Semi-definite representibility. For fun and profit
|
|
- Kimberly Potter
- 5 years ago
- Views:
Transcription
1 Semi-definite representation For fun and profit March 9, 2010
2 Introduction Any convex quadratic constraint x T C T Cx a + b T x Can be recast as the linear matrix inequality (LMI): ( ) In Cx (Cx) T a + b T 0 x If you introduce extra variables, you can express a great variety of polynomial constraints.
3 For example: {(x, y) R 2 : x 2 + y 4 1} Is the projection onto x, y space of {(x, y, w) R 3 : x 2 + w 2 1, y 2 w} Which is expressible as In other words: 1 x 1 w x w 1 ( 1 y y w 1 x 1 w x w 1 0 ) 1 y y w 0 0
4 LMI-representation This raises the question of which convex sets S R n can be expressed as an LMI, for some A 0,, A n symmetric nxn matrices S = {x R n : A 0 + A 1 x A n x n 0} Without loss of generality, 0 S 0, so A 0 0, and we can take A 0 = I.
5 LMIr: necessary conditions Consider f (x) = det(a 0 + A 1 x A n x n ). The zero-set of f divides R n into connected components. Claim: S 0 is the connected component containing 0. Since f (x) > 0 on S 0 and 0 S 0, S 0 must at least contain the connected component and cannot contain more. Thus S has to be semi-algebraic: S = {x R n : g 1 (x) 0,, g m (x) 0}, and the problem is (basically) equivalent to finding A 0,, A n such that f (x) = g 1 (x) g m (x). 1 1 More properly you want f to be in I (δs), the ideal of polynomials vanishing on the boundary.
6 LMIr: rigid convexity condition Let x R n and consider f (tx) = det(i + t(a 1 x 1 + A n x n )) = t n det(i /t + (A 1 x A n x n )) But A 1 x 1 + A n x n is symmetric so we have its characteristic polynomial at 1/t, which can only have real roots. This property of f (x) is called rigid convexity. It is necessary for LMI representibility (Helton and Vinnikov) and sufficient for n = 2. The problem is still open for n > 2. For example, f (x, y) = 2 x 4 y 4 has (i, i) as a root so is not rigidly convex, so does not have a determinantal representation, and the region x 4 + y 4 2 is not LMI representible. If you drop the requirement that the A 0,, A n are symmetric, then all polynomials have a determinantal representation (Helton, McCullough, Vinnikov) that can be explicitly constructed (Quarez).
7 Lifted LMI, Semidefinite representation This leads to the concept of a lifted representation, adding extra variables to make the problem LMI representible. Say S is Semidefinite representible (SDr) if there exist symmetric matrices A, B 1,, B n, C 1,, C M if x S y R M such that A + B i x i + C i y i 0 Example, the region x 4 + y 4 2. x 4 + y 4 2 u 2 + v 2 2, u 2 x, v 2 y 1 u 1 v u v 2 1 u u x 1 v v y 0
8 Which sets are SDr? Conjecture: all convex semi-algebraic sets (convex regions cut out by polynomial inequalities) are SDr. This is known (Helton Nie) in the case where you have either positive curvature or sos-convexity.
9 Positive curvature This is different from the Hessian being positive definite. Definition from differential geometry. For nondegenerate defining functions g i, we want the Hessian to be negative definite on the tangent space: v T 2 g i (x)v > 0 for all 0 v g i (x)
10 Positive curvature - example S = {x R n + : g(x) = x 1 x 2 x n 1 0. S is convex. g(x) g(x) + 1 = ( 1 x 1 1 x x n ) T 2 g(x) g(x) + 1 = 0 1 x 1 x x 1 x n 1 x 1 x x n 1 x n. 1 x 1 x n... 1 x n 1 x n 0 The Hessian has trace 0 so is not positive definite. However: 2 g(x) + g(x) g(x) T = (g(x) + 1) diag( 1 x1 2,, 1 xn 2 ) 0 The result follows.
