I think this is probably obvious but I can not find a formal proof or any other reference on this online. My question is - If there exists a finite number of local minima/maxima of a non-convex function in the feasible region, and if we check each of them one by one and find the optimum minima/maxima (using SOSC?), can we call that global minima/maxima with sufficiency?
If YES, My follow up question on that is: I am trying to find an analytical solution to a problem. So is Lagrange multiplier/KKT a good way to go?
An example of objective function may be -> sin(x)*sin(y)
Let the feasible region be $\mathbb{R}$ and consider $f(x) = x(x-1)(x+1)$.
There are finitely many local optimal but the global optimum doesn't exists.