v2007.11.26 - Convex Optimization

286 CHAPTER 4. SEMIDEFINITE PROGRAMMING4.6.0.0.3 Example. Tractable polynomial constraint.The ability to handle rank constraints makes polynomial constraints(generally nonconvex) transformable to convex constraints. All optimizationproblems having polynomial objective and polynomial constraints can bereformulated as a semidefinite program with a rank-1 constraint. [212]Suppose we require3 + 2x − xy ≤ 0 (676)Assign⎡ ⎤x [ x y 1]G = ⎣ y ⎦ =1[ X zz T 1]∆=⎡ ⎤x 2 xy x⎣ xy y 2 y ⎦∈ S 3 (677)x y 1The polynomial constraint (676) is equivalent to the constraint set (B.1.0.2)tr(GA) ≤ 0[ X zG =z T 1rankG = 1](≽ 0)(678)in symmetric variable matrix X ∈ S 2 and variable vector z ∈ R 2 where⎡ ⎤0 − 1 12A = ⎣ − 1 0 0 ⎦ (679)21 0 3Then the method of convex iteration from4.4.1 is applied to implement therank constraint.4.6.0.0.4 Example. Boolean vector feasible to 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(680)This nonconvex problem demands a Boolean solution [x i = ±1, i=1... N ].