11 Convexity How do you tell if a function is convex? Very hard (since even determining if a polynomial is always positive is hard). There is a more manageable positivity condition: sum of squares. We say that a function f (x) is sos-convex if there is sum polynomial matrix M(x) such that H f (x) = M T (x)m(x) If f (x) is sos-convex you can find M(x) by solving an SDP. There are convex functions that are not sos-convex (Ahmadi and Parrilo): p(x) = 32x x 6 x x 6 1 x x 4 1 x x 4 1 x 2 2 x x 4 1 x x 2 1 x 4 2 x x 2 1 x 2 2 x x 2 1 x x x 6 2 x x 4 2 x x 2 2 x x 8 3
12 LMI Construction. Natural SDP relaxation (Lasserre and Parrilo) Suppose S is convex given in the form S = {x R n : g 1 (x) 0,, g m (x) 0} Make this linear by setting y α = x α for each multi-index α.then the inequality: g 1 (x) 0. = g 0 + x i g i + x α A α g m (x) Becomes 0 g 0 + x i g i + y α A α = g(x, y)
13 LMI Construction cont d Now we need a constraint that requires y α to basically be x α. Let m d (x) be the vector of monomials of degree up to d Then m d (x) = ( 1 x 1 x 2... x 2 1 x 1 x 2... x d n 0 m d (x)m d (x) T = A 0 + x i A i + x α A α And since this is rank 1 one may expect the given region to be small. Let M d (x, y) = A 0 + x i A i + y α A α Our SDP relaxation is now R = {x : ys.t.g(x, y) 0, M d (x, y) 0} In many cases you can show that R = S for sufficiently large d. Usually sufficient to take 2d to be the highest degree of the g i. ) T
14 SDP relaxation: example Consider the set satisfying g(x) 0 where g(x) = 1 (x1 4 + x 2 4 x 1 2x 2 2 ). Direct computation: ( ) ( ) ( ) 2 x x1 g(x) = x x 2 ( ) 12 4 And 0. Hence g is sos-concave. The SDP 4 12 constructed is 1 y 40 y 04 + y x 1 x 2 y 20 y 11 y 02 x 1 y 20 y 11 y 30 y 21 y 12 x 2 y 11 y 03 y 21 y 12 y 03 y 20 y 30 y 21 y 40 y 31 y 22 0 y 11 y 21 y 12 y 31 y 22 y 13 y 02 y 12 y 03 y 22 y 13 y 04 The number of auxiliary variables grows exponentially in m, but you can utilize sparseness to cut this down a bit (Helton, Nie).
15 Some useful lemmas Given R 1,, R n SDr Intersection. R 1 R n is SDr Minkowski sum. R R n = {r r n : r i R i for all i} is SDr Union. conv(r 1 R n ) = {a 1 r a n r n : r i R i, a i 0, a a n = 1} is SDr. Consequence: it is enough to just have SDr locally. Theorem: R is SDr if and only if each x R has a neighbourhood that is SDr. By compactness only finitely many neighbourhoods are necessary to cover R, then apply the union result to this finite number of local SDr s. You also have flexibility in picking the defining polynomials
16 Proof of relaxation Basic idea of the proofs: R contains S. If R contains some other point a then we can seperate a from S by a linear functional l T x. Let l be the minimum of l T x on S, attained at some u S. Note l T a < l There exist Lagrange multipliers λ 1,, λ m 0 such that l = i λ i g i (u). Define f l (x) := l T x l λ i g i (x) Then f l (x) is convex and nonnegative polynomial such that f l (u) = 0 and f l (u) = 0. Use properties of the g i to express f l in a nice form.
17 R = S for sos-concave case If the g i are sos-concave then integrating the Hessian twice, you get that f l (x) is SOS, ie for some symmetric matrix W 0 Or with the y α : l T x l = λ i g i (x) + [x dg ] T W [x dg ] l T x l = λ i L gi (y) + Tr(W M dg (y)) 0 Which contradicts the separation of a. We get a similar case in general, but we need some bounded degree positivstellensatz to tell us that l T x l is in the quadratic module generated by g i, with bounded degree.
