A Survey of the Trust Region Subproblem Within a Semidefinite 
Programming Framework
    
      Henry Wolkowicz
    Department of Combinatorics and Optimization
    University of Waterloo 
      hwolkowicz@uwaterloo.ca
          
Key words:
 Trust Regions, Semidefinite programming, Duality, Unconstrained
Minimization.

ABSTRACT:
The trust region subproblem  (the minimization of a quadratic
objective subject to one quadratic constraint and denoted TRS) 
has many applications in diverse areas, e.g. function minimization,
sequential quadratic programming,
regularization, ridge regression, and discrete optimization.
In particular, it determines the step in
trust region algorithms for function minimization. Trust region algorithms
are popular for their strong convergence properties. However, a
drawback has been the inability to exploit sparsity as well as the 
difficulty in dealing with the so-called hard case. 
These concerns have been addressed by recent advances in the theory and
algorithmic development.

In this talk we provide a survey of recent advances for TRS. 
We emphasize large scale problems
and robustness.  This is done using semidefinite
programming (SDP) and the modern primal-dual
approaches as a unifying framework.