Weak notions of nondegeneracy in nonlinear semidefinite programming

The constraint nondegeneracy condition is one of the most relevant and useful qualifications in nonlinear semidefinite programming. It can be characterized terms any fixed orthonormal basis the, let us say, $$\ell $$ -dimensional kernel matrix, by linear independence a set (\ell +1)/2$$ derivative vectors. We show that this requirement equivalently formulated smaller set, vectors, considering all bases instead. This allows to identify not are for qualification defined, giving rise strictly weaker variant related global convergence an external penalty method. use some these ideas revisit approach Forsgren (Math Program 88, 105–128, 2000) exploiting sparsity structure transformation constraints define qualification, which led develop another relaxed notion using simpler transformation. If zeros derivatives function at given point considered, instead itself neighborhood point, we obtain even connects Forsgren’s ours.