v2010.10.26 - Convex Optimization

4.3. RANK REDUCTION 3054.3.3.0.1 Example. Aδ(X) = b .This academic example demonstrates that a solution found by rank reductioncan certainly have rank less than Barvinok’s upper bound (272): Assume agiven vector b belongs to the conic hull of columns of a given matrix A⎡⎤⎡ ⎤A =⎢⎣−1 1 8 1 1−3 2 812−9 4 8141319Consider the convex optimization problem⎥⎦∈ R m×n , b =⎢⎣11214⎥⎦∈ R m (724)minimize trXX∈ S 5subject to X ≽ 0Aδ(X) = b(725)that minimizes the 1-norm of the main diagonal; id est, problem (725) is thesame asminimize ‖δ(X)‖ 1X∈ S 5subject to X ≽ 0(726)Aδ(X) = bthat finds a solution to Aδ(X)=b. Rank-3 solution X ⋆ = δ(x M) is optimal,where (confer (684))⎡ 2 ⎤1280x M=5⎢ 128 ⎥(727)⎣ 0 ⎦90128Yet upper bound (272) predicts existence of at most a(⌊√ ⌋ )8m + 1 − 1rank-= 22(728)feasible solution from m = 3 equality constraints. To find a lowerrank ρ optimal solution to (725) (barring combinatorics), we invokeProcedure 4.3.1.0.1: