Nonsingularity of FB system and constraint nondegeneracy in semidefinite programming (Q1938082)

For a primal linear programming problem \(\min <c,x> s.t. A(x)=b\), \(x \in K\), with dual \(\max <b,y> s.t. z=c-A'(y) \in K\) in the cone \(K\) of nonnegative quadratic forms, also called semidefinite cone, the first-order optimality conditions are \(A(x)=b\), \(A'(y)+z=c\), \(x \in K\), \(z \in K\), \(<x,z> = 0\), where the latter constraint is quadratic. This constraint can be equivalently replaced by a semidefinite complementarity function \(\Phi\) such that \(\Phi(x,z) = 0\) if and only if \(x \in K\), \(z \in K\), \(<x,z> = 0\). The paper investigates the regularity of a specific complementarity function, using techniques of nonsmooth analysis, arguing that a suitable smoothing Newton method could ensure local quadratic convergence without the classical strict complementarity condition. Numerical experiments are reported, but they are not reproducible.