Questions tagged [logical-constraints]

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

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2
votes
2answers
77 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
38 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 ...
1
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1answer
193 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
votes
2answers
428 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}=...
5
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2answers
632 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 ...
5
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1answer
152 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 ...
2
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1answer
260 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
55 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
79 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
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1answer
60 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
39 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
741 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
60 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
vote
2answers
169 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
261 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
223 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
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1answer
148 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
97 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
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1answer
108 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
180 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
124 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
151 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
171 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
58 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
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1answer
194 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
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2answers
384 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
185 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
191 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
97 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
123 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
87 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
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1answer
145 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
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1answer
67 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 constraint in optimization?

Once I learned from some post that the strict equality constraint in optimization problem does not make much sense. We should always use $\le$ constraint. How far this is true. If I must have a ...
10
votes
1answer
118 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
205 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
2k views

If-then constraints in MIP programming

For continuous variables $x$ and $y$, the constraints are: ...
7
votes
1answer
238 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 ...
9
votes
1answer
98 views

Binary variable to count appearances

Let $x \in \mathbb{R}^n$ be an optimization variable. Now, at a constraint, I would like to count how many times a value, say $2$, appears in $x$ decision. I think we can have a binary variable $y_i$...
9
votes
2answers
233 views

Common structures in Gurobi - Python

I'm new to Gurobi in Python and I was wondering if someone knows how to code some common structures of linear constraints. I'm trying to understand how you'll code something like the following ...
12
votes
2answers
343 views

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 ...
11
votes
1answer
295 views

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 ...
6
votes
1answer
2k views

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 ...
7
votes
2answers
292 views

Difference between Chance constraints and logical constraints

A logical constraint combines linear constraints using logical operators, such as logical-and, logical-or, negation (that is, not), conditional statements (that is, if ... then ...) to express complex ...
4
votes
0answers
96 views

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} ...
22
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3answers
3k views

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?
16
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1answer
810 views

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?
11
votes
2answers
919 views

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 $y=1$ (...
12
votes
4answers
785 views

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 ...
25
votes
3answers
5k views

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 ...