Questions tagged [logical-constraints]

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

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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
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1answer
51 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
111 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
227 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
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2answers
134 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
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1answer
86 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
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1answer
61 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
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1answer
116 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
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1answer
42 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
68 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
59 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
805 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
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1answer
91 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
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6answers
198 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
476 views

If-then constraints in MIP programming

For continuous variables $x$ and $y$, the constraints are: ...
7
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1answer
180 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
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1answer
91 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
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2answers
156 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
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2answers
289 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
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1answer
231 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
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1answer
887 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
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2answers
256 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
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0answers
79 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
1k 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
251 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
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2answers
453 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
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4answers
620 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 ...
22
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3answers
2k 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 ...