The talk starts with a general introduction of the convex
quadratic semidefinite programming problem (QSDP), followed by a number of
interesting examples arising from finance, statistics and computer sciences.
We then discuss the concept of primal nondegeneracy for QSDP and show that
some QSDPs are nondegenerate and others are not even under the linear
independence assumption. The talk then moves on to the algorithmic side by
introducing the dual approach and how it naturally leads to Newton's method,
which is quadratically convergent for degenerate problems. On the
implementation side of the Newton method, we stress that direct methods for
the linear equations in Newton's method are impossible simply because the
equations are quite large in size and dense. Our numerical experiments use
the conjugate gradient method, which works quite well for the nearest
correlation matrix problem. We also discuss difficulties for us to find
appropriate preconditioners for the linear system encountered. The talk
concludes in discussing some other available methods and some future topics.