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In this paper, we propose a new sequential quadratic semidefinite programming(SQSDP) method for solving degenerate nonlinear semidefinite programs (NSDPs),in which we produce iteration points by solving a sequence of stabilizedquadratic semidefinite programming (QSDP) subproblems, which we derive from theminimax problem associated with the NSDP. Unlike the existing SQSDP methods,the proposed one allows us to solve those QSDP subproblems inexactly, and eachQSDP is feasible. One more remarkable point of the proposed method is thatconstraint qualifications (CQs) or boundedness of Lagrange multiplier sequencesare not required in the global convergence analysis. Specifically, withoutassuming such conditions, we prove the global convergence to a point satisfyingany of the following: the stationary conditions for the feasibility problem,the approximate-Karush-Kuhn-Tucker (AKKT) conditions, and the trace-AKKTconditions. Finally, we conduct some numerical experiments to examine theefficiency of the proposed method.

A stabilized sequential quadratic semidefinite programming method for degenerate nonlinear semidefinite programs