v2007.11.26 - Convex Optimization

4.3. RANK REDUCTION 2534.3.3.0.1 Example. Aδ(X) = b .This academic example demonstrates that a solution found by rank reductioncan certainly have rank less than Barvinok’s upper bound (233): Assume agiven vector b∈ R m belongs to the conic hull of the columns of a given matrixA∈ R m×n ;⎡⎤ ⎡ ⎤A =⎢⎣−1 1 8 1 1−3 2 812−9 4 8141319Consider the convex optimization problem⎥⎦ , b =⎢⎣11214⎥⎦ (621)minimize trXX∈ S 5subject to X ≽ 0Aδ(X) = b(622)that minimizes the 1-norm of the main diagonal; id est, problem (622) is thesame asminimizeX∈ S 5 ‖δ(X)‖ 1subject to X ≽ 0Aδ(X) = b(623)that finds a solution to Aδ(X)=b. Rank-3 solution X ⋆ = δ(x M) is optimal,where (confer (581))⎡ 2 ⎤1280x M=5⎢ 128 ⎥(624)⎣ 0 ⎦90128Yet upper bound (233) predicts existence of at most a(⌊√ ⌋ )8m + 1 − 1rank-= 22(625)feasible solution from m = 3 equality constraints. To find a lowerrank ρ optimal solution to (622) (barring combinatorics), we invokeProcedure 4.3.1.0.1: