I am facing a problem where I need to linearize the constraints to get an LP formulation.
The objective:
$ max_{x\in X, s_i \in R, \lambda \leq 0} \sum_{j=1}^N s_j + \epsilon\lambda $
s.t.
$ s_j + \lambda ||\xi_j - \xi_i||^2 \geq min_{l=1..L} \{a_l \xi_i^Tx + b_l\}$ $\forall i,j$
If you have any idea of how to linearize this constraint, please tell me. If the inequality was in the other way around, I could do it but now I'm stuck.
To model $y = \min z$ using a linear model, one simple model is to enumerate the possible cases, either $y = z_1$ or $y = z_2$ etc, and the constraint that $y$ must be smaller than all elements.
Introduce $n$ binary variables $d_i$, and the combinatorial model is
$$ \sum_{i=1}^n d_i = 1\\ d_1 \rightarrow y = z_1\\ d_2 \rightarrow y = z_2\\\vdots\\ d_n \rightarrow y = z_n\\ y \leq z $$
The implications are modelled using standard big-M strategy $-M(1-d_i) \leq y-z_i \leq M(1-d_i)$