Certificates for this sort of questions are discussed in my paper

H. Schichl and A. Neumaier, [Transposition theorems and qualification-free optimality conditions][1], Siam J. Optimization 17 (2006), 1035-1055.

They can be obtained numerically by [semidefinite programming][2]. 


  [1]: https://www.mat.univie.ac.at/~neum/ms/trans.pdf
  [2]: https://en.wikipedia.org/wiki/Semidefinite_programming