Questions tagged [integer-programming]

For questions about mathematical optimization problems involving binary or general integer variables.

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27
votes
3answers
527 views

Feeding known lower bounds to solvers

Given an optimization problem that aims at minimizing some objective function, a lower bound that is valid for all optimal solutions, and your solver of choice: For what theoretical and/or practical (...
25
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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 ...
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?
18
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1answer
587 views

What's the difference between Lagrangian relaxation and Lagrangian decomposition?

What is the difference between Lagrangian relaxation and Lagrangian decomposition? Are they the same thing?
18
votes
1answer
263 views

Combinatorial Optimization: Metaheuristics, CP, IP — “versus” or “and”?

"Recently" someone asked on Twitter whether "people still use genetic algorithms for integer programs". The "majority answer", i.e., 1 out of 1, was: "Yes" . So, my follow-up question is: With all ...
17
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5answers
3k 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 ...
17
votes
3answers
807 views

Variable fixing based on a good feasible solution

Suppose you have a combinatorial optimization problem that is formulated as a mixed integer linear program (minimization). The problem size is denoted $n$ and the expected $n$ is around $100$. The ...
17
votes
3answers
2k views

Can an integer optimization problem be convex?

I'm trying to wrap my head around an apparent paradox that I've come across while trying to learn more about optimization algorithms: On one hand several sources state that convex optimization is ...
17
votes
1answer
208 views

Family of hard instances for Gomory's cutting plane algorithm

Is there a variant of integer programs for which Gomory's cutting plane algorithm demonstrably takes a superpolynomial number of iterations?
16
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7answers
2k views

Does there exist an aggregation of videos on optimization?

Is there a website or otherwise maintained list of talks regarding mathematical optimization? This would be a big help for the community it seems. I'm most interested in those relating to integer ...
15
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3answers
1k views

How does the search space affect the speed of an ILP solver?

Let us suppose we have an optimization problem which we have modeled as an ILP. Suppose we solve this problem using some set of constraints which restricts the search space. Let us suppose we model ...
15
votes
2answers
718 views

Divisibility constraints in integer programming

In the study of a certain pure mathematical problem (related to infinite-dimensional Lie algebras) I found myself in a situation where it would be very desirable to be able to solve an integer ...
15
votes
1answer
553 views

Duality in mixed integer linear programs

I know that the standard duality theory for the linear programming problem does not hold for mixed integer linear programming problems. I was wondering why an integer program does not have a dual ...
14
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4answers
278 views

Does this $0-1$ integer program have any speciality?

Given matrix $A \in \{0,1\}^{m \times n}$ and vector $b \in (\mathbb{Z^+})^m$, where $\mathbb{Z^+}$ is the set of positive integers, $$\begin{array}{ll} \text{maximize} & c^\top x\\ \text{subject ...
14
votes
3answers
2k views

A variant of the Multiple Traveling Salesman Problem

I am trying to find a reference (or a reformulation) of a variant of the multiple Traveling Salesman Problem, where multiple agents need to visit each vertex in a graph with minimal cost. Most of the ...
13
votes
2answers
828 views

How to choose between high number of binary variables or fewer number of integer (not only 0 and 1) variables in a IP formulation?

When I have to write the formulation of an IP, I usually have the choice between writing $i\times j$ binary variables with two indices such as $ x_{i,j} $ or, writing $j$ integer variables $x_i$. Is ...
13
votes
1answer
771 views

What is quadratization?

In the context of discrete optimization, what exactly does it mean to "quadratize" a function? The term seems to be used mainly by operations researchers, in my experience.
13
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1answer
202 views

Symmetry-breaking ILP constraints for square binary matrix

Setup I have a binary $N \times N$ matrix. The objective is to minimize the number of ones in the matrix, subject to various constraints. This leads to symmetries by rotating 90 degrees and/or ...
12
votes
3answers
1k views

Allocating credit card points

I’m interested in the idea behind this in general, so I thought this would be the best place to post, though I have a practical and semi-urgent need of allocating the points on my credit card towards ...
12
votes
2answers
496 views

Expressing a chain of boolean ORs using ILP

How to express a chain of OR operations in an ILP in which each expression is a less than or equal constraint and the left hand side variable in all inequalities is always the same? All the variables ...
12
votes
2answers
335 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 ...
12
votes
1answer
120 views

How to reduce recursion when using Gomory cutting planes to solve an integer program?

Consider the following simple integer program $$\begin{array}{ll} \text{maximize} & 3 x_1 - x_2\\ \text{subject to} & 3x_1 - x_2 \leqslant 3 \\ & -5x_1 - 4x_2 \leqslant -10 \\ & ...
12
votes
0answers
167 views

Integrality gap in bilevel binary linear programming problem

I have a bilevel max-min optimization problem over binary variables, with constraints expressed using linear inequalities. The inner (minimization) problem is $$ \begin{alignat}2 \min\limits_x&\...
11
votes
2answers
867 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 $y=1$ (...
11
votes
2answers
462 views

Finding an optimal set without forbidden subsets

Given $n$ items, I want to select a set items $S\subseteq\{1,2,\dots,n\}$ that maximize profit. The profit of item $i\in\{1,2,\dots,n\}$ is given by $p_i$ and may be assumed to be non-negative. ...
11
votes
2answers
227 views

Generalized Assignment Problem as the sub-problem

I was wondering what is the state-of-the-art for solving the Generalized Assignment Problem (GAP) and if there are special cases that are polynomially solvable? Moreover, is there any usage of this ...
11
votes
1answer
285 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 ...
11
votes
2answers
191 views

Is deciding the presence of mixed-integer points in the relative interior of a polyhedron in NP?

