v2010.10.26 - Convex Optimization

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 353minimizey∈R 2n 〈a ⋆ , y〉subject to 0 ≼ y ≼ 1y T 1 = 2n − k(525)where x ⋆ = [I −I ]a ⋆ ; from which it may be construed that any vector1-norm minimization problem has equivalent expression in a nonnegativevariable.4.6 Cardinality and rank constraint examples4.6.0.0.1 Example. Projection on ellipsoid boundary. [52] [148,5.1][247,2] Consider classical linear equation Ax = b but with constraint onnorm of solution x , given matrices C , fat A , and vector b∈R(A)find x ∈ R Nsubject to Ax = b‖Cx‖ = 1(779)The set {x | ‖Cx‖=1} (2) describes an ellipsoid boundary (Figure 13). Thisis a nonconvex problem because solution is constrained to that boundary.AssignG =[ Cx1] [ ][x T C T 1] X Cx=x T C T 1[ Cxx T C T Cxx T C T 1]∈ S N+1 (780)Any rank-1 solution must have this form. (B.1.0.2) Ellipsoidally constrainedfeasibility problem (779) is equivalent to:findX∈S Nx ∈ R Nsubject to Ax = b[X CxG =x T C T 1rankG = 1trX = 1](≽ 0)(781)