An algorithm for nonsymmetric conic optimization inspired by MOSEK

Abstract

We analyse the scaling matrix, search direction, and neighbourhood used in MOSEK's algorithm for nonsymmetric conic optimization [J. Dahl and E.D. Andersen, A primal-dual interior-point algorithm for nonsymmetric exponential-cone optimization, preprint (2019)]. It is proven that these can be used to compute a near-optimal solution to the homogeneous self-dual model in polynomial time. This provides a theoretical foundation for MOSEK's nonsymmetric conic algorithm. The main steps in the analysis are sandwiching MOSEK's scaling matrix between the primal and dual barrier's Hessians, and using this information to carefully check all the neighbourhood conditions after a small, improving step is taken.