
How to minimize a sum
Posted:
Oct 10, 2013 10:15 PM


If v_i is (sigma_ i)^ 2, the global minimum of sum v_i w_i ^2 occurs when w_i = c/v_i , and choose c to make sum v_i = 1: that is, c = 1/sum (1/v). This follows easily by writing the problem as minimize sum v_i w_i ^ 2, subject to sum w _i = 1, and solving via the Lagrange multiplier method. (Note that for F >= 0, minimizing sqrt(F) is the same as minimizing F itselfthat is, the optimal w_i are the same.)
On the other hand, if you want to MAXIMIZE, you cannot use calculus methods: the solution is to put w = 1 on the largest v.

