v2007.09.13 - Convex Optimization

4.4. RANK-CONSTRAINED SEMIDEFINITE PROGRAM 287Figure 72: Logo of Massachusetts Institute of Technology may be regardeda rank-5 matrix when comprising its white boundary. (Rank of StanfordUniversity logo is much higher;)...4.4.4 rank-constraint conclusionWe find that this direction matrix idea works well and quite independentlyof desired rank or affine dimension.There exists a common thread through all these Examples: that being,convex iteration with a direction matrix as normal to a linear regularization.But each problem type (per Example) possesses its own idiosyncrasies thatslightly modify how a rank-constrained optimal solution is actually obtained.The ball packing problem in Chapter 5.4.2.2.3 requires a problem sequencein a progressively larger number of balls to find a good initial value for thedirection matrix, whereas many of the examples in this chapter require aninitial value of 0. The sparsest solution Example 4.4.3.0.1 wants a directionmatrix corresponding to a rank-2 search when a rank-1 solution is desired.Finding a feasible Boolean vector in Example 4.4.3.0.5 requires a procedureto detect stalls, when other problems have no such requirement; and so on.Nevertheless, this idea of direction matrix is good because of its simplicity:When one is confronted with a problem otherwise convex if not for a rankconstraint, then that constraint becomes a linear regularization term in theobjective. Some work remains in refining initial value of the direction matrixin the regularization because poor initialization of the convex iteration canlead to an erroneous result.