Questions tagged [indicator-constraints]
For questions on constraints controlled by binary variables.
37
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3
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3
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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 $(...
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5
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1
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101
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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 ...
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1
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Robust way to implement $(x=0) \Rightarrow (y=0)$, with $x$ nonnegative and $y$ binary
I am formulating a MILP in which there is a continuous variable x and a binary variable $y$.
In the program formulation there are the following constraints:
$Ay\leq x \leq By$ (with $0\leq A\leq B$). ...
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2
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how can I modify my LP to activate the most constraints possible?
Suppose I have a linear program (LP) that has many optimal solutions. Of those optima, I want to find the optimum that activates (aka, "pegs" or "bumps") the largest number of ...
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1
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131
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If-then condition formulation to avoid variable multiplication
I'm trying to formulate the following logic:
If $y_i =1$, then $c_i \leq x_i$
If $y_i =0$, then $c_i \leq 0$
Where $y_i$, $c_i$, and $x_i$ are decision variables.
The easy way would be to write:
$$c_i ...
- 133
4
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1
answer
136
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Can you calculate the mean of some MIP variables using linear constraints?
got a lingering question from a graduate course in integer programming that's been bugging me ever since.
Is it possible to find the mean of some variables in a MIP without resorting to quadratic ...
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3
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1
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119
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Conditional constraint with a strict inequality
It's almost this question: Formulating the conditional constraint
But there they have non-strict inequality. I have $x_i$ a boolean decision var and $Q_i$ as a nonnegative integer decision variable ...
- 143
1
vote
0
answers
92
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Multiple conditions in indicator constraint of Gurobi
I'm working on a vehicle routing problem, in which a vehicle need to pick up amounts of things in some nodes. I'm trying to solve it by Gurobi optimizer. Except for meeting the time windows of each ...
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5
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1
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361
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Model "if and only if" indicator constraints in Linear programming
Apologies if this question has been asked, but I haven't been able to find it. I'm modelling something with Gurobi and want to do the following:
\begin{align}\text{cond} < \dfrac{1}{3} &\iff x =...
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2
votes
1
answer
108
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CPLEX Indicator Constraints in Java API
I'm using the Java API of CPLEX (12.6.1 version) to solve a MILP problem.
This is how I create 'normal' constraints:
...
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4
votes
3
answers
226
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How to couple a binary variable to a continuous variable to indicate values greater 0
I have a continuous variable $x_t$. A binary variable $b_t$ should be coupled to $x_t$ such that $b_t$ has the value $1$ if $x_t$ has a value greater than $0$ and $b_t$ has the value $0$ if $x_t$ has ...
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1
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161
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MILP constrained by the minimum number of satisfied constraints
I have an MILP where we have
$$
t_k = \sum_i P_i\cdot C_{ik} : P_i\ \in \{0,1\}, C_{ik} \in I^+
$$
and our model is constrained by the number of times $t_k$ is bigger than a certain value $T_k$.
$$
\...
0
votes
1
answer
282
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if-else query depending on optimization variable in Gurobi (Java)
I am looking for the most elegant solution to the following problem:
I have an if-else query that depends on my optimization variable $x_i$.
If $a \leq b_i + x_i$, the parameter $c$ should take the ...
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votes
2
answers
382
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If-then constraint with continuous variables
I was usually using if-then constraints with integer variables but ended up using continuous variables and got confused. I have variables $x_{ij}\in\mathbb{R}_{\geq 0}$, and would like to force the ...
- 1,229
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1
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64
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Logical equivalencies to modeling an indicator decision variable in transportation problem
I am formulating a model that seeks to minimize the cost of shipping goods from factories to warehouses, where the cost of shipping is independent of the type or amount of goods being shipped (except ...
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1
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2
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147
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Modeling the product of two variables
Suppose we have two continuous nonnegative variables $X_{1}$ and $X_{2}$ both bounded by the number $M$ from above.
I would like to model the following:
If $X_{1} > 0$ then $X_{2} = 0$
If $X_{2} &...
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1
vote
1
answer
89
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How to model this chain of logical implication II
I would like to seek some advice on modeling the following (chain of) logical implication:
For instance $\omega_{xz}$ might indicate precedence, i.e., $x$, $z$ being the nodes $x$ and $z$, ...
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1
vote
1
answer
81
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How to model this chain of logical implication
I would like to seek some advice on modeling the following (chain of) logical implication:
For instance $\omega_{xy}$ might indicate precedence, i.e., $x$, $y$ being the nodes $x$ and $y$, ...
