# Solve a Convex optimization Problem which Involves Non Linear Constraints (Using $\log \left( \cdot \right)$ Function)

I want to solve the following optimization problem

$\text{min.}_{x,y}\ \sum_{i=1}^{N} x_i+\sum_{i=1}^{N} y_i$
subject to
$\sum_{i=1}^{N}\log_2(1+a_{i}x_{i}) \geq A$
$\sum_{i=1}^{N} a_{i}y_{i}\geq B$
$x_{i}+y_{i}\leq c_{i}$
$x_{i} \geq0$
$y_{i}\geq 0$
where $(A,B,c_i\text{and}\ a_i) \in \mathbb{R}^{+}$. Knowing that $a_1\geq a_{2} \geq \ldots,\geq a_N$ and assume that $c_1=c_2=\ldots,=c_N$. Note that $a_{i}$ are constants.

The given optimization problem is convex because the objectiveis linear and all the constraints are convex (except the log-function which is concave). The problem I think can be solve using the Lagrange multipliers so the problem will be as follows

$\text{min}_{x_i,y_i,\lambda\geq0,\mu\geq0,\mathbf{\nu}\geq0,\mathbf{\alpha}\geq0,\boldsymbol{\beta}\geq0}~ ~ \mathcal{L}$

where $\mathcal{L}~$ is
$\mathcal{L}=\sum_{i=1}^{N}x_i+\sum_{i=1}^{N}y_i+\lambda(A-\sum_{i=1}^{N}\log_2(1+a_{i}x_{i}))+\mu(B-\sum_{i=1}^{N} a_{i}y_{i})+\nu(x_i+y_i-c_i)-\sum_{i}^{N}\alpha_{i}x_i-\sum_{i}^{N}\beta_{i}y_i$.
What should I do next? do I need to find $\frac{d(\mathcal{L})}{d(x_i)}=0~$ and $\frac{d(\mathcal{L})}{d(y_i)}=0$, If this is the answer, how can I calculate each of $x_i$ and $y_i$?

Any help is highly appreciated.

Thanks

Since it is probable you won't be able to find a closed form solution for the KKT you will have to employ a numerical method.

Those which seems to fit this problem are:

• Agreed, this is going to require a numerical solution. – Michael Grant Mar 26 '18 at 15:36
• Thank you both for your comments. This means the problem has no analytical solution, right? – user841315 Mar 26 '18 at 16:10
• Only by this comment I can see you're optimistic guy :-). – Royi Mar 26 '18 at 16:30
• @user841315 yes, that's what we are saying. – Michael Grant Mar 26 '18 at 20:38