Questions tagged [logical-constraints]

For questions about constraints that can be expressed in (usually propositional) logic.

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3
votes
1answer
207 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 ...
3
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1answer
152 views

Complicated constraint with logical operators in PuLP

I have an optimization problem that I am trying to solve with PuLP. All the variables are Booleans. The variables that are "selected" will be true, all others false. Objective function is ...
2
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1answer
226 views

mixed integer programming with if then statement for two binary sequences

I have two random binary sequences of the same size, denoted as P1 and P2 respectively here. Let's say they are both the size of ten, like P1 = [1,0,1,1,0,0,0,1,1,1], P2 = [0,1,1,0,0,1,0,0,1,1]. I ...
3
votes
1answer
100 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 ...
2
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1answer
75 views

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 $B_i \in \{0,1\}$ ...
1
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1answer
50 views

Represent the minimum between two terms as a continuous constraint

Let's consider the following minimization problem: \begin{align} \min_{x,a,b}&\quad X\tag1\\ \text{s.t.}&\quad X = \min(A,B)\tag2\end{align} with $A,B$ functions that depend on $X$. Is there a ...
4
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1answer
123 views

Constrain Mixed-Integer problem such that a graph is fully connected

I have a problem (see my questions about Architectural layouts which poses an interesting abstract question) where there exists an implicit (symmetric) graph whose values in the adjacency matrix are ...
3
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2answers
126 views

Assistance in formulating binary constraint(s)

I would like to seek some advice on modeling the following logical condition: Given two groups of binary decision variables $A_{i}, i=1...n,$ and $B_{j}, j=1...m$. $A_{i}=1- B_{j}, \forall i, \forall ...
5
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2answers
188 views

In a MIP, how to force a decision variable to be zero unless the sum of specific other decision variables is equal to a certain number?

In an MIP, how can I formulate a constraint such that a decision variable is only greater (or equal to) zero if (and only if) the sum of different decision variables is equal to something. I'm working ...
5
votes
2answers
188 views

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+\...
3
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2answers
74 views

Write in ILP: If $x$ within range then $s=1$, else $0$

How can write the following function in LP: $$ s= \begin{cases} 1 & 1 \leq x \leq C \\ 0 & \text{otherwise} \end{cases} $$ where $x$ takes only non-negative integers and $C$ is some large ...
5
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1answer
307 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 =...
3
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2answers
87 views

MILP modelling on minimal disturbance of right-hand-side to make a linear system infeasible

I try to model the following problem: given $z\in\{0,1\}^m$ and a linear system $Ax\le b(z), x\in\Bbb R^n, A\in\Bbb R^{d\times n}, b(z)\in\Bbb R^{d}$, where $b(z)$ means that some entries of $b$ are ...
2
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1answer
42 views

How to write the following constraint? [duplicate]

I would like to write the following constraint, where $varBuyWater$ and $varSellWater$ are decision variables on how much water to buy and to sell. However, I do not want the solver to buy and sell ...
3
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1answer
215 views

Linearize x different of y in ILP

I am surprised I couldn't find an already written answer for my question in the internet. I want to linearize $x$ different of $y$ for two nonegative integer decision variables. I am not considering ...
4
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2answers
437 views

Modeling a constraint such that a set of binary decision variables do not equate to 1 simultaneously

I would like to seek some advice on modeling the following logical condition: I would like to ensure that a group of binary variables do not equate to 1 simultaneously, i.e., $\omega_{1}=1, \omega_{2}=...
6
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2answers
714 views

How to transform this logical if-then constraint?

Consider the binary variables $x, y, z \in \{0,1\}$. I'd like to formulate the two if-then constraints: $$ x + y \geq 2 \implies z = 0, \tag{1} $$ $$ x + y \leq 1 \implies z = 1. \tag{2} $$ Constraint ...
6
votes
1answer
190 views

ILP Constraint to ensure exactly one constraint from a set of constraints is satisfied

Consider several Integer (0/1) ILP variables, i.e., Boolean variables, $x_i$'s. Consider an ILP constraint $x_1 + x_2 + x_3 \geq 1$ and another constraint $x_4 + x_5 + x_6 \geq 1$. I would like to ...
3
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1answer
319 views

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 ...
1
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1answer
67 views

Priority Constraint

Suppose I have the following set of binary variables: $X_i$: $I$ ranges from {1,..,4} Highest priority among the three variables $X$ , $Y$ and $Z$ $Y_j$: $J$ ranges from {1,..,3} $Z_k$: $K$ ranges ...
1
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1answer
85 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 $z$, ...
1
vote
1answer
70 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 $y$, ...
0
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0answers
48 views

Linearizing max constraint Problem [duplicate]

I want to linearize a max constraint as below: In which x_(i,t),are binary decision variables and T is a constant. How can I linearize this constraint?
6
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4answers
760 views

Can this be formulated as one inequality

I have two binary variables $x_1$ and $x_2$ and a non-negative continuous variable $y$. In addition, I have the following two parameters $u>q>0$. I would like to formulate the following ...
2
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1answer
61 views

distance specific constraint

I have some points with determined coordinates $(a_i,b_i)$. A vehicle can move between these points based on rectangular distance. In more detail, we consider that the path between points is an ...
1
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2answers
243 views

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 ...
1
vote
3answers
315 views

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_{...
3
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3answers
261 views

How to linearize the Min function while letting the binary variable to be fixed for x1==x2 as well?