18 Sources Helton and Nie Semidefinite representation of convex sets Ahmadi and Parrilo A convex polynomial that is not sos-convex Lasserre Representation of nonnegative convex polynomials Brändén Obstructions to determinantal representability Helton and Nie Structured Semidenite Representation of Some Convex Sets Quarez Symmetric Determinantal Representations of Polynomials Helton and Vinnikov Linear matrix inequality representation of sets
Semidefinite Representation of Convex Sets
Mathematical Programming manuscript No. (will be inserted by the editor J. William Helton Jiawang Nie Semidefinite Representation of Convex Sets the date of receipt and acceptance should be inserted later
More informationSemidefinite representation of convex sets and convex hulls
Semidefinite representation of convex sets and convex hulls J. William Helton 1 and Jiawang Nie 2 1 Department of Mathematics, University of California, 9500 Gilman Drive, La Jolla, CA 92093. helton@math.ucsd.edu
More informationDescribing convex semialgebraic sets by linear matrix inequalities. Markus Schweighofer. Universität Konstanz
Describing convex semialgebraic sets by linear matrix inequalities Markus Schweighofer Universität Konstanz Convex algebraic geometry, optimization and applications American Institute of Mathematics, Palo
More informationConvex algebraic geometry, optimization and applications
Convex algebraic geometry, optimization and applications organized by William Helton and Jiawang Nie Workshop Summary We start with a little bit of terminology. A linear pencil is a function L : R g d
More informationNOTES ON HYPERBOLICITY CONES
NOTES ON HYPERBOLICITY CONES Petter Brändén (Stockholm) pbranden@math.su.se Berkeley, October 2010 1. Hyperbolic programming A hyperbolic program is an optimization problem of the form minimize c T x 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 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 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 informationSemidefinite programming and convex algebraic geometry
FoCM 2008 - SDP and convex AG p. 1/40 Semidefinite programming and convex algebraic geometry Pablo A. Parrilo www.mit.edu/~parrilo Laboratory for Information and Decision Systems Electrical Engineering
More informationUnbounded Convex Semialgebraic Sets as Spectrahedral Shadows
Unbounded Convex Semialgebraic Sets as Spectrahedral Shadows Shaowei Lin 9 Dec 2010 Abstract Recently, Helton and Nie [3] showed that a compact convex semialgebraic set S is a spectrahedral shadow if the
More informationFirst Order Conditions for Semidefinite Representations of Convex Sets Defined by Rational or Singular Polynomials
Mathematical Programming manuscript No. (will be inserted by the editor) First Order Conditions for Semidefinite Representations of Convex Sets Defined by Rational or Singular Polynomials Jiawang Nie Received:
More informationOPTIMIZATION plays, as it has for many years, a major
952 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 54, NO 5, MAY 2009 Convex Matrix Inequalities Versus Linear Matrix Inequalities J William Helton, Member, IEEE, Scott McCullough, Mihai Putinar, and Victor
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 informationRepresentations of Positive Polynomials: Theory, Practice, and
Representations of Positive Polynomials: Theory, Practice, and Applications Dept. of Mathematics and Computer Science Emory University, Atlanta, GA Currently: National Science Foundation Temple University
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 informationConvex algebraic geometry, optimization and applications
Convex algebraic geometry, optimization and applications The American Institute of Mathematics The following compilation of participant contributions is only intended as a lead-in to the AIM workshop Convex
More informationChapter 13. Convex and Concave. Josef Leydold Mathematical Methods WS 2018/19 13 Convex and Concave 1 / 44
Chapter 13 Convex and Concave Josef Leydold Mathematical Methods WS 2018/19 13 Convex and Concave 1 / 44 Monotone Function Function f is called monotonically increasing, if x 1 x 2 f (x 1 ) f (x 2 ) It
