# Questions tagged [integer-programming]

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

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### 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 ...
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### 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 ...
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### 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....
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Obtaining the intermediate solutions in AMPL

I know that for some solvers, for example, the constraint programming solver in Google OR-Tools, it is possible to see all the intermediate solutions that the solver finds while it searches for an ...
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### 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 ...
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### 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 ...
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### Linear and Integer programming materials

I was wondering if you could refer me to some online video/text resources to learn linear and integer programming. I am intending to work in the field of data science. I greatly appreciate your kind ...
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### 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 ...
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### Bridge the gap between theory and practice in Integer Programming

I've finished Wolsey's book on Integer programming. It's a theoretic book. I aim to learn how the ideas presented in the book can be applied to solve real-world non-academic problems. I am looking ...
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### 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, ...
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### 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 ...
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### What are good reference books for introduction to operations research?

The reference books should cover the wide range of problem-solving techniques and methods.
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### 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. ...
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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.
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### 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 ...
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### 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 ...
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### 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?
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### 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 feasible solutions, and your solver of choice: For what theoretical and/or practical ...
Consider the following simple integer program $$\begin{array}{ll} \text{maximize} & 3 x_1 - x_2\\ \text{subject to} & 3x_1 - x_2 \leqslant 3 \\ & -5x_1 - 4x_2 \leqslant -10 \\ & ... 20 votes 1 answer 2k views ### What's the difference between Lagrangian relaxation and Lagrangian decomposition? What is the difference between Lagrangian relaxation and Lagrangian decomposition? Are they the same thing? 19 votes 1 answer 464 views ### Combinatorial Optimization: Metaheuristics, CP, IP -- "versus" or "and"? "Recently" someone asked on Twitter whether "people still use genetic algorithms for integer programs". The "majority answer", i.e., 1 out of 1, was: "Yes" . So,... 19 votes 6 answers 7k views ### When to use indicator constraints versus big-M approaches in solving (mixed-)integer programs Various optimization modeling languages and solvers allow for both indicator constraints (see for example here, here and here) and traditional binary variable and big-M approaches can be used to model ... 13 votes 1 answer 258 views ### Symmetry-breaking ILP constraints for square binary matrix Setup I have a binary N \times N matrix. The objective is to minimize the number of ones in the matrix, subject to various constraints. This leads to symmetries by rotating 90 degrees and/or ... 29 votes 3 answers 7k views ### What is the difference between integer programming and constraint programming? At first glance both approaches appear to be very similar. What are the major differences between integer programming and constraint programming? 9 votes 1 answer 484 views ### How to get bounds on ILP optimal solution quality Often, ILP formulations are just too complicated to solve optimally in reasonable time. In those cases, you can still run a solver for some fixed time and simply take the best solution that the solver ... 12 votes 2 answers 3k views ### In an integer program, how can I “activate” a constraint only if a decision variable has a certain value? Suppose we have the constraint$$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 $y=1$ (...
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 ...