v2010.10.26 - Convex Optimization

314 CHAPTER 4. SEMIDEFINITE PROGRAMMING4.4.1.2.2 Exercise. Completely positive semidefinite ⎡ matrix. ⎤ [40]0.50 0.55 0.20Given rank-2 positive semidefinite matrix G = ⎣ 0.55 0.61 0.22 ⎦, find a0.20 0.22 0.08positive factorization G = X T X (900) by solvingfind X ≥ 0X∈R 2×3 [ I Xsubject to Z =X T GrankZ ≤ 2]≽ 0(740)via convex iteration.4.4.1.2.3 Exercise. Nonnegative matrix ⎡ factorization. ⎤17 28 42Given rank-2 nonnegative matrix X = ⎣ 16 47 51 ⎦, find a nonnegative17 82 72factorizationX = WH (741)by solvingfindA∈S 3 , B∈S 3 , W ∈R 3×2 , H∈R 2×3W , H⎡subject to Z = ⎣W ≥ 0H ≥ 0rankZ ≤ 2I W T HW A XH T X T Bwhich follows from the fact, at optimality,⎡ ⎤IZ ⋆ = ⎣ W ⎦ [I W T H ]H T⎤⎦≽ 0(742)(743)Use the known closed-form solution for a direction vector Y to regulate rankby convex iteration; set Z ⋆ = QΛQ T ∈ S 8 to an ordered diagonalization andU ⋆ = Q(:, 3:8)∈ R 8×6 , then Y = U ⋆ U ⋆T (4.4.1.1).