More informationOptimization over Polynomials with Sums of Squares and Moment Matrices
Optimization over Polynomials with Sums of Squares and Moment Matrices Monique Laurent Centrum Wiskunde & Informatica (CWI), Amsterdam and University of Tilburg Positivity, Valuations and Quadratic Forms
More informationExact SDP Relaxations for Classes of Nonlinear Semidefinite Programming Problems
Exact SDP Relaxations for Classes of Nonlinear Semidefinite Programming Problems V. Jeyakumar and G. Li Revised Version:August 31, 2012 Abstract An exact semidefinite linear programming (SDP) relaxation
More informationSum of Squares Relaxations for Polynomial Semi-definite Programming
Sum of Squares Relaxations for Polynomial Semi-definite Programming C.W.J. Hol, C.W. Scherer Delft University of Technology, Delft Center of Systems and Control (DCSC) Mekelweg 2, 2628CD Delft, The Netherlands
More informationLinear Matrix Inequalities vs Convex Sets
Linear Matrix Inequalities vs Convex Sets with admiration and friendship for Eduardo Harry Dym Math Dept Weitzman Inst. Damon Hay Math Dept Florida Texas Igor Klep Math Dept Everywhere in Solvenia Scott
More informationExample: feasibility. Interpretation as formal proof. Example: linear inequalities and Farkas lemma
4-1 Algebra and Duality P. Parrilo and S. Lall 2006.06.07.01 4. Algebra and Duality Example: non-convex polynomial optimization Weak duality and duality gap The dual is not intrinsic The cone of valid
More informationA new look at nonnegativity on closed sets
A new look at nonnegativity on closed sets LAAS-CNRS and Institute of Mathematics, Toulouse, France IPAM, UCLA September 2010 Positivstellensatze for semi-algebraic sets K R n from the knowledge of defining
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 informationRobust and Optimal Control, Spring 2015
Robust and Optimal Control, Spring 2015 Instructor: Prof. Masayuki Fujita (S5-303B) 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 informationHYPERBOLIC POLYNOMIALS, INTERLACERS AND SUMS OF SQUARES
HYPERBOLIC POLYNOMIALS, INTERLACERS AND SUMS OF SQUARES DANIEL PLAUMANN Universität Konstanz joint work with Mario Kummer (U. Konstanz) Cynthia Vinzant (U. of Michigan) Out[121]= POLYNOMIAL OPTIMISATION
More informationSPECTRAHEDRA. Bernd Sturmfels UC Berkeley
SPECTRAHEDRA Bernd Sturmfels UC Berkeley GAeL Lecture I on Convex Algebraic Geometry Coimbra, Portugal, Monday, June 7, 2010 Positive Semidefinite Matrices For a real symmetric n n-matrix A the following
More informationSemidefinite programming lifts and sparse sums-of-squares
1/15 Semidefinite programming lifts and sparse sums-of-squares Hamza Fawzi (MIT, LIDS) Joint work with James Saunderson (UW) and Pablo Parrilo (MIT) Cornell ORIE, October 2015 Linear optimization 2/15
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 informationExact algorithms: from Semidefinite to Hyperbolic programming
Exact algorithms: from Semidefinite to Hyperbolic programming Simone Naldi and Daniel Plaumann PGMO Days 2017 Nov 14 2017 - EDF Lab Paris Saclay 1 Polyhedra P = {x R n : i, l i (x) 0} Finite # linear inequalities
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 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 informationALGEBRAIC DEGREE OF POLYNOMIAL OPTIMIZATION. 1. Introduction. f 0 (x)
ALGEBRAIC DEGREE OF POLYNOMIAL OPTIMIZATION JIAWANG NIE AND KRISTIAN RANESTAD Abstract. Consider the polynomial optimization problem whose objective and constraints are all described by multivariate polynomials.
More informationTangent spaces, normals and extrema
Chapter 3 Tangent spaces, normals and extrema If S is a surface in 3-space, with a point a S where S looks smooth, i.e., without any fold or cusp or self-crossing, we can intuitively define the tangent
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 informationMinimum Ellipsoid Bounds for Solutions of Polynomial Systems via Sum of Squares
Journal of Global Optimization (2005) 33: 511 525 Springer 2005 DOI 10.1007/s10898-005-2099-2 Minimum Ellipsoid Bounds for Solutions of Polynomial Systems via Sum of Squares JIAWANG NIE 1 and JAMES W.
