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Questions tagged [indicator-constraints]

For questions on constraints controlled by binary variables.

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3 votes
1 answer
520 views

Model ```a > 0 implies b = 1```, where a is unbounded above

I want to model if a > 0 then b = 1, where a is an unbounded above continuous variable and ...
3 votes
1 answer
194 views

Reformulate constraints

I have the following constraints and am wondering whether I can formulate the whole thing more narrowly and with fewer constraints. $x_{itk}$ is binary and $u_{it}, v_{itk}\in [0,1]$. $M$ is a Big-M ...
0 votes
2 answers
126 views

Converting a piecewise function to linear equations

I am trying to build a MILP model. In this model, I have a dependent variable (alpha) that its value depends on the value of some other variables (or different combination of some other variables). In ...
0 votes
0 answers
66 views

Why are these two constraint equations not equivalent?

I've made a CP Model of an hospital in ILOG CPLEX and I want to test the performance of the CPLEX version of it. In my CP model, I have the following constraint : ...
1 vote
2 answers
117 views

Priotization rules for variable allocation in linear programming

I’m working on an optimization problem and need help with correctly prioritizing the allocation of certain variables in a constraint. The rules are: Only one of the variables $y_{t}$, $zn_{t}$ and $...
1 vote
1 answer
97 views

Logical conditions

This is similar to question I asked here: Priotization rules for variable allocation in linear programming. In an optimization problem, the goal is to manage the purchase and sale of items under ...
1 vote
1 answer
97 views

Mixed Integer programming, the big M

In the constraints below, why have they used the big M? What do we look for in order to identify the big M in other questions?
1 vote
2 answers
157 views

Matrix lookup modelling variants

As part of a bigger model I have a matrix of variables $x_{ij} \geq 0$ and a "selector" set of variables $y_j \in \{0,1\}, \sum_j y_j = 1$. From $x_{ij}$ I'd like to get the variables of ...
1 vote
1 answer
43 views

if else condition with multiple criteria in MIP

I have problem like below Decision variable x1 >= 0 But it depends on selection variable s1 as binary variable If s1 = 0 then x1= 0 and if s1 = 1 then x1>0 how I can write this as constraint for ...
2 votes
1 answer
100 views

Formulation for choosing how many items to manufacture

I am working on a scheduler for a manufacturing plant. I have currently set it up so the decision variables are set up as binary variables: $x_{m,p,s}$ = 1 if machine m is running part p on shift s ...
1 vote
0 answers
91 views

Linearization of Conditional Constraints for MIP using Cplex

I'm currently working on a mixed-integer programming (MIP) problem and I'm trying to implement a set of conditional constraints in CPlex. These constraints involve decision variables that are indexed ...
1 vote
0 answers
43 views

The rule of the slack variable in an indicator constraint

In some cases I have seen, the indicator constraint can be written as indcons(expression, binary_var). Then it is interpreted as follows: $$LHS - slack (\leq = \geq)...
19 votes
6 answers
8k views

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 ...
0 votes
2 answers
118 views

Modelling a binary variable in LPs

I need your help. I'm setting up an LP and I'm trying to find constraints to introduce the binary varibale $b_{ij}$. So it should take the value 0 if the sum of all $a_{ij}$ values to the period t are ...
5 votes
4 answers
884 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 ...
1 vote
1 answer
56 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 ...
-1 votes
1 answer
300 views

[OR-Tools][CP-SAT] Implement an indicator function in a constraint

Good morning ! I would like to implement this constraint using CP-SAT (see image below). x_i,j is a boolean variable, a and b are given. The problem is that I don't know how to implement the ...
3 votes
1 answer
135 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....
2 votes
2 answers
573 views

Using indicator constraint with two variables

I want to use the sum of two binary decision variables (when their sum equals to one) as the condition of Model.AddGenConstrIndicator https://www.gurobi.com/documentation/current/refman/...
3 votes
1 answer
103 views

Modelling Question

Let $W^C_t$, $W_t$ be binary variables and $p$ an integer variable with $1 \leq p \leq 3$ The variables are related through the following equation: $$W^C_t = \sum_{\theta=1}^{p} W_{t-\theta}$$ I can ...
1 vote
1 answer
192 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} = ...
6 votes
2 answers
507 views

Is this constraint with an indicator function nonlinear?

We have two variables $x\geq0$ and $y\in\mathbb{Z}^{0+}$. We have this constraint in our model $$x = \sum_{i = 0}c_i \mathbb{1}_{\{y=i\}}$$ where $c_i$ is a parameter and $\mathbb{1}_{A} = 1$ if $A$ ...
1 vote
2 answers
177 views

How to express this constraint efficiently?

Let, $\mathcal{C}=\{1,2,\cdots,C\}$, $\mathcal{U}=\{1,2,\cdots,U\}$ $\mathcal{S}_u$ is a subset of $\mathcal{C}$ with $u\in \mathcal{U}$ $d_{u,c}$ is a binary variable with $u=1,2,\cdots,U$ and $c=1,2,...
-2 votes
2 answers
70 views

How to model the following "if, then" constraint

There are two matrices: Matrix $S^{v\times v} \in \{0,1\}$ this is a know parameter. Matrix $X^{v\times p} \in \mathbb{N}$. Elements of $X$ are the variables. I want to model: $$\textrm{if }s_{v v'} =...
3 votes
2 answers
374 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, & \...
1 vote
1 answer
124 views

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 $z$, ...
3 votes
1 answer
308 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 ...
3 votes
1 answer
388 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 ...
3 votes
3 answers
1k 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 $(...
5 votes
1 answer
153 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 ...
2 votes
1 answer
165 views

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 $0\leq A\leq B$). ...
7 votes
2 answers
404 views

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 ...
3 votes
1 answer
191 views

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 ...
4 votes
1 answer
367 views

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 ...
3 votes
1 answer
223 views

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 ...
1 vote
0 answers
139 views

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 ...
5 votes
1 answer
647 views

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 =...
2 votes
1 answer
197 views

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: ...
3 votes
1 answer
504 views

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\...
4 votes
3 answers
1k views

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 ...
5 votes
3 answers
860 views

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 ...
3 votes
2 answers
673 views

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 ...
4 votes
1 answer
589 views

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 $x=3$ ...
2 votes
1 answer
244 views

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
500 views

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 ...
3 votes
1 answer
76 views

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 ...
4 votes
2 answers
332 views

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 $...
1 vote
2 answers
277 views

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} &...
1 vote
1 answer
149 views

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 $y$, ...
2 votes
0 answers
124 views

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, $a_i, b_j \in \mathbb{R}_+$ ...