"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.