How to integrate an indicator function/constraint into the cost function of a linear program?

Question

I have a mathematical model $P$ for which I optimize two cost functions say $F_1$ and $F_2$ subject to a set of constraints $C1$–$C10$.

In $F_2$, I want it to be included only when its expression violates a certain quantity. In fact, $F_2$ involves the product of two decision variables to be specific and its quantity should be multiplied by a penalty parameter say $\sigma$ only when it is greater than another quantity say $\alpha_i^{k,l}$.

$$F_2=\sigma \sum_{\mathcal{R}_i \in \mathcal{R}}\sum_{p^j_{(s^j, d^j)} \in \mathcal{P}} \sum_{(v^i_k, v^i_l) \in \mathcal{L}^i} x^{i, k}_{s^j}\times x^{i, l}_{d^j} d^j_{(s^j, d^j)}.$$

$x$ is a binary decision variable, $d$ is a constant.

How do I model this and what should be the constraints I need to add into the set of existing constraints?

Answer 1

I will simplify the notation to illustrate the idea. You want to minimize $$\sigma \max\left(\sum_{i,j} d_{ij} x_i x_j - \alpha, 0\right).$$ Introduce binary decision variable $y_{ij}$ to represent the product $x_i x_j$, and introduce nonnegative decision variable $z$ to represent the $\max$. Now minimize $\sigma z$ subject to linear constraints \begin{align} y_{ij} &\ge x_i + x_j - 1 \\ z &\ge \sum_{i,j} d_{ij} y_{ij} - \alpha \end{align}