joint work with Takayuki Okuno
The LP-Newton method solves the linear optimization problem (LP) by repeatedly projecting a current point onto a certain relevant polytope. In this talk, we extend the LP-Newton method to the second-order cone optimization problem (SOCP) via a linear semi-infinite optimization problem (LSIP) reformulation. In the extension, we produce a sequence by projection onto polyhedral cones constructed from LPs obtained by finitely relaxing the LSIP. We show the global convergence of the proposed algorithm under mild assumptions, and investigate investigate its efficiency through numerical experiments.
joint work with Nicolas Fuentealba
In this work we study nonconvex trust region problems with an additional quadratic constraint. For this class of problems we propose a family of relaxations, and study some key properties regarding convexity, exactness of the relaxation, sufficient condition for exactness, etc. For the case that the additional quadratic constraint is not convex , we generalize sufficient conditions for exactness of the relaxation proposed originally for trust region problems with an additional conic and discuss some examples for which trivial extensions of the sufficient conditions do not apply.
joint work with Lek-Heng Lim
We introduce a conic embedding condition that gives a hierarchy of cones and cone programs. This condition is satisfied by a large number of convex cones including the cone of copositive matrices, the cone of completely positive matrices, and all symmetric cones. We discuss properties of the intermediate cones and conic programs in the hierarchy. In particular, we demonstrate how this embedding condition gives rise to a family of cone programs that interpolates between LP, SOCP, and SDP.