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We have the optimization problem $$\begin{align} \min_{[a_1,a_2,\ldots,a_N]}&\prod_{i=1}^{N}\left[1-{A\over\prod_{n=1}^{R}\left(\lambda_n+{\sum_{u\ne i}|C_i\cdot h_u|^2a_u+B\over a_{i}}\right) }\right]\end{align}$$ and constraints $$P_\max>a_1>a_2>\cdots>a_N>0,$$ and $$\gamma_{i}={|C_i\cdot h_i|^2a_{i}\over\sum_{u\ne i}|C_i\cdot h_u|^2a_u+D}\ge\Gamma_0,~\forall i\in\mathcal{U}$$
description:
- $A,B,D,\Gamma_0,P_\max$ and $\lambda_n$'s are all constants independent to users.
- $C_i$'s and $h_u$'s are constants dependent on each user $i$, and $|C_i\cdot h_u|^2$ is products of $C_i$ and $h_u$.
- $a_i$'s are optimization variables, it has only two constraints.
- In a single term of object function, for example $$\left[1-{A\over\prod\limits_{n=1}^{R}\left(\lambda_n+{\sum_{u\ne 1}|C_1\cdot h_u|^2a_u+B\over a_{1}}\right) }\right],$$it consists of variable $a_1$ but also other variables, so does in other terms of object. This makes the problem to be highly coupling.
How to solve this problem? Can you give me some tips or tutorials?
Thanks a lot!