This is a very good question and can be answered more concretely if we decide to restrict ourselves to certain types of problems.
Suppose we have a system that we want to maximize or minimize that is subject to certain equality and inequality constraints and that the systems in question are all polynomial.
We can then find a global optimal solution using the Karush-Kuhn-Tucker criteria (or alternatively the Fritz-John conditions).
Consider the following contrived example.
Suppose we want to maximize $x_1^2 + 2x_1x_2 + x_2x_3 + 2x_1 - 4$ subject to points on the unit sphere, i.e. $x_1^2 + x_2^2 + x_3^2 - 1$. This actually reduces down to using LaGrange multipliers.
We then consider the zero locus (the set of common zeroes, also known as the variety) of the following system of equations:
$f_1(x_1,x_2,x_3,\mu_1) = 2x_1 + 2x_2 + 2 + 2\mu_1x_1 = 0$
$f_2(x_1,x_2,x_3,\mu_1) = 2x_1 + x_3 + 2\mu_1x_2 = 0$
$f_3(x_1,x_2,x_3,\mu_1) = x_2 + 2\mu_1x_3 = 0$
$f_4(x_1,x_2,x_3,\mu_1) = x_1^2 + x_2^2 + x_3^2 - 1 = 0$
Solving this system with any software package that does homotopy continuation (possibly HOMPACK or Phcpack) gives us two solutions that are approximately
$(-.7179056815169287, .6495969519526578, -.2502703187745829, 1.29779063520861)$
and
$(.9205188417273060, .3831260888917872, .07654712297337447, -2.502550546707357)$.
(with the last coordinate representing the lagrange multiplier)
Plugging these into the equation we want to maximize gives us $-6.015696316725544$ for the first solution and $-0.576930611565337$ for the second.
As such, $(.9205188417273060, .3831260888917872, .07654712297337447)$ is the approximate point on the sphere that maximizes our potential function.