Given $P = \{x\in\mathbb R^n: Ax \leq b\}$, I want to decide if $(\mathbb Z^\ell \times \mathbb R^{n-\ell}) \cap \operatorname{relint}(P)$ is non-empty. Is this problem in NP? One idea is to check ...
11
votes
1answer
96 views

Computational complexity to compute an IIS

How hard is it to compute an irreducible infeasible subset (IIS) for a linear program? What about an integer program (e.g., removing the integrality constraint on a single variable may be enough to ...
10
votes
6answers
2k views

Nonlinear integer (0/1) programming solver

I have the following optimisation problem.\begin{align}\max&\quad\sum_i\sum_j\sum_k x_{ji}y_{kj} \operatorname{cost}(i,k)\\\text{s.t.}&\quad\sum_j x_{ji}=1\quad\forall i\\&\quad\sum_k y_{...
10
votes
3answers
377 views

Are there explainability approaches in optimization?

In the machine learning community there is the big topic of explainability, where you want to make the solution of ML models explainable or derive explainable models. This is also interesting for ...
10
votes
2answers
857 views

Decision Variable Value from a Set (Gurobi)

Is there a way to set a decision variable to take values from a set? Example: decision variable $x \in \{0,50,100\}$ So this variable can only take one of these three values and not more. I have ...
10
votes
1answer
111 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 ...
10
votes
3answers
143 views

Theoretical results on performance of branch-and-bound

Are there any theoretical results on the performance of branch-and-bound, even for a subset of instances of a particular discrete optimization problem? As an example, does there exist a result of ...
10
votes
1answer
171 views

Finding the linear functions defining a polyhedron through integer data?

Let's say I have a bunch of linear functions $f_1,\cdots,f_n$ in $k$ variables; then $f_1,\cdots, f_n\le0$ defines a polyhedron $P$ in the $k$-dimensional space. What I'm looking for is going the ...
10
votes
2answers
185 views

Use integer/quadratic programming to maximize consecutive zeros in a binary array

A binary array $t = [t_1, t_2, t_3, t_4, t_5]$ with each element a binary integer variable taking values 0 or 1. You can think this vector as slots with 1 representing the slot being taken and 0 ...
10
votes
1answer
432 views

TSP problem: traveller does not visit all nodes - Google OR-tools

Context: I am dealing with a kind of scheduling problem, in which I have a set of tasks and machines. All tasks must be assigned to machines (not necessary all of them). In addition to that, I must ...
9
votes
2answers
407 views

How to maximize “contrast” between nodes on a graph?

I have an undirected graph such as the one shown below. I can make up to 3 choices about the color of each node. The edge weights are equal to the difference between the nodes, given by the "...
9
votes
1answer
213 views

How to get bounds on ILP optimal solution quality

Often, ILP formulations are just too complicated to solve optimally in reasonable time. In those cases, you can still run a solver for some fixed time and simply take the best solution that the solver ...
9
votes
1answer
214 views

Should I factor in time as a parameter or a variable in a scheduling problem with MILP?

I am trying to formulate a problem that will spit out an optimal schedule for my tasks to be completed. To keep the information confidential, I will refer to my tasks as papers that need to be written....
9
votes
1answer
114 views

Equivalence of formulations

I have a simple model such as: \begin{align}\max&\quad z=X_1+X_2+X_3+X_4\\\text{s.t.}&\quad y_1+y_2+y_3+y_4=2\\&\quad X_1 \leq y_1\\&\quad X_2 \leq y_1+y_2\\&\quad X_3 \leq y_2+...
9
votes
1answer
160 views

Static stochastic knapsack problem: unbounded version

In the static stochastic knapsack problem (SSKP) the weights $w_i$ of the items are distributed according to a probability distribution. Each item $i \in I$ can be selected at most once. So, ...
9
votes
0answers
74 views

What to do with cuts (constraints) when a fixation is contrary to a RHS in a ILP / LP relaxation?

I am trying to understand an algorithm in a paper by Crévits et al. (2012)1 (see algorithm 2, the cuts I'm referring to are from the reduced costs). It uses a series of successive cuts on a linear ...
8
votes
4answers
2k views

Why is there not a feasible solution for a MIP?

Is there a way to see why a solver (OR-Tools, CPLEX, Gurobi) cannot find a feasible solution when solving a MIP? By that I mean, is there a possibility to show at which constraint and exact indices ...
8
votes
4answers
313 views

Understanding integer programming solvers

I would like to verify if I understand the nature or workings of integer programming solvers. My understanding is that for integer programming problems like the knapsack problem or the traveling ...
8
votes
2answers
1k views

How to formulate problems in the language of mathematical programming?

The question says it all. I am having difficulties formulating general problems (meaning no numbers just variables). When I read the solution, I understand but I can't figure how to formulate myself ...
8
votes
3answers
1k views

Open source MILP solver for quick “good enough” solution

I have a problem that I have already posted elsewhere in OR.stack, but the question is focused around a large binary MILP (about 1 million decision variables). Ultimately, I am more time constrained ...
8
votes
1answer
318 views

When using docplex.cp is it possible to get all feasible solutions?

I would like to solve an ILP and get all feasible solutions (even the worst one). How could I do that using docplex.cp? I've seen a similar question in: Using CPLEX "solution pool" to count ...
8
votes
1answer
622 views

How to linearize the multiplication of an integer and a binary integer variable?

I have the following constraints \begin{align}\sum_{i=1}^{N}{x_it_i}&= M\\\sum_{i=1}^{N}{t_i}&\le S\end{align} where $x_i\ge 0$ is an integer variable, $t_i\in\{0,1\}$ is a binary variable ...
8
votes
0answers
87 views

Automatic detection of SOS variables and constraints

We've been working on a new feature for Octeract Engine, namely to automatically extract SOS structure from a model and then exploit it. While the literature is quite rich on what to do with SOS once ...