More informationMonotone Function. Function f is called monotonically increasing, if. x 1 x 2 f (x 1 ) f (x 2 ) x 1 < x 2 f (x 1 ) < f (x 2 ) x 1 x 2
Monotone Function Function f is called monotonically increasing, if Chapter 3 x x 2 f (x ) f (x 2 ) It is called strictly monotonically increasing, if f (x 2) f (x ) Convex and Concave x < x 2 f (x )
More informationEconS 301. Math Review. Math Concepts
EconS 301 Math Review Math Concepts Functions: Functions describe the relationship between input variables and outputs y f x where x is some input and y is some output. Example: x could number of Bananas
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 informationAn Exact Jacobian SDP Relaxation for Polynomial Optimization
An Exact Jacobian SDP Relaxation for Polynomial Optimization Jiawang Nie May 26, 2011 Abstract Given polynomials f(x), g i (x), h j (x), we study how to imize f(x) on the set S = {x R n : h 1 (x) = = h
More informationx +3y 2t = 1 2x +y +z +t = 2 3x y +z t = 7 2x +6y +z +t = a
UCM Final Exam, 05/8/014 Solutions 1 Given the parameter a R, consider the following linear system x +y t = 1 x +y +z +t = x y +z t = 7 x +6y +z +t = a (a (6 points Discuss the system depending on the
More informationMoments and Positive Polynomials for Optimization II: LP- VERSUS SDP-relaxations
Moments and Positive Polynomials for Optimization II: LP- VERSUS SDP-relaxations LAAS-CNRS and Institute of Mathematics, Toulouse, France Tutorial, IMS, Singapore 2012 LP-relaxations LP- VERSUS SDP-relaxations
More informationSolving Global Optimization Problems with Sparse Polynomials and Unbounded Semialgebraic Feasible Sets
Solving Global Optimization Problems with Sparse Polynomials and Unbounded Semialgebraic Feasible Sets V. Jeyakumar, S. Kim, G. M. Lee and G. Li June 6, 2014 Abstract We propose a hierarchy of semidefinite
More informationConstrained Optimization Theory
Constrained Optimization Theory Stephen J. Wright 1 2 Computer Sciences Department, University of Wisconsin-Madison. IMA, August 2016 Stephen Wright (UW-Madison) Constrained Optimization Theory IMA, August
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 non-negative integers. For α N n, define the
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 informationCONVEXITY IN SEMI-ALGEBRAIC GEOMETRY AND POLYNOMIAL OPTIMIZATION
CONVEXITY IN SEMI-ALGEBRAIC GEOMETRY AND POLYNOMIAL OPTIMIZATION JEAN B. LASSERRE Abstract. We review several (and provide new) results on the theory of moments, sums of squares and basic semi-algebraic
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 informationarxiv:math/ v1 [math.oc] 11 Jun 2003
February 1, 2008 arxiv:math/0306180v1 [math.oc] 11 Jun 2003 LINEAR MATRIX INEQUALITY REPRESENTATION OF SETS J. William Helton and Victor Vinnikov Mathematics Dept. UCSD Ben Gurion Univ. of the Negev La
More informationd A 0 + m t k A k 0 whenever λ min (B k (x)) t k λ max (B k (x)) for k = 1, 2,..., m x n B n (k).
MATRIX CUBES PARAMETERIZED BY EIGENVALUES JIAWANG NIE AND BERND STURMFELS Abstract. An elimination problem in semidefinite programming is solved by means of tensor algebra. It concerns families of matrix
More informationLecture 5 : Projections
Lecture 5 : Projections EE227C. Lecturer: Professor Martin Wainwright. Scribe: Alvin Wan Up until now, we have seen convergence rates of unconstrained gradient descent. Now, we consider a constrained minimization
More informationSemidefinite Representation of the k-ellipse
arxiv:math/0702005v1 [math.ag] 31 Jan 2007 Semidefinite Representation of the k-ellipse Jiawang Nie Pablo A. Parrilo Bernd Sturmfels February 2, 2008 Abstract The k-ellipse is the plane algebraic curve
More informationComputation of the Joint Spectral Radius with Optimization Techniques
Computation of the Joint Spectral Radius with Optimization Techniques Amir Ali Ahmadi Goldstine Fellow, IBM Watson Research Center Joint work with: Raphaël Jungers (UC Louvain) Pablo A. Parrilo (MIT) R.
