T Y = 12 X 22 Y12 T Y 22 Then X 0 = Y 0 f(t) = sinh(c κ πt/2) with c κ so that f(1) = 1 g(t) = sin(c κ πt/2) f and g are as in Lemma since sinh(t) = k=0 t 2k+1 (2k + 1)! sin(t) = ( 1) k t 2k+1 (2k + 1)! k=0
31 X is optimal sol and Y is as in lemma Y 0 and Y ii = 1 We can apply rounding to feas. Y to get y E y T Ay = 2 π trace(a sin 1 (Y )) = 2 π trace(s sin 1 (Y 12 )) = c κ trace(sx 12 ) At least c κ times best possible value c κ trace(sx 12 ) E y T Ay trace(sx 12 )
32 Generalized randomization approach max trace(ax) s.t. X 0 X ii = 1 (1) v N(0, X ) (2) x i = sgn(v i ) Sometimes E X f 0 (X) α p
33 Applications of semidefinite programming 1 Control and system theory 2 Combinatorial and nonconvex optimization 3 Spectral estimation & superresolution
34 Spectral estimation Sparse superposition of tones s(t) = j c j e i2πωjt (+ noise) ω j [0, 1] Observe samples d = s(t), t T n = {0, 1,..., n} Problem How do we find frequencies and amplitudes?
35 Convex programming approach If ω Ω with Ω finite, natural procedure min c 1 s.t. Ac = d A tω = e i2πωt (t, ω) T n Ω But Ω = [0, 1]... Proposal Recover signal by solving min c TV subject to Ac = d totalvariation norm c TV = sup j c(b j ) with sup over all finite partitions {B j } of [0, 1] Linear mapping (Ac)(t) = e i2πωt c(dω) Continuum of decision variables!
36 Superresolution Swap time and frequency x = j c j δ τj c j C, τ j [0, 1] Wish to recover x: spike locations and amplitudes Only have lowfrequency data d d k = j c j e i2πktj k = n/2, n/2 + 1,..., n/2 Recovery Linear mapping min x TV subject to Ax = d (Ax)(k) = e i2πkt x(dt) Continuum of decision variables!
37 Formulation as a finitedimensional problem Primal problem min x TV s. t. Ax = y Infinitedimensional variable x Finitely many constraints Semidefinite representability (A c)(t) 1 for all t [0, 1] equivalent to Dual problem (1) there is Q Hermitian s. t. [ ] Q c c 0 1 (2) trace(q) = 1 max Re y, c s. t. A c 1 Finitedimensional variable c Infinitely many constraints (A c)(t) = c k e i2πkt k n/2 (3) sums along superdiagonals vanish, n j i=1 Q i,i+j = 0 for 1 j n 1
38 Semidefinite representability P (t) = n 1 k=0 c ke i2πkt P (t) 1 for all t [ ] Q c c 0, 1 n j Q i,i+j = i=1 { 1 j = 0 0 j = 1, 2,..., n 1 = (easy part) [ ] Q c c 0 1 Q cc 0 = z cc z z Qz z = (z 0,..., z n 1 ), z k = e i2πkt z Qz = 1 z cc z = c z 2 = p(t) 2
39 How to compute primal solutions? Use complementary slackness Support of x contained in {t : p(t) = 1} Find support and solve leastsquares problem
40 References 1 A. BenTal and A. Nemirovski, Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications, MPSSIAM Series on Optimization 2 S. Boyd, EE 364B, Stanford University 3 Semidefinite Optimization and Convex Algebraic Geometry, Edited by G. Blekherman, P. Parrilo and R. Thomas 4 E. J. Candès, and C. FernandezGranda, Towards a mathematical theory of superresolution. To appear in Comm. Pure Appl. Math
Relaxations 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 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 informationConvex Optimization M2
Convex Optimization M2 Lecture 8 A. d Aspremont. Convex Optimization M2. 1/57 Applications A. d Aspremont. Convex Optimization M2. 2/57 Outline Geometrical problems Approximation problems Combinatorial
More informationLecture 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 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 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 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 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 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 informationAgenda. 1 Cone programming. 2 Convex cones. 3 Generalized inequalities. 4 Linear programming (LP) 5 Secondorder cone programming (SOCP)
Agenda 1 Cone programming 2 Convex cones 3 Generalized inequalities 4 Linear programming (LP) 5 Secondorder cone programming (SOCP) 6 Semidefinite programming (SDP) 7 Examples Optimization problem in
More 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 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 informationAgenda. Interior Point Methods. 1 Barrier functions. 2 Analytic center. 3 Central path. 4 Barrier method. 5 Primaldual path following algorithms
Agenda Interior Point Methods 1 Barrier functions 2 Analytic center 3 Central path 4 Barrier method 5 Primaldual path following algorithms 6 Nesterov Todd scaling 7 Complexity analysis Interior point
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 informationOptimization for Machine Learning
Optimization for Machine Learning (Problems; Algorithms  A) SUVRIT SRA Massachusetts Institute of Technology PKU Summer School on Data Science (July 2017) Course materials http://suvrit.de/teaching.html
More 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 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 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 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 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 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 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 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 informationConvex Optimization & Lagrange Duality
Convex Optimization & Lagrange Duality Chee Wei Tan CS 8292 : Advanced Topics in Convex Optimization and its Applications Fall 2010 Outline Convex optimization Optimality condition Lagrange duality KKT
