I have a practical question about quasi-Newton methods.
In quasi-Newton methods, Hessian matrix is approximated. It seems to be impossible for them to check the second optimality condition. In practice, do those common public domain optimization packages using quasi-Newton methods, like MATLAB fminunc function with BFGS, check the second order optimality condition? If yes, what kind of techniques do they use to compute the hessian matrix? finite difference?
Or there is little chance optimization will terminate at saddle points and no need to check that.