Questions tagged [logical-constraints]
For questions about constraints that can be expressed in (usually propositional) logic.
22
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In an integer program, how I can force a binary variable to equal 1 if some condition holds?
Suppose we have a binary or continuous variable $x$, a binary variable $y$, and a constant $b$, and we want to enforce a relationship like
If $x \gtreqless b$, then $y = 1$.
How can we write this ...
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12
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2
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In an integer program, how can I “activate” a constraint only if a decision variable has a certain value?
Suppose we have the constraint
$$a_1x_1 + \cdots + a_nx_n \gtreqless b,$$
where $a_i$ and $b$ are constants and $x_i$ are decision variables. Suppose also that we want the constraint to hold if $y=1$ (...
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4
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Single reference for Mixed Integer Programming formulations to linearize, handle logical constraints and disjunctive constraints, do Big M, etc?
Is there a single crisp and accessible reference which covers how to generate Mixed Integer Programming formulations to linearize products, handle logical constraints and disjunctive constraints, do ...
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3
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What is the difference between integer programming and constraint programming?
At first glance both approaches appear to be very similar.
What are the major differences between integer programming and constraint programming?
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1
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How to linearize min function as a constraint?
I'm trying to solve an optimization problem including following constraint, and I need to linearize it in a maximization nonlinear programming model. Please help me to reformulate it with mixed ...
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11
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1
answer
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Expressing a chain of boolean ORs using ILP involving different variables
How can I express a chain of OR operations in an ILP, given that each operand is an inequality between two binary variables? I have asked a similar question here: Chain of Boolean ORs. In that ...
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5
votes
2
answers
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How to linearize specific range constraints?
I would like to know about the linearization of the $(If, Then)$ constraints as follows:
$$\begin{array}{l}
\text { If: } \\
15 \leqslant x \leqslant 25 \\
\text { then: } \quad y=\color{blue}{a} x+\...
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4
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0
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Conditional constraint formulation [duplicate]
How can I create constraints to make sure $x=1$ if $k\geq 0$ and $x=0$ if $k<0$, where $x\in \{0,1\}$ and $k\in \mathbb{R}$?
Here is my attempt:
\begin{equation}\label{cons:1}
\begin{aligned}
...
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3
votes
1
answer
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How to enforce logical implication $\sum_j a_j x_j \le b \implies \sum_j c_j x_j \le d$
Some modeling languages and solvers support indicator constraints of the form $$y=\hat{y} \implies \sum_j a_j x_j \le b,$$ where $y$ is a binary decision variable and $\hat{y}\in\{0,1\}$ is a constant....
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2
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1
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Linearize product of $x\cdot y \text{ with } x,y \in \{-1,0,1\}$ for MILP
I have a problem where I have many products between variables drawn out of $\{-1,0,1\}$. Could you suggest a linearization in terms of variables in $\{-1,0,1\}$ or $B_1 - B_2$ where $B_i \in \{0,1\}$ ...
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17
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1
answer
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What is the "big-M" method? And are there two of them?
I’ve seen the "big-$M$ method" referred to in different ways. What is the "big-$M$ method" and why does it seem to mean two different things?
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12
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3
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Does it make sense to use strict equality constraints in optimization?
Once I learned from some post that the strict equality constraint in an optimization problem does not make much sense. We should always use $\le$ constraint. How much truth is in this?
If I must have ...
- 2,211
12
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1
answer
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If-then constraints in MIP programming
For continuous variables $x$ and $y$, the constraints are:
...
- 221
12
votes
2
answers
610
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Expressing an implication as ILP where each implication term comprises a chain of boolean ORs
Consider an implication of the form $A \implies B$ where both $A, B$ comprises a chain of Boolean OR variables. For example, $(a_1 \lor a_2 \lor a_3) \implies (b_1 \lor b_2 \lor b_3)$. How can this ...
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5
votes
2
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How to convert this if-then constraint to MIP constraint?
How to convert this if-then constraint to MIP constraint?
$\text{if } a \geq 0 \text{ then } b=K_1 \text{ else(a <0 ) } \ b =K_2$
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5
votes
2
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Binary variable constraint
The task is to ensure that if $x_i = 1$ for at least $k$ of the possible indices $i$ in $\{1,...,n\}$ then $y = 1$, where $k$ and $n$ are parameters, $x$ is a binary variable vector with $n$ elements, ...
4
votes
2
answers
307
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Model "If, then" constraint
How to model the following "If, then" type constraint?
If $\sum\limits_{i \in I}x_i = 0$ then $\sum\limits_{j \in J}x_{j} = n$
where $x$ are binary variables, $n$ is a known parameter and $...
4
votes
2
answers
525
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Formulating the conditional constraint
I want to develop a model extension of capacitated location problem.
The variables are a binary $x_i$ and a continuous $Q_i$. The following condition must be satisfied:
if $x_i = 0$, $Q_i$ must be ...
3
votes
1
answer
542
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If else condition to MILP
I have following problem:
$c_i = 1$ if $X + \sum_j^N G_j = T$ else $c_i = 0$
Also there is another constraint which upper bounds equation
$X + \sum_j^N G_j \le T$.
$c_i$ is binary
$X, T$ are ...
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3
votes
1
answer
417
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Constraint on groups of variables
Assume a LP/MILP with a large number of variables.
It is easy to formulate constraints to group variables such that each variable in a group takes the same value, if we know which variables are in a ...
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2
votes
1
answer
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Piecewise constraint using big-M notation
I have a piecewise constraint that I am having a hard time converting using big-M modelling. The context is a gym owner that is updating membership costs subject to churn restrictions. The owner can ...
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1
vote
3
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How can I linearize this IF-THEN constraint?
Let
$P_{t,u}; t=1,2,\ldots,T, u=1,2,\ldots,U$ be known values
$\alpha$ is also a known parameter
$X_{t,u}$ an optimization variable
I have the following constraint: IF $P_{t,u}\geq\alpha$, THEN $X_{...
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