  In 2020, Yamakawa and Okuno proposed a stabilized sequential quadratic
semidefinite programming (SQSDP) method for solving, in particular, degenerate
nonlinear semidefinite optimization problems. The algorithm is shown to
converge globally without a constraint qualification, and it has some nice
properties, including the feasible subproblems, and their possible inexact
computations. In particular, the convergence was established for
approximate-Karush-Kuhn-Tucker (AKKT) and trace-AKKT conditions, which are two
sequential optimality conditions for the nonlinear conic contexts. However,
recently, complementarity-AKKT (CAKKT) conditions were also consider, as an
alternative to the previous mentioned ones, that is more practical. Since few
methods are shown to converge to CAKKT points, at least in conic optimization,
and to complete the study associated to the SQSDP, here we propose a revised
version of the method, maintaining the good properties. We modify the previous
algorithm, prove the global convergence in the sense of CAKKT, and show some
preliminary numerical experiments.
