v2010.10.26 - Convex Optimization

298 CHAPTER 4. SEMIDEFINITE PROGRAMMING⎛⎡y ⋆ ⎜⎢= round⎝⎣1 − 1 √21 − 1 √21⎤⎞⎥⎟⎦⎠ =⎡⎢⎣001⎤⎥⎦ (700)(infeasible, with or without rounding, with respect to original problem (680))whereas solving semidefinite program (692) produces⎡⎤1 1 −1 1round(G ⋆ ) = ⎢ 1 1 −1 1⎥⎣ −1 −1 1 −1 ⎦ (701)1 1 −1 1with sorted eigenvaluesλ(G ⋆ ) =⎡⎢⎣3.999999650572640.00000035942736−0.00000000000000−0.00000001000000⎤⎥⎦ (702)Truncating all but the largest eigenvalue, from (688) we obtain (confer y ⋆ )⎛⎡⎤⎞⎡ ⎤0.99999999625299 1x ⋆ = round⎝⎣0.99999999625299 ⎦⎠ = ⎣ 1 ⎦ (703)0.00000001434518 0the desired minimal cardinality Boolean result.4.2.3.1.3 Exercise. Minimal cardinality Boolean art.Assess general performance of standard-practice approximation (697) ascompared with the proposed semidefinite program (692). 4.2.3.1.4 Exercise. Conic independence.Matrix A from (682) is full-rank having three-dimensional nullspace. Findits four conically independent columns. (2.10) To what part of proper coneK = {Ax | x ≽ 0} does vector b belong?4.2.3.1.5 Exercise. Linear independence.Show why fat matrix A , from compressed sensing problem (518) or (523),may be regarded full-rank without loss of generality. In other words: Is aminimal cardinality solution invariant to linear dependence of rows?