Could someone give a solution to this problem as well as an interpretation of the results? I am not sure how to deal with inequality constraints.
Part 1:
Find the minimum value of $f(x) = |x|^2$
Subject to the constraints $ x^i \geq 0 \quad (i=1,2, ..., n) $
and $x^1 + x^2 + ... + x^n =1$
Part2:
Interpret the result geometrically.
For a 2D vector, the vector with minimum distance from origin is the one that is obtained by intersection of the line x + y = 1 and x = y (1/2, 1/2).
Same can be visualized for the triangle formed by 3D plane x + y + z = 1.
I am not sure but for n dimensional case it appears to be (1/n, 1/n, ... 1/n) and hence the value of f(x) = 1/n.