More informationA geometric proof of the spectral theorem for real symmetric matrices
0 0 0 A geometric proof of the spectral theorem for real symmetric matrices Robert Sachs Department of Mathematical Sciences George Mason University Fairfax, Virginia 22030 rsachs@gmu.edu January 6, 2011
More informationd + x u1 y v p 1 (x, y) = det
SEMIDEFINITE REPRESENTATION OF THE K-ELLIPSE JIAWANG NIE, PABLO A. PARRILO, AND BERND STURMFELS Abstract. The k-ellipse is the plane algebraic curve consisting of all points whose sum of distances from
More informationthat a broad class of conic convex polynomial optimization problems, called
JOTA manuscript No. (will be inserted by the editor) Exact Conic Programming Relaxations for a Class of Convex Polynomial Cone-Programs Vaithilingam Jeyakumar Guoyin Li Communicated by Levent Tunçel Abstract
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 informationThe Algebraic Degree of Semidefinite Programming
The Algebraic Degree ofsemidefinite Programming p. The Algebraic Degree of Semidefinite Programming BERND STURMFELS UNIVERSITY OF CALIFORNIA, BERKELEY joint work with and Jiawang Nie (IMA Minneapolis)
More informationMATH2070 Optimisation
MATH2070 Optimisation Nonlinear optimisation with constraints Semester 2, 2012 Lecturer: I.W. Guo Lecture slides courtesy of J.R. Wishart Review The full nonlinear optimisation problem with equality constraints
More informationThe convex algebraic geometry of rank minimization
The convex algebraic geometry of rank minimization Pablo A. Parrilo Laboratory for Information and Decision Systems Massachusetts Institute of Technology International Symposium on Mathematical Programming
More informationSystems of Equations and Inequalities. College Algebra
Systems of Equations and Inequalities College Algebra System of Linear Equations There are three types of systems of linear equations in two variables, and three types of solutions. 1. An independent system
More informationLinear and non-linear programming
Linear and non-linear 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 informationTMA 4180 Optimeringsteori KARUSH-KUHN-TUCKER THEOREM
TMA 4180 Optimeringsteori KARUSH-KUHN-TUCKER THEOREM H. E. Krogstad, IMF, Spring 2012 Karush-Kuhn-Tucker (KKT) Theorem is the most central theorem in constrained optimization, and since the proof is scattered
More informationALGEBRA: From Linear to Non-Linear. Bernd Sturmfels University of California at Berkeley
ALGEBRA: From Linear to Non-Linear Bernd Sturmfels University of California at Berkeley John von Neumann Lecture, SIAM Annual Meeting, Pittsburgh, July 13, 2010 Undergraduate Linear Algebra All undergraduate
More informationSARD S THEOREM ALEX WRIGHT
SARD S THEOREM ALEX WRIGHT Abstract. A proof of Sard s Theorem is presented, and applications to the Whitney Embedding and Immersion Theorems, the existence of Morse functions, and the General Position
More informationarxiv:math/ v2 [math.oc] 25 May 2004
arxiv:math/0304104v2 [math.oc] 25 May 2004 THE LAX CONJECTURE IS TRUE A.S. Lewis, P.A. Parrilo, and M.V. Ramana Key words: hyperbolic polynomial, Lax conjecture, hyperbolicity cone, semidefinite representable
More informationThe Geometry of Semidefinite Programming. Bernd Sturmfels UC Berkeley
The Geometry of Semidefinite Programming Bernd Sturmfels UC Berkeley Positive Semidefinite Matrices For a real symmetric n n-matrix A the following are equivalent: All n eigenvalues of A are positive real
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 informationSOS-Convex Lyapunov Functions with Applications to Nonlinear Switched Systems
SOS-Convex Lyapunov Functions with Applications to Nonlinear Switched Systems Amir Ali Ahmadi and Raphaël M. Jungers Abstract We introduce the concept of sos-convex Lyapunov functions for stability analysis
More informationTrust Region Problems with Linear Inequality Constraints: Exact SDP Relaxation, Global Optimality and Robust Optimization
Trust Region Problems with Linear Inequality Constraints: Exact SDP Relaxation, Global Optimality and Robust Optimization V. Jeyakumar and G. Y. Li Revised Version: September 11, 2013 Abstract The trust-region