More 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 informationThere are several approaches to solve UBQP, we will now briefly discuss some of them:
3 Related Work There are several approaches to solve UBQP, we will now briefly discuss some of them: Since some of them are actually algorithms for the Max Cut problem (MC), it is necessary to show their
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 informationThree Recent Examples of The Effectiveness of Convex Programming in the Information Sciences
Three Recent Examples of The Effectiveness of Convex Programming in the Information Sciences Emmanuel Candès International Symposium on Information Theory, Istanbul, July 2013 Three stories about a theme
More informationRank minimization via the γ 2 norm
Rank minimization via the γ 2 norm Troy Lee Columbia University Adi Shraibman Weizmann Institute Rank Minimization Problem Consider the following problem min X rank(x) A i, X b i for i = 1,..., k Arises
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 informationCOM Optimization for Communications 8. Semidefinite Programming
COM524500 Optimization for Communications 8. Semidefinite Programming Institute Comm. Eng. & Dept. Elect. Eng., National Tsing Hua University 1 Semidefinite Programming () Inequality form: min c T x s.t.
More informationTowards a Mathematical Theory of Superresolution
Towards a Mathematical Theory of Superresolution Carlos FernandezGranda www.stanford.edu/~cfgranda/ Information Theory Forum, Information Systems Laboratory, Stanford 10/18/2013 Acknowledgements This
More informationSumofSquares Method, Tensor Decomposition, Dictionary Learning
SumofSquares Method, Tensor Decomposition, Dictionary Learning David Steurer Cornell Approximation Algorithms and Hardness, Banff, August 2014 for many problems (e.g., all UGhard ones): better guarantees
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 informationModule 04 Optimization Problems KKT Conditions & Solvers
Module 04 Optimization Problems KKT Conditions & Solvers Ahmad F. Taha EE 5243: Introduction to CyberPhysical Systems Email: ahmad.taha@utsa.edu Webpage: http://engineering.utsa.edu/ taha/index.html September
More 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 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 information1 The independent set problem
ORF 523 Lecture 11 Spring 2016, Princeton University Instructor: A.A. Ahmadi Scribe: G. Hall Tuesday, March 29, 2016 When in doubt on the accuracy of these notes, please cross chec with the instructor
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 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 15 Newton Method and SelfConcordance. October 23, 2008
Newton Method and SelfConcordance October 23, 2008 Outline Lecture 15 Selfconcordance Notion Selfconcordant Functions Operations Preserving Selfconcordance Properties of Selfconcordant Functions Implications
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 ChuanHao Guo a,, YanQin Bai
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 informationA CONIC DANTZIGWOLFE DECOMPOSITION APPROACH FOR LARGE SCALE SEMIDEFINITE PROGRAMMING
A CONIC DANTZIGWOLFE DECOMPOSITION APPROACH FOR LARGE SCALE SEMIDEFINITE PROGRAMMING Kartik Krishnan Advanced Optimization Laboratory McMaster University Joint work with Gema Plaza Martinez and Tamás
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 informationFINDING LOWRANK SOLUTIONS OF SPARSE LINEAR MATRIX INEQUALITIES USING CONVEX OPTIMIZATION
FINDING LOWRANK SOLUTIONS OF SPARSE LINEAR MATRIX INEQUALITIES USING CONVEX OPTIMIZATION RAMTIN MADANI, GHAZAL FAZELNIA, SOMAYEH SOJOUDI AND JAVAD LAVAEI Abstract. This paper is concerned with the problem
More informationExample: feasibility. Interpretation as formal proof. Example: linear inequalities and Farkas lemma
41 Algebra and Duality P. Parrilo and S. Lall 2006.06.07.01 4. Algebra and Duality Example: nonconvex polynomial optimization Weak duality and duality gap The dual is not intrinsic The cone of valid
More information Wellcharacterized problems, minmax relations, approximate certificates.  LP problems in the standard form, primal and dual linear programs
LPDuality ( Approximation Algorithms by V. Vazirani, Chapter 12)  Wellcharacterized problems, minmax relations, approximate certificates  LP problems in the standard form, primal and dual linear programs
More informationBBM402Lecture 20: LP Duality
BBM402Lecture 20: LP Duality Lecturer: Lale Özkahya Resources for the presentation: https://courses.engr.illinois.edu/cs473/fa2016/lectures.html An easy LP? which is compact form for max cx subject to
More information1. Introduction. Consider the following quadratic binary optimization problem,
ON DUALITY GAP IN BINARY QUADRATIC PROGRAMMING XIAOLING SUN, CHUNLI LIU, DUAN LI, AND JIANJUN GAO Abstract. We present in this paper new results on the duality gap between the binary quadratic optimization
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 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 informationEE363 homework 7 solutions
EE363 Prof. S. Boyd EE363 homework 7 solutions 1. Gain margin for a linear quadratic regulator. Let K be the optimal state feedback gain for the LQR problem with system ẋ = Ax + Bu, state cost matrix Q,
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 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 information14. Duality. ˆ Upper and lower bounds. ˆ General duality. ˆ Constraint qualifications. ˆ Counterexample. ˆ Complementary slackness.
