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Unformatted text preview: 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 the SDP in standard form: p * := max X C,X : A i ,X = b i , i = 1 ,...,m, X , (13.1) where C,A i are given symmetric matrices, A,B = Tr AB denotes the scalar product between two symmetric matrices, and b ∈ R m is given. 13.1.2 Dual problem At first glance, the problem (13.1) is not amenable to the duality theory developed so far, since the constraint X 0 is not a scalar one. Minimum eigenvalue representation. We develop a dual based on a representation of the problem via the minimum eigenvalue, as p * = max X C,X : A i ,X = b i , i = 1 ,...,m, λ min ( X ) ≥ , (13.2) where we have used the minimum eigenvalue function of a symmetric matrix A , given by λ min ( A ) := min Y Y,A : Y , Tr Y = 1 (13.3) to represent the positive semi-definiteness condition X 0 in the SDP. The proof of the above representation of the minimum eigenvalue can be obtained by first showing that we can without loss of generality assume that A is diagonal, and noticing that we can then restrict Y to be diagonal as well. Note that the above representation proves that λ min is concave, so problem (13.2) is indeed convex as written. 13-1 EE 227A Lecture 13 — October 14, 2008 Fall 2008 Lagrangian and dual function. The Lagrangian for the maximization problem (13.2) is L ( X,λ,ν ) = C,X + m i =1 ν i ( b i- A i ,X ) + λ · λ min ( X ) = ν T b + C- m i =1 ν i A i ,X + λ · λ min ( X ) , where nu ∈ R m and λ ≥ 0 are the dual variables. The corresponding dual function g ( λ, ν ) := max X L ( X,λ,ν ) . involves the following subproblem, in which Z = C- ∑ m i =1 ν i A i and λ ≥ 0 are given: G ( λ, Z ) := max X Z,X + λλ min ( X ) . (13.4) For fixed λ ≥ 0, the function G ( · ,λ ) is the conjugate of the convex function- λλ min ....
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This note was uploaded on 09/03/2011 for the course EE 227A taught by Professor Staff during the Spring '11 term at Berkeley.
- Spring '11