v2010.10.26 - Convex Optimization

362 CHAPTER 4. SEMIDEFINITE PROGRAMMINGwhere⎡E =⎢⎣0 0 0 0 1 0 00 0 0 0 0 1 00 0 0 1 0 0 00 0 1 0 0 0 01 0 0 0 0 0 00 1 0 0 0 0 00 0 0 0 0 0 0⎤12 ∈ S7 (807)⎥⎦(Matlab code on Wıκımization) Positive semidefiniteness is optional onlywhen rank-1 constraints are explicit by Theorem A.3.1.0.7. Optimal solutionto problem (803) is not unique: (x, y , z) = (0, 2 , 1).3 34.6.0.0.8 Example. Boolean vector satisfying Ax ≼ b. (confer4.2.3.1.1)Now we consider solution to a discrete problem whose only known analyticalmethod of solution is combinatorial in complexity: given A∈ R M×N andb∈ R M find x ∈ R Nsubject to Ax ≼ bδ(xx T ) = 1(808)This nonconvex problem demands a Boolean solution [x i = ±1, i=1... N ].Assign a rank-1 matrix of variables; symmetric variable matrix X andsolution vector x :[ xG =1] [x T 1] =[ X xx T 1][ xx T xx T 1]∈ S N+1 (809)Then design an equivalent semidefinite feasibility problem to find a Booleansolution to Ax ≼ b :findX∈S Nx ∈ R Nsubject to Ax ≼ bG =[ X xx T 1rankG = 1δ(X) = 1](≽ 0)(810)