I want to solve the following LP: \begin{array}{ll} \underset{x}{\min} &Ax+b\\ &s.t \quad C_ix\leq0 \quad \text{for} \quad i=1...N \end{array}
I know that not all of the constraints could be feasible. So is it a good idea if I rewrite it like this with slack variables $\alpha_i$? \begin{array}{ll} \underset{x,\alpha}{\min} &Ax+b+\lambda\sum{\alpha_i}\\ &s.t \quad C_ix\leq \alpha_i \quad \quad,\alpha_i \geq 0 \quad ,\quad \text{for} \quad i=1...N \end{array}
But then i need to also tune $\lambda$. Is there any better way to solve the above LP without needing to use a hyperparameter like $\lambda$?