v2009.01.01 - Convex Optimization

340 CHAPTER 4. SEMIDEFINITE PROGRAMMING
problem in Chapter 5.4.2.2.3, for example, requires a problem sequence in
a progressively larger number of balls to find a good initial value for the
direction matrix, whereas many of the examples in the present chapter require
an initial value of 0. Finding a feasible Boolean vector in Example 4.6.0.0.6
requires a procedure to detect stalls, when other problems have no such
requirement. The combinatorial Procrustes problem in Example 4.6.0.0.3
allows use of a known closed-form solution for direction vector when solved
via rank constraint, but not when solved via cardinality constraint. Some
problems require a careful weighting of the regularization term, whereas other
problems do not, and so on. It would be nice if there were a universally
applicable method for constraining rank; one that is less susceptible to quirks
of a particular problem type.
Poor initialization of the direction matrix from the regularization can
lead to an erroneous result. We speculate one reason to be a simple dearth
of optimal solutions of desired rank or cardinality; 4.55 an unfortunate choice
of initial search direction leading astray. Ease of solution by convex iteration
occurs when optimal solutions abound. With this speculation in mind, we
now propose a further generalization of convex iteration for constraining rank
that attempts to ameliorate quirks and unify problem types:
4.7 Convex Iteration rank-1
We now develop a general method for constraining rank that first decomposes
a given problem via standard diagonalization of matrices (A.5). This
method is motivated by observation (4.4.1.1) that an optimal direction
matrix can be diagonalizable simultaneously with an optimal variable
matrix. This suggests minimization of an objective function directly in
terms of eigenvalues. A second motivating observation is that variable
orthogonal matrices seem easily found by convex iteration; e.g., Procrustes
Example 4.6.0.0.2.
It turns out that this general method always requires solution to a rank-1
constrained problem regardless of desired rank from the original problem. To
demonstrate, we pose a semidefinite feasibility problem
4.55 Recall that, in Convex Optimization, an optimal solution generally comes from a
convex set of optimal solutions that can be large.