optimization of coefficients with constant sum of inverses Does anybody knows if there is an easy solution to the following problem:
Given $A = [a_1, a_2, ... a_n]$ and K, find B = $[b_1, b_2,...b_n]$ that minimizes $AB^T$ such that $\sum_{i}^{n}\frac{1}{b_i} = K$?

It turns out that if $a\neq 0$ and $n>1$, there is no solution to the problem in that it can have arbitrarily small values.
If $n=1$ the answer is given by the constraint so there is nothing to be done, so suppose $n>1$.
If $a=0$, any $b$ will do, so suppose $a \neq 0$.
So, suppose $a_1 \neq 0$. Then choose $x_2=\cdots = x_{n-1} = \beta$ such that
$K'=K-(n-1)\beta \neq 0$. Then an upper bound for the problem is
$\inf_{{1 \over x_1} + {1 \over x_n} = K'} (a_1 x_1+a_n x_n  )$.
Now choose $x_n = {1 \over K' - {1 \over x_1}} $ to get the upper bound
$\inf_{x \neq {1 \over K'} } (a_1 x + a_n {1 \over K' - {1 \over x}} )$. Now let $x \to (-\operatorname{sgn} a_1) \infty$, and notice that the problem is unbounded below.
