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 $\sum_{i}^{n}\frac{1}{b_i} = K$?

When $A\ne0$ and $n\ge2$, no minimum exists, because $AB^T$ is unbounded below. More specifically, suppose $a_1\ne0$. Let $b_1=-ca_1$ and $b_2=b_3=\ldots=b_n=(n-1)/(K+\frac1{ca_1})$ for some nonnegative $c$ that is greater than some large positive number (so that $K+\frac1{ca_1}\ne0$). Then $\sum_i\frac1b_i=K$ and $b_2,\ldots,b_n$ are bounded. Hence $AB^T$ is equal to $-c|a_1|^2$ plus a bounded quantity, and it approaches $-\infty$ when $c\to+\infty$.
