# Convexify this optimization problem with one nonlinear (bilinear) constraint

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?

• Your constraint on $\rho$ is wrong; it should be $R_\min^2 \vec{1} \preceq \rho \preceq R_\max^2\vec{1}$. You're on the right track though. Replace $0\preceq \theta \preceq \vecone$ with $0\preceq f \preceq \rho$. Then you can get rid of the constraints $f_i=\theta\rho_i$ altogether, and recover $\theta$ later. – Michael Grant Feb 12 '18 at 6:26
• Thank you! That makes sense. – space_voyager Feb 12 '18 at 6:48