sdp094_digest

# sdp094_digest - Introduction to Semidenite Programming(SDP...

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Introduction to Semideﬁnite Programming (SDP) Robert M. Freund and Brian Anthony with assistance from David Craft May 4-6, 2004

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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
± ± 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

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± 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 := .
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sdp094_digest - Introduction to Semidenite Programming(SDP...

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