- 663
2
votes
0
answers
98
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Indicator function for integer variable with inequality constraint
I have $n$ integer variables $\vec{x}$ with the following integer programming problem.
$$
COST = \sum^{n-1}_{i = 0} a_i x_i + \sum^{n-1}_{j=0} b_j I(x_j > 0)
$$
Here, $a_i, b_j \in \mathbb{R}_+$ ...
- 201
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votes
1
answer
199
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Portfolio optimization with indicator function constraint in CVXPY
I have the following portfolio optimization problem that I want to solve using CVXPY: \begin{align}\min_w&\quad w^\top\Pi\\\text{s.t.}&\quad\sum_{i=1}^nw_i=1\\&\quad w^\top\Sigma w\le\...
1
vote
2
answers
297
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Switching of decision variables to be larger than or equal to a decision variable according to an indicator variable value
I would like to seek some advice on modeling the following:
I have two integer decisions variables, $x, x'$, that are either equal or greater than zero and either of them is greater than or equal to a ...
- 663
4
votes
1
answer
190
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Switching of decision variables to be equal to a certain decision variable according to a binary (indicator) variable
I would like to seek some advice on modeling the following:
I have two integer decisions variables, $x, x'$, that are either equal or greater than zero and either of them is to be equated to a third ...
- 663
2
votes
1
answer
77
views
Formulating indicator constraint set
I am having difficulty formulating the indicator constraints for the following:
Consider a set of $A_{n}$ decision variables such that $A_{1},A_{2},⋯,A_{n}<A$. While all of them are integers that ...
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votes
1
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59
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Reformulating to locate the second largest decision variable of a set of decision variables
Consider a set of $A_{vn}$ decision variables such that $A_{v1},A_{v2},\cdots,A_{vn}<A$. While this is the standard formulation finding the maximum value of $A_{vn}$, I would also like to find the ...
- 663
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1
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180
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Linearize sum of continuous and boolean variable
For maximizing the objective function $\sum_i{d_i y_i}+ A x - B \cdot \mathbb{I}_{x>0}$, how can I linearize $ A x - B \cdot \mathbb{I}_{x>0}$ term where $d_i, A$ and $B$ are positive constants ...
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4
votes
2
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314
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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
2
answers
235
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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
3
answers
341
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Faster implementation of "or" constraints in ILP
I have implemented a set of "or" constraints in my ILP using binary decision variables (as in this method). It works fine for smaller problems, but when I try to increase the number of ...
- 141
4
votes
1
answer
468
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Indicator function in math programming
Let $x$ be an integer variable that takes the values $1$, $2$ or $3$.
Let $y_1$ and $y_2$ be binary variables.
I want to express the two following logical constraints:
if $x=2$ then $y_1=1$
if $x=3$ ...
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3
votes
1
answer
137
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How to optimize with "if" constraints
The minimizing problem is the following :
$$ \underset{w}{\operatorname{argmin}} \sum_{i=1}^{n}\left[w_{i}\times (\frac{Vw}{\sigma})_{i} - b_{i}\right]^{2}$$
with $V$ a $n\times n$ matrix (covariance ...
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5
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1
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How are indicator constraints implemented? [duplicate]
I wonder how systems like CPLEX, GUROBI, etc implement indicator constraints. Do they just implement Big M equivalents? If yes, what is then the justification for using them?
Edit
The question does ...
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2
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427
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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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1
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How to transform these conditional constraints to linear integer ones in a more efficient way?
The conditional constraints A and B can be transformed to a set of linear integer constraints as follows:
A) $\text{if} \ x_1=0 \ \text{then} \ d_1=1 \ \text{else} \ d_1= 0\\ x_1\in {\rm I\!R}^{\geq ...
- 284
3
votes
1
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How to fomulate the following conditional constraint in MILP?
How can I formulate the following conditional constraint to a linear constraint using indicator variables? Please note that all variables are continuous and $c \ge 0$
$\text{1: if} \ c=0 \ \& \ ...
- 284
3
votes
2
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383
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Mocking up conditional statements in LP
I would like to know how if condition statements in linear programming can be reformulated using indicator constraints, and hence solved as a mixed integer linear program. Specifically:
1. Is it ...
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12
votes
1
answer
612
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Representing an indicator function: binary variables and "indicator constraints"
I want to represent the indicator function:
$$ \mathbb{1}_{(y=j)}$$
where $y$ is a non negative, integer variable.
My attempt is as follows: define a binary variable:
$$ z_j =\begin{cases}
1 \qquad\...
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19
votes
5
answers
5k
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When to use indicator constraints versus big-M approaches in solving (mixed-)integer programs
Various optimization modeling languages and solvers allow for both indicator constraints (see for example here, here and here) and traditional binary variable and big-M approaches can be used to model ...
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