I have a list x(variable in MIP) of outcomes.
x = [1,-1,-1,-1,1,1,-1,1,1,1,1,-1,1,-1,-1,1]
1 represent a win and -1 represent a loss. In this case the max winning streak is 4 and max losing streak is 3, thus overall max streak is 4.
How can we formulate an MIP where the objective is to minimize maximum overall streak?
How should the constraints look like?
Let $z$ represent the maximum streak length. The problem is to minimize $z$ subject to \begin{align} z &\ge k\left(\sum_{j=t}^{t+k-1} x_j - k + 1\right) &&\text{for all $k,t$}\tag1\\ z &\ge k\left(-\sum_{j=t}^{t+k-1} x_j - k + 1\right) &&\text{for all $k,t$}\tag2 \end{align} Constraint $(1)$ forces $z\ge k$ if $x_t=\dots=x_{t+k-1}=1$. Constraint $(2)$ forces $z\ge k$ if $x_t=\dots=x_{t+k-1}=-1$.
For the other part, just take $x_j = a_j b_j$, where $a_j,b_j,x_j\in\{-1,1\}$ and the value of each $a_j$ is known.