v2007.09.13 - Convex Optimization

476 CHAPTER 7. PROXIMITY PROBLEMSverifiable by observing conic dependencies (2.10.3) among the aggregate ofhalfspace-description normals. The cone membership constraint in (1209a)therefore inherently insures existence of a symmetric hollow matrix. Optimization (1209b) would be a Procrustes problem (C.4) were itnot for the hollowness constraint; it is, instead, a minimization over theintersection of the nonconvex manifold of orthogonal matrices with anothernonconvex set in variable R specified by the hollowness constraint.We solve problem (1209b) by a method introduced in4.4.3.0.4: DefineR = [r 1 · · · r N+1 ]∈ R N+1×N+1 and make the assignment⎡G = ⎢⎣r 1.r N+11⎤[ r1 T · · · rN+1 T 1] ⎥∈ S (N+1)2 +1⎦⎡⎤ ⎡R 11 · · · R 1,N+1 r 1.....∆= ⎢⎣ R1,N+1 T ⎥=⎢R N+1,N+1 r N+1 ⎦ ⎣r1 T · · · rN+1 T 1(1212)⎤r 1 r1 T · · · r 1 rN+1 T r 1.....r N+1 r1 T r N+1 rN+1 T ⎥r N+1 ⎦r1 T · · · rN+1 T 1where R ij ∆ = r i r T j ∈ R N+1×N+1 . Then (1209b) is equivalently expressed:∥ ∥∥∥ N+12∑minimize Υ ii R ii − ΛR ij , r i∥i=1Fsubject to trR ii = 1, i=1... N+1trR ij ⎡= 0,⎤i