v2009.01.01 - Convex Optimization
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v2009.01.01 - Convex Optimization

v2009.01.01 - Convex Optimization v2009.01.01 - Convex Optimization

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342 CHAPTER 4. SEMIDEFINITE PROGRAMMING The fact Λ ≽ 0 ⇔ X ≽ 0 (1353) allows splitting semidefinite feasibility problem (776) into two parts: ( ρ ) minimize Λ ∥ A svec ∑ λ i Q ii − b ∥ i=1 [ R ρ ] (779) + subject to δ(Λ) ∈ 0 ( ρ ) minimize Q ∥ A svec ∑ λ ⋆ i Q ii − b ∥ i=1 (780) subject to Q T = Q −1 The linear equality constraint A svec X = b has been moved to the objective within a norm because these two problems (779) (780) are iterated; equality might only become feasible near convergence. This iteration always converges to a local minimum because the sequence of objective values is monotonic and nonincreasing; any monotonically nonincreasing real sequence converges. [222,1.2] [37,1.1] A rank-ρ matrix X feasible to the original problem (776) is found when the objective converges to 0. Positive semidefiniteness of matrix X with an upper bound ρ on rank is assured by the constraint on eigenvalue matrix Λ in convex problem (779); it means, the main diagonal of Λ must belong to the nonnegative orthant in a ρ-dimensional subspace of R n . The second problem (780) in the iteration is not convex. We propose solving it by convex iteration: Make the assignment ⎡ G = ⎣ ⎡ = ⎣ ⎤ q 1 [q1 T · · · qρ T ] . ⎦ ∈ S nρ q ρ ⎤ ⎡ Q 11 · · · Q 1ρ q 1 q . ... . ⎦ ∆ 1 T · · · q 1 qρ T = ⎣ . ... . Q T 1ρ · · · Q ρρ q ρ q1 T · · · q ρ qρ T ⎤ ⎦ (781)

4.7. CONVEX ITERATION RANK-1 343 Given ρ eigenvalues λ ⋆ i , then a problem equivalent to (780) is ∥ ( ∥∥∥ ρ ) ∑ minimize A svec λ ⋆ Q ij ∈R n×n i Q ii − b ∥ i=1 subject to trQ ii = 1, i=1... ρ trQ ij = 0, i

4.7. CONVEX ITERATION RANK-1 343<br />

Given ρ eigenvalues λ ⋆ i , then a problem equivalent to (780) is<br />

∥ ( ∥∥∥ ρ<br />

) ∑<br />

minimize A svec λ ⋆<br />

Q ij ∈R n×n<br />

i Q ii − b<br />

∥<br />

i=1<br />

subject to trQ ii = 1,<br />

i=1... ρ<br />

trQ ij = 0,<br />

i

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