As discussed here, the min function, i.e $X = \min\{x_1,x_2\}$, can be linearized as follows: \begin{align} X & \le x_1 \\ X & \le x_2 \\ X & \ge x_1 - ...
4
votes
1answer
175 views

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 ...
4
votes
1answer
109 views

Formulating these logical constraint in an ILP

I have these two constraints : $z \leq My$ $t \leq M'y $ where $z$ and $t$ are two integer variables $ z, t\geq 0$, $y$ is a binary variable, and $M$, $M'$ are two big numbers. So basically these ...
4
votes
1answer
146 views

how to apply Big M to model the logic constraint (if-then-else)

I was hoping to get some help in modelling the following logic as an MIP Constraint c_{m,l}^{RC} is binary decision variable. Simplify it:
4
votes
2answers
238 views

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
1answer
146 views

Logical constraint in ILP

I want to write the following constraint: Let $z$ be an integer variable such that $0\le z\le M$, and $t$ be a binary variable where $M$ denotes big-M. The logical constraint is as follows: if $z \...
3
votes
1answer
209 views

How to express this logical constraint for an ILP?

I am trying to write an ILP for a problem but I have this logical constraint and I'm stuck. In my model I have: two binary variables: $x$ and $y$ One Integer variable: $z$ The logical constraint I am ...
3
votes
2answers
216 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 $...
4
votes
1answer
64 views

Maximizing 1-norm: using binary variables to relax non-convexity

It is well-known that when we maximize a 1-norm, e.g., $\|Ax\|_1$, we can use binary variables and obtain a mixed-integer convex problem (otherwise maximizing 1-norm is non-convex). I am mentioning ...
6
votes
1answer
256 views

Convert summation of min functions into linear constraints for optimization

I have the following optimization problem: $$ \mbox{maximize } j^{*} \mbox{ subject to:} \sum_{j^{*}\leq j\leq J} \min({\bf A}_j,{\bf B}_j) \geq \lambda, \lambda \in \mathbb{R} \mbox{ and } {\bf A}_j,{...
4
votes
2answers
409 views

How to convert this if-then constraint to MIP constraint?

How to convert this if-then constraint to MIP constraint? $\text{if } a \geq 0 \text{ then } b=K_1 \text{ else(a <0 ) } \ b =K_2$
4
votes
2answers
210 views

Conditional Constraint in MIP

I need to formulate a conditional constraint for a binary variable z defined as: $z_{i,j,k}$, $\ \ i=1:10 \ , \ j=1:5 \ , \ k=1:3$ If any $z_{i,j,3} = 1$ then $z_{i,j,1} + z_{i,j,2} = 0 \ \ \...
3
votes
1answer
248 views

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 \ \& \ ...
2
votes
1answer
149 views

Same values constraint and grouping of variables

In a linear program, I would like some variables to: 1. Take the same values 2. Group some variables i.e. some variables should take same values or lie within certain percentage. 3. All different ...
4
votes
1answer
132 views

Buying shares in increments

I constructed an optimisation model which objective is to find highest return on available stocks. Now want to add a constraint that allows to buy stocks only in $2000 increments, how can I do it? i.e ...
4
votes
1answer
119 views

Inputting logical constraints into a binary programming model in Gurobi

I am very new to Gurobi and OR in general (I'm in my first class for it now), so apologies if this is a very obvious answer. For a project in that class, I am generating a flight schedule for a ...
3
votes
1answer
243 views

Logical operation on an array in DOCplex

Here is a simple example of logical operation in docplex taken from here. Code below works fine. ...
7
votes
1answer
76 views

Expressing a chain of boolean if-then with logical ANDs using MIP

How to express a chain of boolean If-then as MIP such as: If $(x_{10} \ge b_1$ and $x_{11} \le b_1)$ AND $(x_{20} \ge b_2$ and $x_{21} \le b_2)$... AND... then $y_1 = 1$ else $y_1 = 0$. So basically,...
12
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3answers
1k views

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 ...
10
votes
1answer
180 views

MIP: If integer variable $>0$ it should be equal to other integer variables $>0$

I have an MIP problem where $n$ different types of cars are delivering packages. Sometimes multiple types of cars are required to go to a single location. For example if car $1$ makes two deliveries ...
13
votes
6answers
215 views

How to formulate: each pair of elements in $A$ has one common unit in $B$

We have two sets, $A$ and $B$. Some elements of $A$ must be connected to some elements of $B$, but no element of a given set is connected to another element of the same set. (Think of a bipartite ...
12
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1answer
3k views

If-then constraints in MIP programming

For continuous variables $x$ and $y$, the constraints are: ...
7
votes
1answer
269 views

Logical Constraints Modelling using Big-M formulation

I am trying to model some logical constraints in ILOG. Logical constraints could be given such as: Constraint 1 or Constraint 2, Constraint 3 or Constraint 4, Constraint 5 or Constraint 6. The ...