More informationGlobal Optimization with Polynomials
Global Optimization with Polynomials Geoffrey Schiebinger, Stephen Kemmerling Math 301, 2010/2011 March 16, 2011 Geoffrey Schiebinger, Stephen Kemmerling (Math Global 301, 2010/2011) Optimization with
More informationSPECTRAHEDRA. Bernd Sturmfels UC Berkeley
SPECTRAHEDRA Bernd Sturmfels UC Berkeley Mathematics Colloquium, North Carolina State University February 5, 2010 Positive Semidefinite Matrices For a real symmetric n n-matrix A the following are equivalent:
More informationHYPERBOLICITY CONES AND IMAGINARY PROJECTIONS
HYPERBOLICITY CONES AND IMAGINARY PROJECTIONS THORSTEN JÖRGENS AND THORSTEN THEOBALD Abstract. Recently, the authors and de Wolff introduced the imaginary projection of a polynomial f C[z] as the projection
More informationA positive definite polynomial Hessian that does not factor
A positive definite polynomial Hessian that does not factor The MIT Faculty has made this article openly available Please share how this access benefits you Your story matters Citation As Published Publisher
More informationNotes on the decomposition result of Karlin et al. [2] for the hierarchy of Lasserre by M. Laurent, December 13, 2012
Notes on the decomposition result of Karlin et al. [2] for the hierarchy of Lasserre by M. Laurent, December 13, 2012 We present the decomposition result of Karlin et al. [2] for the hierarchy of Lasserre
More informationOn Polynomial Optimization over Non-compact Semi-algebraic Sets
On Polynomial Optimization over Non-compact Semi-algebraic Sets V. Jeyakumar, J.B. Lasserre and G. Li Revised Version: April 3, 2014 Communicated by Lionel Thibault Abstract The optimal value of a polynomial
More informationA new approximation hierarchy for polynomial conic optimization
A new approximation hierarchy for polynomial conic optimization Peter J.C. Dickinson Janez Povh July 11, 2018 Abstract In this paper we consider polynomial conic optimization problems, where the feasible
More informationLecture 5. 1 Goermans-Williamson Algorithm for the maxcut problem
Math 280 Geometric and Algebraic Ideas in Optimization April 26, 2010 Lecture 5 Lecturer: Jesús A De Loera Scribe: Huy-Dung Han, Fabio Lapiccirella 1 Goermans-Williamson Algorithm for the maxcut problem
More informationOptimality Conditions
Chapter 2 Optimality Conditions 2.1 Global and Local Minima for Unconstrained Problems When a minimization problem does not have any constraints, the problem is to find the minimum of the objective function.
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 informationSeminars on Mathematics for Economics and Finance Topic 5: Optimization Kuhn-Tucker conditions for problems with inequality constraints 1
Seminars on Mathematics for Economics and Finance Topic 5: Optimization Kuhn-Tucker conditions for problems with inequality constraints 1 Session: 15 Aug 2015 (Mon), 10:00am 1:00pm I. Optimization with
More information2 EBERHARD BECKER ET AL. has a real root. Thus our problem can be reduced to the problem of deciding whether or not a polynomial in one more variable
Deciding positivity of real polynomials Eberhard Becker, Victoria Powers, and Thorsten Wormann Abstract. We describe an algorithm for deciding whether or not a real polynomial is positive semidenite. The
More informationLift-and-Project Techniques and SDP Hierarchies
MFO seminar on Semidefinite Programming May 30, 2010 Typical combinatorial optimization problem: max c T x s.t. Ax b, x {0, 1} n P := {x R n Ax b} P I := conv(k {0, 1} n ) LP relaxation Integral polytope
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 informationON SUM OF SQUARES DECOMPOSITION FOR A BIQUADRATIC MATRIX FUNCTION
Annales Univ. Sci. Budapest., Sect. Comp. 33 (2010) 273-284 ON SUM OF SQUARES DECOMPOSITION FOR A BIQUADRATIC MATRIX FUNCTION L. László (Budapest, Hungary) Dedicated to Professor Ferenc Schipp on his 70th
More informationPROFIT FUNCTIONS. 1. REPRESENTATION OF TECHNOLOGY 1.1. Technology Sets. The technology set for a given production process is defined as
PROFIT FUNCTIONS 1. REPRESENTATION OF TECHNOLOGY 1.1. Technology Sets. The technology set for a given production process is defined as T {x, y : x ɛ R n, y ɛ R m : x can produce y} where x is a vector
