The trust region subproblem (the minimization of a quadraticobjective 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 is used to determine the step intrust region algorithms for function minimization. Trust region algorithmsare popular due to their strong convergence properties. However, one ofthe drawbacks 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 andalgorithmic development.

This talk provides an in depth study of TRS and its propertiesas well as a survey of recent advances. This is done using semidefiniteprogramming (SDP) and the modern primal-dualapproaches as a unifying framework.

The SDP framework solves TRS efficiently and robustly; and itshows that TRS is always a well-posedproblem, i.e. the optimal value and an optimum can be calculated to agiven tolerance. This is contrary to statements in the literature which label TRS ill-posed or degenerate, if the so-called hardcase holds. We provide both theoretical and empiricalevidence to illustrate the strength of the SDP and duality approach. In particular, this includes new insights into handling the hard case.