"Hybrid Methods for Minimizing Least Distance Functions
with Semi-Definite Matrix Constraints"

Suliman Al-Homidan

ABSTRACT
 Hybrid methods for minimizing least distance functions
with semi-definite matrix constraints are considered.
One approach is to formulate the problem
as a constrained  least distance problem in which the
 constraint is the intersection of three convex sets.
The Dykstra-Han  projection algorithm can then be used to solve the problem.
This method is globally  convergent but the  rate of convergence is slow.
However, the method does have the capability
of determining the correct rank of the solution matrix,
and this can be done in relatively  few iterations.
If the correct rank of the solution matrix is known, it is shown how to
formulate the problem as a smooth nonlinear  minimization problem,
for which a rapid  convergence  can be  obtained
 by $l_1$SQP method.
Also this paper studies hybrid  methods that attempt to  combine the best
features of both types of method.
An important  feature  concerns  the interfacing
of the component methods.
Thus, it has to be decided  which method  to use  first,
and when  to switch between methods.  Difficulties such as these
are addressed in the paper. Comparative numerical results  are also
reported.