v2007.09.13 - Convex Optimization

478 CHAPTER 7. PROXIMITY PROBLEMSthen we wanted an EDM closest to H in some sense; id est, input-matrix Hwas assumed corrupted somehow. For practical problems, it withstandsreason that such a proximity problem could instead be reformulated so thatsome or all entries of H were unknown but bounded above and below byknown limits; the norm objective is thereby eliminated as in the developmentbeginning on page 265. That redesign (the art, p.8), in terms of theGram-matrix bridge between point-list X and EDM D , at once encompassesproximity and completion problems.A third recourse is to apply the method of convex iteration just like wedid in7.2.2.7.1. This technique is applicable to any semidefinite problemrequiring a rank constraint; it places a regularization term in the objectivethat enforces the rank constraint.The importance and application of solving rank-constrained problemsare enormous, a conclusion generally accepted gratis by the mathematicsand engineering communities. Rank-constrained semidefinite programsarise in many vital feedback and control problems [121], optics [54],and communications [208] [185]. Rank-constrained problems also appearnaturally in combinatorial optimization. (4.4.3.0.7)