Binary variables in time series: integer linear programming

Question

I'm working on a problem and I can't seem to find an easy solution to it. It's about an optimization problem, concerning a time series.

I have a binary variable $\alpha_t$ for $t \in [0, 24[$. I also have an extra constraint, which states that $$\sum_{t=0}^{23} \alpha_t \geq 14.$$ The problem is that I want to add an extra constraint that if a certain $\alpha_t = 1$, then either $$\alpha_{t-1} = \alpha_{t+1} = 1$$ or $$\alpha_{t-1} = \alpha_{t-2} = 1$$ or $$\alpha_{t+1} = \alpha_{t+2} = 1, $$ i.e. at least 3 consecutive times $\alpha$ needs to be 1. It can be 4 times, it can be 5, but it has to be at least 3 times.

The most intuitive idea is probably this: $$\alpha_t = 1 \Rightarrow \alpha_t + \alpha_{t+1} + \alpha_{t + 2} = 3,$$ but from a certain $t$, this will result that all $\alpha_t = 1$.

I also tried big M constraints, but for larger consecutive times ( $\geq 3)$, this becomes almost impossible to write down/implement.

Answer 1

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.

Answer 2

I think I've got it:

use the reasoning in this post Integer linear programming constraint for maximum number of consecutive ones in a binary sequence. Here, we have to look at the $\alpha_t$ as zero's instead of ones. At this point, you can impose a maximum of consecutive zero's.

If the variable however has a value of one, then you can use big M constraints to set the sum of the next 3, equal to 3.

Answer 3

One method is to let $x_t$ denote the starting indices and $y_t$ denote the ending indices of the sequences of ones. For example, if $x=(0,1,0,0,0,1,0)$ and $y=(0,0,0,1,0,0,1)$, the sequence is $\alpha=(0,1,1,1,0,1,1)$. You get the following constraints:

1. number of starting indices equals number of ending indices: $$\sum_t x_t = \sum_t y_t$$

2. cannot end a sequence unless it was started at least 3 periods prior: $$y_i \leq \sum_{t=1}^{i-2}x_t-y_t \quad \forall i$$

3. cannot start a new sequence before the previous one is closed: $$x_i \leq 1- \sum_{t=1}^{i-1}(x_t-y_t) \quad \forall i$$

4. relating $\alpha$ to $x,y$: $$\alpha_i = \sum_{t=1}^{i}x_t - \sum_{t=1}^{i-1}y_t \quad \forall i$$

Answer 4

Reading your question I think that you want

$$\alpha_t = 1 \implies \alpha_{t+1} + \alpha_{t+2} = 2 \vee \alpha_{t-1} + \alpha_{t+1} = 2 \vee \alpha_{t-2} + \alpha_{t-1} = 2$$

not

$$\alpha_t = 1 \implies \alpha_t + \alpha_{t+1} + \alpha_{t + 2} = 3, ~ \forall t\in [0, n-2]$$

or, equivalently,

$$\alpha_t = 1 \implies \alpha_{t+1} + \alpha_{t + 2} = 2, ~ \forall t\in [0, n-2]$$

In this case, the answer should be

$$ \alpha_t \implies \alpha_{t+1} \wedge \alpha_{t + 2}, ~ \forall t\in [0, n-2]$$

$$ \neg\alpha_t \vee (\alpha_{t+1} \wedge \alpha_{t + 2}), ~ \forall t\in [0, n-2]$$

$$ (\neg\alpha_t \vee \alpha_{t+1}) \wedge (\neg\alpha_t \vee \alpha_{t + 2}), ~ \forall t\in [0, n-2]$$

rewriting this sentence in binary variables, the constraints are

$$ \begin{align} (1-\alpha_t) + \alpha_{t+1} \geq 1, ~ \forall t\in [0, n-2]\\ (1-\alpha_t) + \alpha_{t+2} \geq 1, ~ \forall t\in [0, n-2] \end{align} $$

Another case

OK, consider this logical sentence

$$\alpha_t \implies (\alpha_{t+1} \wedge \alpha_{t+2}) \vee (\alpha_{t-1} \wedge \alpha_{t+1}) \vee (\alpha_{t-2} \wedge \alpha_{t-1})$$

$$\neg\alpha_t \vee (\alpha_{t+1} \wedge \alpha_{t+2}) \vee (\alpha_{t-1} \wedge \alpha_{t+1}) \vee (\alpha_{t-2} \wedge \alpha_{t-1})$$

