v2010.10.26 - Convex Optimization

390 CHAPTER 4. SEMIDEFINITE PROGRAMMING4.8 Convex Iteration rank-1We now develop a general method for constraining rank that first decomposesa given problem via standard diagonalization of matrices (A.5). Thismethod is motivated by observation (4.4.1.1) that an optimal directionmatrix can be simultaneously diagonalizable with an optimal variablematrix. This suggests minimization of an objective function directly interms of eigenvalues. A second motivating observation is that variableorthogonal matrices seem easily found by convex iteration; e.g., ProcrustesExample 4.6.0.0.2.It turns out that this general method always requires solution to a rank-1constrained problem regardless of desired rank from the original problem. Todemonstrate, we pose a semidefinite feasibility problemfindX ∈ S nsubject to A svec X = bX ≽ 0rankX ≤ ρ(869)given an upper bound 0 < ρ < n on rank, vector b ∈ R m , and typically fatfull-rank⎡ ⎤svec(A 1 ) TA = ⎣ . ⎦∈ R m×n(n+1)/2 (650)svec(A m ) Twhere A i ∈ S n , i=1... m are also given. Symmetric matrix vectorizationsvec is defined in (56). Thus⎡ ⎤tr(A 1 X)A svec X = ⎣ . ⎦ (651)tr(A m X)This program (869) states the classical problem of finding a matrix ofspecified rank in the intersection of the positive semidefinite cone witha number m of hyperplanes in the subspace of symmetric matrices S n .[26,II.13] [24,2.2]Express the nonincreasingly ordered diagonalization of variable matrixn∑X QΛQ T = λ i Q ii ∈ S n (870)i=1