More informationRank-one LMIs and Lyapunov's Inequality. Gjerrit Meinsma 4. Abstract. We describe a new proof of the well-known Lyapunov's matrix inequality about
Rank-one LMIs and Lyapunov's Inequality Didier Henrion 1;; Gjerrit Meinsma Abstract We describe a new proof of the well-known Lyapunov's matrix inequality about the location of the eigenvalues of a matrix
More informationHyperbolic Polynomials and Generalized Clifford Algebras
Discrete Comput Geom (2014) 51:802 814 DOI 10.1007/s00454-014-9598-1 Hyperbolic Polynomials and Generalized Clifford Algebras Tim Netzer Andreas Thom Received: 14 December 2012 / Revised: 8 January 2014
More informationConvex Optimization & Parsimony of L p-balls representation
Convex Optimization & Parsimony of L p -balls representation LAAS-CNRS and Institute of Mathematics, Toulouse, France IMA, January 2016 Motivation Unit balls associated with nonnegative homogeneous polynomials
More informationDEVELOPMENT OF MORSE THEORY
DEVELOPMENT OF MORSE THEORY MATTHEW STEED Abstract. In this paper, we develop Morse theory, which allows us to determine topological information about manifolds using certain real-valued functions defined
More informationarxiv: v1 [math.oc] 6 Mar 2009
A convex polynomial that is not sos-convex Amir Ali Ahmadi and Pablo A. Parrilo arxiv:090.87v1 [math.oc] 6 Mar 2009 Abstract A multivariate polynomial p(x) = p(x 1,...,x n ) is sos-convex if its Hessian
More informationEC /11. Math for Microeconomics September Course, Part II Problem Set 1 with Solutions. a11 a 12. x 2
LONDON SCHOOL OF ECONOMICS Professor Leonardo Felli Department of Economics S.478; x7525 EC400 2010/11 Math for Microeconomics September Course, Part II Problem Set 1 with Solutions 1. Show that the general
More informationBoston College. Math Review Session (2nd part) Lecture Notes August,2007. Nadezhda Karamcheva www2.bc.
Boston College Math Review Session (2nd part) Lecture Notes August,2007 Nadezhda Karamcheva karamche@bc.edu www2.bc.edu/ karamche 1 Contents 1 Quadratic Forms and Definite Matrices 3 1.1 Quadratic Forms.........................
More informationLecture 6 Positive Definite Matrices
Linear Algebra Lecture 6 Positive Definite Matrices Prof. Chun-Hung Liu Dept. of Electrical and Computer Engineering National Chiao Tung University Spring 2017 2017/6/8 Lecture 6: Positive Definite Matrices
More informationCHAPTER 2: CONVEX SETS AND CONCAVE FUNCTIONS. W. Erwin Diewert January 31, 2008.
1 ECONOMICS 594: LECTURE NOTES CHAPTER 2: CONVEX SETS AND CONCAVE FUNCTIONS W. Erwin Diewert January 31, 2008. 1. Introduction Many economic problems have the following structure: (i) a linear function
More informationNon-Commutative Subharmonic and Harmonic Polynomials
Non-Commutative Subharmonic and Harmonic Polynomials by Joshua A. Hernandez Dr. J. William Helton, Honors Advisor UCSD Department of Mathematics June 9, 2005 1 Contents 1 Introduction 3 1.1 Non-Commutative
More informationCalculus and optimization
Calculus an optimization These notes essentially correspon to mathematical appenix 2 in the text. 1 Functions of a single variable Now that we have e ne functions we turn our attention to calculus. A function
More informationMoments and Positive Polynomials for Optimization II: LP- VERSUS SDP-relaxations
Moments and Positive Polynomials for Optimization II: LP- VERSUS SDP-relaxations LAAS-CNRS and Institute of Mathematics, Toulouse, France EECI Course: February 2016 LP-relaxations LP- VERSUS SDP-relaxations
More informationConvex optimization. Javier Peña Carnegie Mellon University. Universidad de los Andes Bogotá, Colombia September 2014
Convex optimization Javier Peña Carnegie Mellon University Universidad de los Andes Bogotá, Colombia September 2014 1 / 41 Convex optimization Problem of the form where Q R n convex set: min x f(x) x Q,
More informationarxiv:math/ v3 [math.oc] 5 Oct 2007
arxiv:math/0606476v3 [math.oc] 5 Oct 2007 Sparse SOS Relaxations for Minimizing Functions that are Summations of Small Polynomials Jiawang Nie and James Demmel March 27, 2018 Abstract This paper discusses
More information