Questions tagged [logical-constraints]
For questions about constraints that can be expressed in (usually propositional) logic.
111
questions
3
votes
2
answers
338
views
How to model a binary variable?
I am trying to find a constraint for the following relationship, but am failing a bit at it right now. I want to find a linear constraint that does the following. The binary variable $switch_{ot}$ is ...
- 31
5
votes
4
answers
829
views
Rewriting if-then constraints of binary summations
Suppose both $x_{i,j}^{ab}$ and $y_{i,j}^a$ are binaries. Then how can I rewrite the following if-then in linear form?
$\sum_b x_{i,j}^{ab} \ge 1 \implies \sum_{i,j} y_{i,j}^a = 0$
I was thinking of ...
- 163
1
vote
1
answer
49
views
Assistance in formulating implication constraints for inequalities
I would like to seek some advice on modeling the following logical implications, where $\delta$ is a binary variable, $D_{j}$ and $A_{j}$ are nonnegative discrete variables, and $p_{j}$ are ...
- 707
4
votes
3
answers
480
views
How can I formulate this 'if-then' constraint problem?
I have five integer variables, and I need to write some constraints on them:
$x_0$ , $[x_1, x_2, x_3, x_4 ]$. $1 \leq x_i \leq 3$
if $x_0 =1$ then no constraint on $[x_1, x_2, x_3, x_4 ]$
if $x_0 =...
- 1,120
2
votes
1
answer
38
views
A tighter relaxation of the mix logical constraints
Suppose the following logical form there exists.
$$Iff: (x_{j,m} \land x_{k,m}) \implies ((C_{j} \leq S_{k}) \lor (C_{k} \leq S_{j}))$$
This is well-known as a no_overlap_constraint in the parallel ...
- 7,569
3
votes
2
answers
326
views
How to model C1 = C2 implies b1 = b2
Suppose $C_1 \ge 0$, $C_2 \ge 0$ are continuous variables and $b_1$, $b_2$ are binary variables.
How could I model the following?
$C_1 = C_2 \implies b_1 = b_2$, the opposite does not hold.
- 2,132
2
votes
2
answers
349
views
Representing "and"/"or" relationships into constraints
How do I represent the following constraints in terms of equations?
For each i, either (1) holds or (2) holds-
$$
x_{1}^i \geq x_{2}^i \quad and \quad x_{3}^i \geq x_{4}^i \quad {(1)}
$$
$$
x_{1}^i \...
- 77
3
votes
1
answer
61
views
How to enforce logical implication $\sum_j a_j x_j \le b \implies \sum_j c_j x_j \le d$
Some modeling languages and solvers support indicator constraints of the form $$y=\hat{y} \implies \sum_j a_j x_j \le b,$$ where $y$ is a binary decision variable and $\hat{y}\in\{0,1\}$ is a constant....
- 28.1k
3
votes
3
answers
214
views
Equivalence between constraints in ILP
Let's have binary variables $x$ and $y$. I'd like to define a helping binary variable $z$ such that
$$ z = 1 \; \;\; \mathrm{iff} \; \; \; x + y = 2.$$
If I wanted to express the equivalence between ...
- 133
2
votes
0
answers
54
views
The linearization of the logical constraints
I know the logical constraints can be linearized by either the logical representation of whose relation, (for pure binary variables e.g. CNF/DNF) or for general form by using Big-M formulation. As I ...
- 7,569
4
votes
2
answers
283
views
The linearization of the (Iff-and-only-Iff) expression
I am trying to linearize the following expression without using the Big-M formulation, but I cannot convert it. I am willing to know if there exists an efficient way to do that?
$$ Iff \quad (w=1) \...
- 43
0
votes
1
answer
57
views
Maximization problem with preferences on variables
Consider the following trivial, theoretical model:
$$
\max x+2y+3z \qquad s.t.
$$
$$
x \leq b_x
$$
$$
y \leq b_y
$$
$$
z \leq b_z
$$
$$
x+y+z = 1
$$
$$
x,y,z \in \{0, 1\}
$$
and $b_x$, $b_y$ and $b_z$ ...
- 927
2
votes
2
answers
214
views
Linearizing a disjunctive expression into MILP
I want to linearize the following disjunctive form.
$$\left[\begin{gathered}w_{1}\\x \geq a\end{gathered}\right] \vee \left[\begin{gathered}w_{2}\\x \geq b\end{gathered}\right]$$
where $w_1$ and $w_2$...
- 7,569
3
votes
3
answers
567
views
Modelling if elif else conditions as MIP
I have 4 variables. Xl6, Xs6, Pl6, Ps6. I have a constant C as well.
Xl6 and Xs6 are binary whereas Pl6 and Ps6 are integers. Also, all variables can take only positive values.
I have to implement ...
- 181
3
votes
3
answers
264
views
Set null the next set of N values
I'm dealing with a problem I already modelled by using linear programming. The already existing constraints set at 1 groups of contiguous variables (for ex: ...
