Understanding KKT Conditions for Minimization Problems with Constraints

[hoffmann]

what does it mean to write out the kkt conditions and find x* for the following problem:

minimize $$f(x) = \sum x_i$$ subject to $$\prod x_i = 1$$ and $$x_i \geq 0$$ for 1<= i <= n. the bounds on the sum and product are from i = 1 to n.

 

[HallsofIvy]

Well, what is the "kkt" (Karush–Kuhn–Tucker) theorem?

 

[hoffmann]

basically the kkt conditions need to be satisfied if the solution is optimal. you have the two constraints as your functions (say g and h) -- both these and the objective function need to be stationary, dual and primal feasible, and satisfy complementary slackness.

anyway, so i think there are two cases for the product to be equal to one: one is when all the x_i are equal to 1 and the other is when the product of the x_i's somehow approaches 1. in the first case, the sum would just give n since all the x_i's equal 1, and the second case...well I'm not so sure.

am i thinking about this problem in the right way?