How to solve an optimization problem with a sum constraint?

Question

I am currently working on an optimization problem that can be mathematically represented as:

\begin{aligned} \min_{\{f_{k}\}} & \sum_{k=1}^{K} \frac{q_{k}}{f_{k}} \\ \text{s.t.} & \sum_{k=1}^{K} f_{k} = F \end{aligned} In this formulation, $\{f_{k}\}$ are positive decision variables that I need to optimize, $\{q_{k}\}$ are known positive constants, $F$ is a positive integer, and $K$ is the number of variables.

I am looking for guidance on how to approach solving this optimization problem. Specifically, I would like to know:

What type of optimization problem is this, and what methods can be used to solve it? Are there any analytical solutions available for this type of problem? How can I handle the sum constraint effectively in the optimization process? Any insights, suggestions, or references to relevant literature would be greatly appreciated. Thank you in advance for your help!

Answer 1

I'm assuming that the $f_k$ are positive. This can be done in an elementary way using the Cauchy-Schwarz inequality, which gives

$$\left( \sum \frac{q_k}{f_k} \right) \left( \sum f_k \right) \ge \left( \sum \sqrt{q_k} \right)^2.$$

This gives that the minimum is $\boxed{ \frac{(\sum \sqrt{q_k})^2}{F} }$ and occurs (using the known equality case of Cauchy-Schwarz) exactly when $\frac{q_k}{f_k}$ is proportional to $f_k$, or equivalently when $q_k$ is proportional to $f_k^2$, which gives $f_k = c \sqrt{q_k}$ for some suitable constant $c$ (which we can compute is $\frac{F}{\sum \sqrt{q_k}}$).

In general this sort of constrained optimization can be done, at least in principle, using Lagrange multipliers. You can check that you get the same answer this way.