sdp094_digest

sdp094_digest - Introduction to Semidenite Programming(SDP...

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
Introduction to Semidefinite Programming (SDP) Robert M. Freund and Brian Anthony with assistance from David Craft May 4-6, 2004
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
1 Outline Slide 1 Alternate View of Linear Programming Facts about Symmetric and Semidefinite Matrices SDP SDP Duality Examples of SDP Combinatorial Optimization: MAXCUT Convex Optimization: Quadratic Constraints, Eigenvalue Problems, log det( X ) problems Interior-Point Methods for SDP Application: Truss Vibration Dynamics via SDP 2 Linear Programming 2.1 Alternative Perspective Slide 2 LP : minimize c · x s.t. a i · x = b i ,i =1 ,...,m n x ∈± + . n c · x means the linear function j =1 c j x j n ± + := { x n | x 0 } is the nonnegative orthant. n ± + is a convex cone . K is convex cone if x, w K and α, β 0 αx + βw K . Slide 3 LP : minimize c · x s.t. a i · x = b i n x + . “Minimize the linear function c · x , subject to the condition that x must solve m given n equations a i · x = b i , and that x must lie in the convex cone K = ± + .” 2.1.1 LP Dual Problem Slide 4 m LD : maximize y i b i i =1 m s.t. y i a i + s = c i =1 n s + . 1
Background image of page 2
± ± ² ³ ± For feasible solutions x of LP and ( y, s )o f LD , the duality gap is simply m m c · x y i b i = c y i a i · x = s · x 0 i =1 i =1 Slide 5 If LP and LD are feasible, then there exists x and ( y ,s ) feasible for the primal and dual, respectively, for which m c · x y i b i = s · x =0 i =1 3 Facts about the Semidefinite Cone Slide 6 If X is an n × n matrix, then X is a symmetric positive semidefinite (SPSD) matrix if X = X T and v T Xv 0 for any v ∈± n If X is an n × n matrix, then X is a symmetric positive definite (SPD) matrix if X = X T and v T > 0 for any v n ,v ² 4 Facts about the Semidefinite Cone Slide 7 S n denotes the set of symmetric n × n matrices S n + denotes the set of (SPSD) n × n matrices. S n ++ denotes the set of (SPD) n × n matrices. Let X, Y S n . Slide 8 X ³ 0” denotes that X is SPSD X ³ Y denotes that X Y ³ 0 X ´ 0” to denote that X is SPD, etc. Remark: S n = { X S n | X ³ 0 } is a convex cone. + 5 Facts about Eigenvalues and Eigenvectors Slide 9 If M is a square n × n matrix, then λ is an eigenvalue of M with corresponding eigenvector q if Mq = λq and q ² . Let λ 1 2 ,...,λ n enumerate the eigenvalues of M . 6 about Eigenvalues and Eigenvectors Slide 10 2 n The corresponding eigenvectors q 1 ,q ,...,q of M can be chosen so that they are orthonormal, namely ( i ) T ( ) ( i ) T ( ) j q q for i ² = j, and q q i =1 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
± ² ± ² ³ ´ µ Define: 2 n Q := q 1 q ··· q Then Q is an orthonormal matrix: Q T 1 Q = I, equivalently Q T = Q Slide 11 λ 1 2 ,...,λ n are the eigenvalues of M 1 2 n q ,q ,...,q are the corresponding orthonormal eigenvectors of M 2 n Q := q 1 q q Q T Q = equivalently Q T = Q 1 Define D : 0 λ 1 0 0 λ 2 D := .
Background image of page 4
Image of page 5
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 28

sdp094_digest - Introduction to Semidenite Programming(SDP...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online