For $a,b,c\in\mathbb{R}$ with $a>0,b>0,c>0$ fixed and $\xi_{1},\xi_{2},\ldots,\xi_{N},\mu \in\mathbb{R}$ also fixed, I need to solve the following optimization problem:
$$ \left\{\begin{array}{lll} {\displaystyle \inf_{\beta\in\mathbb{R},\:s\in\mathbb{R}^{N},\: r\in\mathbb{R}^{N}}} & {a+\frac{1}{N}\sum_{i=1}^{N}s_{i} } & \\ \mbox{subject to} & {\displaystyle b\beta^{2}+ c(\mu-\xi_{i})(\beta+\mu-\xi_{i})+r_{i}= s_{i} } & \forall i\leq N \\ & r_{i} \geq 0 & \forall i\leq N \end{array}\right. \tag{$\bigstar$}$$
Remark: I do not want to use solvers, my idea is to use Lagrange multipliers to determine exactly the optimal value.
My attempt: Note that ($\bigstar$) is a problem with Equality and Inequality Constraints, so everything in this link works for ($\bigstar$) (That's what I think). Therefore, following the steps outlined in that link we have that the Lagrangian is:
$$\mathcal{L}(\beta,s,r,\lambda,\alpha)=a+\frac{1}{N}\sum_{i=1}^{N}s_{i}+\sum_{i=1}^{N}\lambda_{i}(s_{i}-b\beta^{2}- c(\mu-\xi_{i})(\beta+\mu-\xi_{i})-r_{i})+\sum_{i=1}^{N}\alpha_{i}r_{i}.$$
According to the link to solve ($\bigstar$) I must solve the following system: $$ \left\{\begin{array}{rclr} \sum_{i=1}^{N}\lambda_{i}\left(-2b\beta-c(\mu-\xi_{i}) \right) &=& 0 \\ \frac{1}{N}+\lambda_{i}&=&0 & \forall i\leq N \\ -\lambda_{i}+\alpha_{i}&=&0 & \forall i\leq N\\ \alpha_{i}r_{i} &=&0 & \forall i\leq N\\ \alpha_{i}&\geq& 0 & \forall i\leq N\\ b\beta^{2}+ c(\mu-\xi_{i})(\beta+\mu-\xi_{i})+r_{i}-s_{i} &=&0 & \forall i\leq N\\ r_{i}&\geq& 0 & \forall i\leq N\\ \end{array}\right. $$
But this system has problems, the second and third equations are contradictory with $\alpha_{i}\geq 0$. I do not know what I am doing wrong, I would appreciate your help.