CS/ECE/ISyE 524 Introduction to Optimization Spring 2016 17 14. Duality ˆ Upper and lower bounds ˆ General duality ˆ Constraint qualifications ˆ Counterexample ˆ Complementary slackness ˆ Examples ˆ Sensitivity
More informationConvex Optimization and SVM
Convex Optimization and SVM Problem 0. Cf lecture notes pages 12 to 18. Problem 1. (i) A slab is an intersection of two half spaces, hence convex. (ii) A wedge is an intersection of two half spaces, hence
More informationECE G: Special Topics in Signal Processing: Sparsity, Structure, and Inference
ECE 18898G: Special Topics in Signal Processing: Sparsity, Structure, and Inference Lowrank matrix recovery via convex relaxations Yuejie Chi Department of Electrical and Computer Engineering Spring
More informationLecture 10. Semidefinite Programs and the MaxCut Problem Max Cut
Lecture 10 Semidefinite Programs and the MaxCut Problem In this class we will finally introduce the content from the second half of the course title, Semidefinite Programs We will first motivate the discussion
More informationSparse Optimization Lecture: Dual Certificate in l 1 Minimization
Sparse Optimization Lecture: Dual Certificate in l 1 Minimization Instructor: Wotao Yin July 2013 Note scriber: Zheng Sun Those who complete this lecture will know what is a dual certificate for l 1 minimization
More informationSupport Detection in Superresolution
Support Detection in Superresolution Carlos FernandezGranda (Stanford University) 7/2/2013 SampTA 2013 Support Detection in Superresolution C. FernandezGranda 1 / 25 Acknowledgements This work was
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 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 informationComplexity of Deciding Convexity in Polynomial Optimization
Complexity of Deciding Convexity in Polynomial Optimization Amir Ali Ahmadi Joint work with: Alex Olshevsky, Pablo A. Parrilo, and John N. Tsitsiklis Laboratory for Information and Decision Systems Massachusetts
More informationCS229T/STATS231: Statistical Learning Theory. Lecturer: Tengyu Ma Lecture 11 Scribe: Jongho Kim, Jamie Kang October 29th, 2018
CS229T/STATS231: Statistical Learning Theory Lecturer: Tengyu Ma Lecture 11 Scribe: Jongho Kim, Jamie Kang October 29th, 2018 1 Overview This lecture mainly covers Recall the statistical theory of GANs
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. 2934 A semidefinite relaxation scheme for quadratically constrained quadratic problems with an additional linear constraint M. Salahi Semidefinite
More informationGrothendieck s Inequality
Grothendieck s Inequality Leqi Zhu 1 Introduction Let A = (A ij ) R m n be an m n matrix. Then A defines a linear operator between normed spaces (R m, p ) and (R n, q ), for 1 p, q. The (p q)norm of A
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 informationLecture 6. Foundations of LMIs in System and Control Theory
Lecture 6. Foundations of LMIs in System and Control Theory Ivan Papusha CDS270 2: Mathematical Methods in Control and System Engineering May 4, 2015 1 / 22 Logistics hw5 due this Wed, May 6 do an easy
More information12. Interiorpoint methods
12. Interiorpoint methods Convex Optimization Boyd & Vandenberghe inequality constrained minimization logarithmic barrier function and central path barrier method feasibility and phase I methods complexity
More informationLecture: 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: 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 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 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 informationDuality of LPs and Applications
Lecture 6 Duality of LPs and Applications Last lecture we introduced duality of linear programs. We saw how to form duals, and proved both the weak and strong duality theorems. In this lecture we will
More informationECE 8201: Lowdimensional Signal Models for Highdimensional Data Analysis
ECE 8201: Lowdimensional Signal Models for Highdimensional Data Analysis Lecture 7: Matrix completion Yuejie Chi The Ohio State University Page 1 Reference Guaranteed MinimumRank Solutions of Linear
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 informationDecentralized Control of Stochastic Systems
