Justification of the use of gradient method insolving an unconstrained multivariable function in optimization?

Answer 1

Otherwise known as the hill climbing method in numerical analysis? Worked well for me in a process modelling program.

Answer 2

The gradient method is not necesarily a hill-climbing method, it depends how it is implemented.

If the obejct function is differentiable, the local optima have zero gradient. This criteria works unless the function is unlimited an goes to infinity when you approach the boarders of the domain. So you need to check for that.

If you algorithm for finding roots in the gradient can find only one solution (like a hill-climbing algortihm), it works only if there is only one local optimum, which is (then) also the global optimum. If this is not guarenteed, you must try with many different start guesses to make sure you hit the global optimum.