Questions tagged [integer-programming]
For questions about mathematical optimization problems involving binary or general integer variables.
133
questions
16
votes
2answers
563 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 ...
13
votes
2answers
317 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 ...
16
votes
1answer
513 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 ...
19
votes
1answer
244 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 ...
12
votes
2answers
736 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 $y=1$ (...
24
votes
3answers
4k 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 ...
14
votes
4answers
258 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 ...
9
votes
1answer
213 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....
17
votes
3answers
729 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 ...
10
votes
3answers
138 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
166 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 ...
12
votes
2answers
217 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 ...
9
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 ...
11
votes
2answers
777 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 ...
18
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 ...
15
votes
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 ...
11
votes
2answers
190 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 ...
6
votes
1answer
128 views
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 ...
5
votes
2answers
273 views
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 ...
11
votes
1answer
264 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 ...
5
votes
2answers
386 views
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 ...
12
votes
2answers
471 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 ...
9
votes
1answer
154 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, ...
18
votes
1answer
200 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?
15
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 ...
11
votes
2answers
460 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.
...
13
votes
1answer
723 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
votes
1answer
197 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 ...
11
votes
1answer
88 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 ...
27
votes
3answers
456 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 (...
12
votes
1answer
115 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 \\ & ...
18
votes
1answer
452 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?
9
votes
1answer
196 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 ...
