Necessity of Hessian Matrix

Question

Hessian Matrix helps determine the saddle points, and the local extremum of a function.

Source: https://machinelearningmastery.com/a-gentle-introduction-to-hessian-matrices/

Hessian Matrix is used in Newton methods to optimize functions. Of what use is a Hessian Matrix in optimization when it cannot give global minima (which is what we look for when optimizing functions)?

Answer 1

Some algorithms (1) only require function evaluations, (2) only functions+gradients, or (3) both functions, gradients and second derivatives. Algorithms using second derivatives have potentially faster convergence.

Global solvers work very differently from local solvers. Many global solvers need to know the functional form of the objective and constraints.