Decentralized Control of Stochastic Systems Sanjay Lall Stanford University CDCECC Workshop, December 11, 2005 2 S. Lall, Stanford 2005.12.11.02 Decentralized Control G 1 G 2 G 3 G 4 G 5 y 1 u 1 y 2 u
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 informationLINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM
LINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM Unless otherwise stated, all vector spaces in this worksheet are finite dimensional and the scalar field F is R or C. Definition 1. A linear operator
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 informationSemidefinite Programming Duality and Linear Timeinvariant Systems
Semidefinite Programming Duality and Linear Timeinvariant Systems Venkataramanan (Ragu) Balakrishnan School of ECE, Purdue University 2 July 2004 Workshop on Linear Matrix Inequalities in Control LAASCNRS,
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 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 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 informationConvex Optimization of Graph Laplacian Eigenvalues
Convex Optimization of Graph Laplacian Eigenvalues Stephen Boyd Stanford University (Joint work with Persi Diaconis, Arpita Ghosh, SeungJean Kim, Sanjay Lall, Pablo Parrilo, Amin Saberi, Jun Sun, Lin
More informationA Continuation Approach Using NCP Function for Solving MaxCut Problem
A Continuation Approach Using NCP Function for Solving MaxCut Problem Xu Fengmin Xu Chengxian Ren Jiuquan Abstract A continuous approach using NCP function for approximating the solution of the maxcut
More informationAcyclic Semidefinite Approximations of Quadratically Constrained Quadratic Programs
2015 American Control Conference Palmer House Hilton July 13, 2015. Chicago, IL, USA Acyclic Semidefinite Approximations of Quadratically Constrained Quadratic Programs Raphael Louca and Eilyan Bitar
More informationHomework Set #6  Solutions
EE 15  Applications of Convex Optimization in Signal Processing and Communications Dr Andre Tkacenko JPL Third Term 111 Homework Set #6  Solutions 1 a The feasible set is the interval [ 4] The unique
More 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 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 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 informationLecture 17 (Nov 3, 2011 ): Approximation via rounding SDP: MaxCut
CMPUT 675: Approximation Algorithms Fall 011 Lecture 17 (Nov 3, 011 ): Approximation via rounding SDP: MaxCut Lecturer: Mohammad R. Salavatipour Scribe: based on older notes 17.1 Approximation Algorithm
More informationPrimalDual InteriorPoint Methods. Javier Peña Convex Optimization /36725
PrimalDual InteriorPoint Methods Javier Peña Convex Optimization 10725/36725 Last time: duality revisited Consider the problem min x subject to f(x) Ax = b h(x) 0 Lagrangian L(x, u, v) = f(x) + u T
More informationarzelier
COURSE ON LMI OPTIMIZATION WITH APPLICATIONS IN CONTROL PART II.1 LMIs IN SYSTEMS CONTROL STATESPACE METHODS STABILITY ANALYSIS Didier HENRION www.laas.fr/ henrion henrion@laas.fr Denis ARZELIER www.laas.fr/
More informationConstrained Optimization and Lagrangian Duality
CIS 520: Machine Learning Oct 02, 2017 Constrained Optimization and Lagrangian Duality Lecturer: Shivani Agarwal Disclaimer: These notes are designed to be a supplement to the lecture. They may or may
More informationMIT Algebraic techniques and semidefinite optimization May 9, Lecture 21. Lecturer: Pablo A. Parrilo Scribe:???
MIT 6.972 Algebraic techniques and semidefinite optimization May 9, 2006 Lecture 2 Lecturer: Pablo A. Parrilo Scribe:??? In this lecture we study techniques to exploit the symmetry that can be present
More informationSummary of the simplex method
MVE165/MMG630, The simplex method; degeneracy; unbounded solutions; infeasibility; starting solutions; duality; interpretation AnnBrith Strömberg 2012 03 16 Summary of the simplex method Optimality condition:
More informationLargest dual ellipsoids inscribed in dual cones
Largest dual ellipsoids inscribed in dual cones M. J. Todd June 23, 2005 Abstract Suppose x and s lie in the interiors of a cone K and its dual K respectively. We seek dual ellipsoidal norms such that
More information