How to construct my mixed integer programming problem with constraint of minimum consecutive ones

Question

My target is to formulate a binary sequence with fixed size $N$ = 10, such as $[1, 0, 0, 0 ,1, 1, 0, 1, 0, 0]$. However, I want to constrain this sequence so that when 1 appears, it has to appear at least three times; This is partial problem when I formulate my traffic signal optimization problem, in which 1/0 indicates green/red. I need to add such constraint to emulate minimum green time in the real world.

Answer 1

Define a two sets of binary variables : variables $x_i$ take value $1$ if and only if the $i^{th}$ term of the sequence equals $1$, and variable $y$ that takes value $1$ if and only if one of the terms equals $1$.

You want to enforce

$$ y=1 \quad \Longrightarrow \quad \sum_i{x_i} \ge 3 \\ x_i = 1 \quad \Longrightarrow \quad y=1 $$

You can do this with \begin{align*} 3 y &\le \sum_i{x_i} \\ x_i &\le y \quad \forall i \end{align*}

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It is not clear in your question if the ones have to be consecutive or not. If they do, then you need to forbid all the patterns that do not satisfy this. @ErwinKalvelagen shows us how to achieve this here:

One simple way to enforce a run length of at least three, is to forbid patterns 010 and 0110. This can be modeled as:

$$ -x_t + x_{t+1} - x_{t+2} \le 0 $$

and

$$ -x_t + x_{t+1} + x_{t+2} - x_{t+3} \le 1 $$

A little bit of thought is needed to decide what to do at the borders, especially the first time period.

A different approach is detailed here.