Wikipedia suggests that:
"Methods based on real algebraic geometry" are some of the "most successful general strategies" for solving global optimization problems.
Could someone suggest an reference for learning about how algebraic geometry can be used to solve optimization problems? Preferably one that doesn't assume advanced knowledge beyond that of ye average graduate with a Math degree.
Of course, if someone would like to add a short explanation of their own as to why and how this method is used, they are more than welcome.
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.