- 33
-1
votes
2
answers
84
views
How to apply smooth approximation to non-linear complementarity constraints?
$P =$
$ x, if U \geq U^{max} $
$ y, if U^{up} < U < U^{max} $
$ z, if U^{down} < U < U^{up} $
$ \alpha, if U^{min} < U < U^{down} $
$ \beta, if U \leq U^{min} $
Where $P$, and $U$ ...
2
votes
1
answer
119
views
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 ...
- 43
4
votes
1
answer
333
views
How can I transform this logical if-then constraint? [duplicate]
I want to know how to transform this logical if-then constraint? If $B=1$,then $A \ge C$, else $A=0$, where $A$ and $B$ are decision variables and $C$ are constants. $B$ is binary variable and $A\ge 0$...
- 43
1
vote
1
answer
162
views
How to write this logical expression with Gurobi + Java, or express it as a big-m formulation
I am trying to write the following expression in Gurobi+Java or Gurobi+python, if it is more practical It could be expressed as a big-M formulation.
\begin{equation} \label{const4}
\text{D}_{uv} =
...
- 11
5
votes
3
answers
195
views
MIP constraint with sum of decision variables having certain value : $\sum_{i=1}^nx_i = 2 \implies \delta = 1$
I want to formulate a MIP constraint such that :
$$\sum_{i=1}^nx_i = 2 \implies \delta = 1$$
$x_i, \delta \in \{0, 1\}$.
My problem is that delta should be one when this sum is exactly 2 and not ...
- 235
2
votes
1
answer
390
views
If/then constraint formulation
Let's assume we have event $i=1,2,\cdots,k$, denoted as $\text{event}_i$. We know for a fact that $\text{event}_i$ is smaller then $\text{event}_{i+1}$ i.e., $\text{event}_i \leq \text{event}_{i+1}$. ...
2
votes
1
answer
118
views
Conditional constraint for binary
Could you please check where I might be wrong?
Task is:
If $z=1$, then either $x=1$ or $y=1$
My approach:
If $z=1$, then $x+y=1$
$\implies x+y\le1$
$\implies x+y\ge1$
If $z=0$, then $x+y\ge0 - M\cdot(...
3
votes
2
answers
196
views
Can we use continuous variables instead of binary variables in this NLP problem?
The following problem is defined with binary variables $a_{i1}, a_{i2}, a_{i3}, k_1$ and $k_2$.
Is it possible to avoid binary variables and to only work with continuous variables? How would we ought ...
- 1,011
4
votes
3
answers
420
views
How to transform a logical constraint with integer variables?
Consider the binary variables $x_1, x_2 \in \{0,1\}$ and the integer variable $y \in \mathbb{Z}$ with $0 \leq y \leq 3$.
I'd like to formulate the following logical constraint:
$$
x_1 = 1 \wedge y \...
- 385
2
votes
1
answer
170
views
Binary variable constraint for condition
I am trying to solve the following task:
If $x=1$ or $y=0$ then $z=0$
My approach:
If $z=0$ then $x+y \le 2 + Mz \implies x+y \le 2+2z \quad$
where $M = 2$
If $z=1$ then $x+y=1 \\
\implies
x+y \le 1, ...
0
votes
1
answer
54
views
Conditional constraint for binary variables
I would appreciate any help to solve the following task:
If $y=1$ then $x_i=1$ for at least $k$ of the possible indices $i\in\{1,\cdots,n\}$ where $k$ and $n$ are parameters, $x$ is a binary variable ...
2
votes
2
answers
73
views
How to model this?
$i$ is a set $1$ to $n$.
$j$ is a set $1$ to $m$.
$j$ and $k$ are from the same set such that $j\neq k$.
$c_{ij}$ is a parameter.
$x_{ij}$ and $y_{j}$ are binary variables.
How to model: If
$$c_{ij}\...
user5245
5
votes
2
answers
238
views
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, ...
3
votes
1
answer
263
views
Binary variables constraint
What constraints would you write to ensure that if $x = 1$ then $y = 0$ where $x, y$ are binary variables?
Until now I only learnt how to build the constraint with 3 binary variables, therefore the ...
4
votes
1
answer
69
views
Another difficult constraint for an ILP
How can I add to this ILP with all binary variables (again related to this question):
$$\min \sum_{1\leq i<j\leq n-1-h} t_{i,j}$$
$$\sum_{i=1}^{n-1-h} a_{k,i} \ge \lfloor (n-1)/2\rfloor \qquad \...
- 393
5
votes
3
answers
611
views
Constraint for two binary vectors to be different
If I have a matrix $A$ of binary variables $a_{i,j}$, $1 \le i \le n$, $1 \le j \le m$, how can I enforce in an Integer Linear Program with binary variables, the condition that every two columns must ...
- 393
3
votes
2
answers
98
views
How to model if-then?
$i$ is a set $1$ to $n$.