After some operations ...

$$(\neg\alpha_t \vee \alpha_{t-2} \vee \alpha_{t+1}) \wedge (\neg\alpha_t \vee \alpha_{t-1} \vee \alpha_{t+1}) \wedge (\neg\alpha_t \vee \alpha_{t-1} \vee \alpha_{t+2})$$

the constraints for $t\in [2, n-2]$ are

$$ \begin{align} (1-\alpha_t) + \alpha_{t-2} + \alpha_{t+1} \geq 1 \\ (1-\alpha_t) + \alpha_{t-1} + \alpha_{t+1} \geq 1 \\ (1-\alpha_t) + \alpha_{t-1} + \alpha_{t+2} \geq 1 \end{align} $$

You need to fix the cases $t=0, t=1, t=n-1, t=n$ using the same idea. For $t\in\{0,n\}$ you can use the first set of equations presented in this text.

$$ \begin{align} (1-\alpha_0) + \alpha_{1} \geq 1 \\ (1-\alpha_0) + \alpha_{2} \geq 1 \\ (1-\alpha_n) + \alpha_{n-1} \geq 1 \\ (1-\alpha_n) + \alpha_{n-2} \geq 1 \end{align} $$

For $t\in\{1,n-1\}$

$$\alpha_1 \implies (\alpha_{2} \wedge \alpha_{3}) \vee (\alpha_{0} \wedge \alpha_{2})$$

$$\neg\alpha_1 \vee (\alpha_{2} \wedge \alpha_{3}) \vee (\alpha_{0} \wedge \alpha_{2}) $$

$$(\neg\alpha_1 \vee \alpha_{0} \vee \alpha_{3}) \wedge (\neg\alpha_{1} \wedge \alpha_{2}) $$

and

$$\alpha_{n-1} \implies (\alpha_{n-2} \wedge \alpha_{n-3}) \vee (\alpha_{n} \wedge \alpha_{n-2})$$

$$\neg\alpha_{n-1} \vee (\alpha_{n-2} \wedge \alpha_{n-3}) \vee (\alpha_{n} \wedge \alpha_{n-2}) $$

$$(\neg\alpha_{n-1} \vee \alpha_{n} \vee \alpha_{n-3}) \wedge (\neg\alpha_{n-1} \wedge \alpha_{n-2}) $$

resulting in these constraints

$$ \begin{align} (1-\alpha_1) + \alpha_{0} + \alpha_{3}\geq 1 \\ (1-\alpha_1) + \alpha_{2} \geq 1 \\ (1-\alpha_{n-1}) + \alpha_{n} +\alpha_{n-3} \geq 1 \\ (1-\alpha_{n-1}) + \alpha_{n-2} \geq 1 \end{align} $$

finally

$$ \left\{\begin{align} & (1-\alpha_0) + \alpha_{1} \geq 1 & \\ & (1-\alpha_0) + \alpha_{2} \geq 1 & \\ & (1-\alpha_1) + \alpha_{0} + \alpha_{3}\geq 1 & \\ & (1-\alpha_1) + \alpha_{2} \geq 1 & \\ & (1-\alpha_t) + \alpha_{t-2} + \alpha_{t+1} \geq 1, & \forall t\in [2,n-2] \\ & (1-\alpha_t) + \alpha_{t-1} + \alpha_{t+1} \geq 1, & \forall t\in [2,n-2] \\ & (1-\alpha_t) + \alpha_{t-1} + \alpha_{t+2} \geq 1, & \forall t\in [2,n-2] \\ & (1-\alpha_{n-1}) + \alpha_{n} +\alpha_{n-3} \geq 1 & \\ & (1-\alpha_{n-1}) + \alpha_{n-2} \geq 1 & \\ & (1-\alpha_n) + \alpha_{n-1} \geq 1 & \\ & (1-\alpha_n) + \alpha_{n-2} \geq 1 & \end{align}\right. $$

These constraints cover all cases correctly. There is no counterexample.