Second-order necessary points of the nonsymmetric squared-variable reformulation (SSV or DSS) correspond exactly to weak second-order necessary points of the original semidefinite program (NSDP or BC), without constraint qualifications or strict complementarity.
Local minima and convergence in low-rank semidefinite programming
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On Squared-Variable Formulations for Nonlinear Semidefinite programming
Second-order necessary points of the nonsymmetric squared-variable reformulation (SSV or DSS) correspond exactly to weak second-order necessary points of the original semidefinite program (NSDP or BC), without constraint qualifications or strict complementarity.