Presentation Title: Primal-dual interior-point methods with asymmetric barriers

Abstract:
Very often, in the standard optimization problem with general cone constraints, the natural self-concordant barriers for primal and dual cones are not self-conjugate. Geometric programming and the power cones provide us with the most important examples. This primal-dual asymmetry destroys the good properties of the primal-dual barrier function. As a result, the standard machinery of the long-step infeasible-start primal-dual methods does not work.
In this talk, we show that even in the asymmetric case, the interior-point methods remain a powerful computational tool. We discuss several potential-reduction and path following primal-dual schemes. We show that some of them can be implemented in a matrix-free way. This opens a possibility for a direct competition of these polynomial-time methods with recently revived fast gradient schemes, which are applicable to very large optimization problems.