I have the following nonlinear optimization problem:
$$ \begin{aligned} & \underset{R,\theta,f,s}{\text{minimize}} && \sum_{i=1}^m L_iR_i^2 \\ & \text{subject to} && R_\min\mathbf{1}\preceq R\preceq R_\max\mathbf{1} \\ &&& Af = \begin{bmatrix}-s \\ c\end{bmatrix} \\ &&& 0\preceq\theta\preceq\mathbf{1} \\ &&& 0\preceq s\preceq S_\max\mathbf{1} \\ &&& f_i = \theta_i R_i^2\quad i=1,...,m % In #3, begin each extra line with &&& for correct alignment \end{aligned}% $$
where $R\in\mathbb R^m$, $f\in\mathbb R^m$, $s\in\mathbb R^k$, $c\in\mathbb R^{n-k}_+$. Apart from the optimization variables $R$, $\theta$, $f$ and $s$, the rest are given constant parameters.
This problem must be solved using convex optimization methods. My problem is the nonlinear constraint $f_i=\theta_iR_i^2$. An easy thing to do straight away is to define $\rho_i:=R_i^2$, which gives:
$$ \begin{aligned} & \underset{R,\theta,f,s}{\text{minimize}} && \sum_{i=1}^m L_i\rho_i \\ & \text{subject to} && R_\min^2\mathbf{1}\preceq \rho\preceq R_\max^2\mathbf{1} \\ &&& Af = \begin{bmatrix}-s \\ c\end{bmatrix} \\ &&& 0\preceq\theta\preceq\mathbf{1} \\ &&& 0\preceq s\preceq S_\max\mathbf{1} \\ &&& f_i = \theta_i \rho_i\quad i=1,...,m % In #3, begin each extra line with &&& for correct alignment \end{aligned}% $$
However the bilinear constraint $f_i=\theta_i\rho_i$ remains. How can one get rid of it?