$j$ is a set $1$ to $m$.
$j$ and $k$ are from the same set such that $j\neq k$.
$c_{ij}$ is a parameter.
$x_{ij}$ is a binary variable.
How to model: If
$$c_{ij}\cdot x_{ij} \...
user5245
1
vote
3
answers
194
views
How to linearize this if-then constraint?
If $x \ge 1$ then $y = y + x$. And, if $x \le 0$ then $y = y$, where $x$ and $y$ are non-negative integer decision variables. I am using GLPK solver.
How do I linearize this if-then constraint?
3
votes
2
answers
227
views
How to model logic constraint: $y=1$ if $a\le x\le b$ and $y=0$ otherwise?
I am trying to formulate indicator-type of constraints. $y$ is binary $0$ or $1$ and $x$ is a continuous variable.
$$ y =
\begin{cases}
1, & \text{ if } a \leq x \leq b \\
0, & \...
- 53
3
votes
1
answer
240
views
Unexpected runtime error
I have a model that contains the sets (both ordered and Days also circular): Days and Personnel, and I'm trying to make a restriction run exaclty once per Day for every member of Personnel. I'm using ...
4
votes
1
answer
175
views
Creating constraints dynamically in pyomo abstract model
I have a networkX graph with few nodes and these nodes have attributes such as "demand".
...
- 69
3
votes
2
answers
62
views
Constrained shift assignment problem using ilogcp solver can´t find an optimal solution
I´ve been trying to solve a contrained assignation problem given a set of constraints based in a real-world problem. I modeled the problem in AMPL as follows:
...
1
vote
1
answer
178
views
PULP : Constraint violation - multiple of expected value
I am developing a linear programming model in PULP that sends material from Origin Facilities to Transfer and Final Treatment Facilities. The objective function is based on summing the transport and ...
- 11
11
votes
0
answers
131
views
which method has been used to automatically reformulate logical constraints in a standard MIP solver?
There are many of the formulations to linearize logical constraints by introducing new auxiliary binary or any appropriate variables and adding the corresponding constraints to the model. It can be ...
- 7,569
3
votes
1
answer
268
views
Piecewise function with two variables
I have a square like region centered at the origin, which is divided into 4 sub-regions. Region 1 can formed from by the diagonal of a square, $x + y \leq 0$. Region 2 is formed by joining the center ...
- 133
3
votes
1
answer
239
views
gurobi bigM constraint vs. epsilon
I am new to mathematical programming and I am trying to implement case specific constrains in Gurobi with Python.
I am wondering about how I can implement my constraints in the fastest or most common ...
- 147
3
votes
1
answer
135
views
Disjunctive Constraint , Using Binary Variable to Replace a If or condition
I am trying to use a binary variable based on an inequality.
The value of binary variable $q $ is 1 or 0 based on the following equation.
[
$q $ =
\begin{cases}
0,& \text{if } b \geq \pi ,\\
1,...
2
votes
3
answers
349
views
How to formulate if-then for two sums in an integer program
I have two sets of Boolean variables, $x_1, \dots, x_n$ and $y_1, \dots, y_m$ and a positive integer $b$. I would like to add the constraint:
$$\text{If }\sum_i x_i = b \text{ then }\sum_i y_i > b$...
- 131
3
votes
3
answers
778
views
Converting if conditions to linear constraints
I have an optimization problem and I want to convert the following if conditions to linear constraints:
If $(y_1 > U_1)$ and $(m_1)$ and $(E_1)$ then $x_1=1$
If $(y_2 > U_2)$ and $(m_2)$ and $(...
- 77
5
votes
1
answer
139
views
Binary variable to indicate zero probabilities
I have a finite probability distribution $p_1, p_2, \ldots, p_n$ (but these are variables of an optimization problem). Moreover, we have monotonicity, $p_1 \geq p_2 \geq \cdots \geq p_n$.
Assume we ...
- 3,808
3
votes
1
answer
212
views
Disjunctive equality constraints: modelling
I have the following constraint in my model:
$$x = 0 \lor \sum_{i=1}^n y_i = 3$$
where $x$ and $y_i$ are all binary variables.
How this can be linearized by means of big-M notation?
Should I include ...
- 33
7
votes
4
answers
914
views
is prime? in Operations Research
Is there a way to linearize is prime? in Operations Research?
is prime(n) being true if $n$ is a prime number or false otherwise....
- 583
2
votes
1
answer
64
views
Ensure that scheduled repeating maintenance has to be completed
I'm trying to model the scheduling of maintenance in some machines, and was wondering how I could ensure that, if maintenance is planned to start in period $t$, then it has to be carried out until ...
- 516
2
votes
1
answer
281
views
Gurobi add constraint which two variables cant be zero at the same time
I'm currently using Gurobi in python and trying to add a constraint that the variable will not equal to 0 at the same time, say a != 0 or b != 0, both a and b can ...
- 145
3
votes
1
answer
371
views
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 ...
- 333