Problem Description:
I have a mixed integer linear optimization problem (MILP) with objective function $$\min_{x_{jt}} \sum_\omega \sum_j \sum_t y_{jt}(\omega)$$
I know $L_{jt}\le x_{jt}\le U_{jt}$ with $L_{jt}\ge 0$ and $L_{jt}\le U_{jt}$. We consider $b_{jt}(\omega)\ge 0$ known and provided as input data.
We also know $x_{jt},y_{jt}\in \mathbb{R}^1$ and $0 \le y_{jt} \le y_{j,t-1} + b_{jt}$ for $t\ge 2$ with $0\le y_{j1} \le b_{j1}$ when $t=1$.
Question:
Is it possible to use a MILP reformulation technique to address equality constraints of the following form? I would like to linearize these if possible.
$$y_{jt}(\omega) = \max( b_{jt}(\omega)-x_{jt},0), \quad t=1,\forall j,\forall \omega$$
$$y_{jt}(\omega) = \max( b_{jt}(\omega)+y_{j,t-1}(\omega)-x_{jt},0), \quad t\ge2,\forall j,\forall \omega$$
Best Answer
Since you already know that $y$ is nonnegative, you just need to add the constraint $$y_{j1}(\omega) \ge b_{j1}(\omega)-x_{j1}\, \forall j,\forall \omega.$$ Since the objective function minimizes the sum of the $y$ variables, you can be sure that $y_{j1}(\omega)$ will be no larger than it needs to be.