New answers tagged logical-constraints
6
votes
Accepted
Is it possible to do a linearization without introducing new variables?
You want to enforce
$$\bigvee_m x_{i,j}^{m,r} \implies w_j^r = w_i^r + \sum_{m} y_j^{m,r} - \sum_{m} z_j^{m,r} \quad \text{for all $r,i,j$}$$
Rewriting in conjunctive normal form leads to linear big-M ...
- 29.8k
3
votes
Accepted
Formulation of binary constraint with the least binary variables for linear programming
You want to impose the following two logical constraints:
$$
\begin{align}
\delta_{t-1} = 1 \wedge \delta_t = 0 &\implies \beta_t = 1 \tag{1} \\
\neg (\delta_{t-1} = 1 \wedge \delta_t = 0) &\...
- 1,477
2
votes
Linearizing if else conditions in ILP
Besides @RobPratt's answer, the first condition would be (for simplicity I omitted indices $i$ and $j$ and continued with only two $y$ variables:
$$ x \implies (y_1 \oplus y_2) $$
$$ \lnot x \lor (y_1 ...
- 8,187
2
votes
Accepted
Linearizing if else conditions in ILP
Your first constraint enforces more than was asked. When $X_{ij}=0$, it forces $\sum_k Y_{jk}=0$, hence $Y_{jk}=0$ for all $k$. To enforce only $$X_{ij}=1 \implies \sum_k Y_{jk}=1,$$ you can instead ...
- 29.8k
2
votes
Accepted
Activating a sequence of the binary variables in a multi-dimensional array
Introduce an ordering $\text{ord}: I \to \{1,\dots,|I|\}$ and impose
$$x_i \le x_j$$
for all $i\in I$ and $j\in I$ such that $\text{ord}_i + 1 = \text{ord}_j$.
An alternative, less efficient, approach ...
- 29.8k
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