Let $\xi_{1},\xi_{2},\ldots,\xi_{N}\in\mathbb{R}^{m}$, $\mu,\varepsilon\in\mathbb{R}$ with $\varepsilon>0$ (These are not variables, they are constant). We consider the following optimization problem:
$$ \left\{\begin{array}{lll} {\displaystyle \inf_{x\in\mathbb{R}^{m},\: \beta,\lambda\in\mathbb{R}}} & {\displaystyle \lambda\varepsilon +\frac{\beta^{2}}{4(\lambda-1)}+\frac{\beta\lambda}{\lambda-1}\left(\frac{1}{N}\sum_{i=1}^{N}(\mu-\left\langle \xi_{i},x \right\rangle)\right) + \frac{\lambda}{\lambda-1}\left(\frac{1}{N}\sum_{i=1}^{N}(\mu-\left\langle\xi_{i},x\right\rangle)^{2}\right) } & \\ \mbox{subject to} & \lambda \geq 1& \\ & \left\langle e,x \right\rangle=1 & \\ & x_{i}\geq 0 \quad \forall i\leq N& \end{array}\right. \tag{$\bigstar$}$$ where $e=(1,1,\ldots,1)\in\mathbb{R}^{m}$ and $\langle\cdot,\cdot\rangle$ is the dot product.
The question: What solver can I use to solve ($\bigstar$)? Please, I would appreciate if your answer includes the way in which this problem is entered in the solver that you propose, for me this is the most important part of this post.
Remark: I do not mean symbolic calculation, I need a solver that when I give values to $\xi_{1},\xi_{2},\ldots,\xi_{N}$, $\mu$ and $\varepsilon$ this will solve the problem ($\bigstar$).
I try to solve ($\bigstar$) in CVX, but this solver does not work on this problem since it involves multiplication between the variables of the problem, that is a problem in this solver.
I also tried a solver called Gloptipoly3, this did not work either.
Now, if someone is able to solve ($\bigstar$) in an analytical way, their solution would be very appreciated although I will settle for you suggesting a solver and the way to enter the problem.
You can use interior point methods to find a local minimum. If you're using Matlab, you can use fmincon.