Please, can you help to solve the following convex optimization problem to get a closed-form solution?
I tried to solve it using Lagrange multiplier but it's not easy. I can't get the closed-form solution to the value of x and y.
\begin{equation}\label{MINPower} \min \; x + y \end{equation} subject to \begin{equation} \begin{split} C1 &: (\left | a\right | \sqrt{x} + \left | b \right | \sqrt{y} )^2 \geq c, \\ C2 &: 0 \leq x , y \leq c, \\ \end{split} \end{equation}
My trial
$x+y-\lambda_1 \left | a\right | ^2 x - \lambda_1 \left | b \right |^2 y - 2\lambda_1 \left | a\right | \left | b\right | \sqrt{xy} + \lambda_1 c - \lambda_2 x - \lambda_3 y + \lambda_4 x - \lambda_4 c + \lambda_5 y - \lambda_5 c= 0$
Differentiate with respect to $x$ -- > $ 1 + \lambda_1 \left | a\right | ^2 + \lambda_1 \left | a\right | \left | b\right | \sqrt{y} x^{-1/2} - \lambda_2 + \lambda_4 = 0$
Differentiate with respect to $y$ -- > $ 1 +\lambda_1 \left | b\right | ^2 + \lambda_1 \left | a\right | \left | b\right | \sqrt{x} y^{-1/2} - \lambda_3 + \lambda_5 = 0$
$-\lambda_1 \left | a\right | ^2 x - \lambda_1 \left | b \right |^2 y - 2\lambda_1 \left | a\right | \left | b\right | \sqrt{xy} + \lambda_1 c = 0$
$- \lambda_2 x = 0$
$- \lambda_3 y = 0$
$ \lambda_4 x - \lambda_4 c = 0$
$ \lambda_5 y - \lambda_5 c = 0$
Thanks for your help in advance.