Local convergence of primal-dual interior point methods for nonlinear semi-definite optimization using the family of Monteiro-Tsuchiya directions

Abstract

The recent advance of algorithms for nonlinear semi-definite optimization problems, called NSDPs, is remarkable. Yamashita et al. first proposed a primal-dual interior point method (PDIPM) for solving NSDPs using the family of Monteiro-Zhang (MZ) search directions. Since then, various kinds of PDIPMs have been proposed for NSDPs, but, as far as we know, all of them are based on the MZ family. In this paper, we present a PDIPM equipped with the family of Monteiro-Tsuchiya (MT) directions, which were originally devised for solving linear semi-definite optimization problems as were the MZ family. We further prove local superlinear convergence to a Karush-Kuhn-Tucker point of the NSDP in the presence of certain general assumptions on scaling matrices, which are used in producing the MT scaling directions.

Interior-point methods for second-order stationary points of nonlinear semidefinite optimization problems using negative curvature

“…Yamakawa and Yamashita [65] developed a different PDIPM based on the shifted barrier KKT conditions for NSDPs. Okuno [49] analyzed local convergence of a PDIPM using the family of Monteiro-Tsuchiya directions. We re-emphasize that the existing NSDP algorithms including the above PDIPMs have neither convergence guarantees to an SOSP nor iteration complexities even for a KKT point.…”

“…The proposed algorithm gradually decreases the weight parameter ν in ψ µ,ν to zero, whereas it is fixed in many PDIPMs [30,49,65,67]. This device is required for technical reasons concerning the proofs.…”