Questions tagged [integer-programming]

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

23 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
12
votes
0answers
172 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&\...
8
votes
0answers
89 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 ...
7
votes
0answers
92 views

Characterization for total dual integrality

A problem I study reduces to whether the polyhedron $P=\{\mathbf{x}\mid A\mathbf{x}=\mathbf{1}, \mathbf{x}\geq0\}$ is integral ($A$ is a matrix with coefficients in $\{0,1\}$). I know that the ...
6
votes
0answers
114 views

Help in solving resource allocation optimization problem

I've been pondering on this question for some work optimization, and I need some help in being directed to the right direction. I have multiple customers that require an amount of $X$, $Y$ and $Z$ ...
5
votes
0answers
84 views

Bounding the size of the dual solution

Given an primal optimization with bounded feasible set: $\max \{cx: Ax \leq b\}$. The feasible region of the dual is $D = \{y:y^\top A = c^\top, y \geq 0\}$. If the primal feasbile region is a ...
5
votes
0answers
140 views

When is there at least an integral point in a polyhedron?

This problem comes from a problem of economics. Let $x\in [0,1]^n$. $\{x_1,x_2,\ldots,x_n\}$ is partitioned into ${S_1, S_2,\ldots,S_k}$ such that $\sum_{x_i\in S_j}x_i\leq 1$ for each $1\leq j\leq k$....
5
votes
0answers
82 views

Are there any good models for min-max vehicle routing problem?

I am trying to model a min-max VRP problem with multiple delivery vehicles and I have come up with a model using branch and cut but I do not think it is strong enough as it takes lot of time to ...
5
votes
0answers
79 views

How to communicate number of integer combinations to a user

I'm working on a nifty little feature for our next release, i.e., to print the number of possible integer combinations left during branch and bound. This is really handy for the user because they ...
4
votes
0answers
64 views

Continue on “Is there a known MILP to schedule routes after routes are made”

I have made some progress on my previous question (Is there a known MILP to schedule routes after routes are made). I have derived the sets of the problem, which are: 1) Itineraries of vehicle: $i \in ...
3
votes
0answers
77 views

How to linearize a max min objective function?

Let us suppose that I have a $\max \min$ objective function that only depends on one set of variables: $\underset{x}\max \underset{y}\min dy$ Associated with the linear set of constraints and right ...
3
votes
0answers
104 views

0 1 solution of linear programming problem with only equality constraints

I have a linear programming problem $LP$ where all the variables $x_{i}$ take value in $\left[0, 1\right]$ (that is $0\leq x_{i} \leq 1$). All the constraints are as follow: $a_{1}+a_{2}+a_{3}=1$ that ...
3
votes
0answers
67 views

Optimal Seat Allocation Problem

I have to do an operations research assignment based on optimal seat allocation. The problem goes something like this. There are 5 rooms in an office each with a separate seating capacity. We now have ...
3
votes
0answers
52 views

Flexible Job Shop with Preemption

I'm trying to solve a flexible job shop problem variant that has precedence constraints on jobs along with a few other issues. We have a MIP formulation and also a simulated annealing algorithm to ...
3
votes
0answers
40 views

How to find all covers and minimal covers?

Consider a constraint of type $$c_1x_1+c_2x_2+\cdots+c_nx_n\leq C$$ with $x_i$ binary. We call a cover a subset of the $n$ indices such that the sum of the corresponding coefficients is higher than ...
3
votes
0answers
63 views

Theoretical aspect of using extended formulation

If I can show a polyhedron Y is an extended formulation of polyhedron X and every extreme point in Y is integral, does that automatically imply the projection of Y onto the variable space of X gives ...
3
votes
0answers
34 views

Linear functions in Lenstra's algorithm

I had asked this question at MathOverflow and was pointed here. I'm working on implementing Lenstra's algorithm. At the bottom of p.5 (at "construct $n+1$ linear functions"), he says to ...
2
votes
0answers
77 views

Indicator function for integer variable with inequality constraint

I have $n$ integer variables $\vec{x}$ with the following integer programming problem. $$ COST = \sum^{n-1}_{i = 0} a_i x_i + \sum^{n-1}_{j=0} b_j I(x_j > 0) $$ Here, $a_i, b_j \in \mathbb{R}_+$ ...
2
votes
0answers
96 views

Condition for an integer program and its linear relaxation to have the same value

Let $A$ be a $(0,1)$-matrix where no row or column is a zero vector, and consider the following optimization programs \begin{align}(1):\min&\quad y\cdot1\\\text{s.t.}&\quad yA\ge w\\&\quad ...
2
votes
0answers
152 views

Operation hours optimization for circular schedule

Here is my problem. A store has X = 15 electical devices with the ability to work non-stop, fully charged, up to 8 hours. Their battery charge lasts 2 hours and the operating hours of the store differ ...
1
vote
0answers
67 views

How can I set the solution of gurobi to be a multiple of 10 instead of all integers?

For example, the solution for gurobi has two solutions, as follows: [10,20,50,70] [55,79,30,80] I only want to output solutions that contain only multiples of 10. The sample example as follow: ...
1
vote
0answers
45 views

Unifying constraint matrices in sparse situations

$\DeclareMathOperator\Set{Set}$ Let $Set=\{x\in\mathbb Z^{n}:\exists y\in\mathbb Z^m\text{ satisfying } A[x,y]'\leq b\}$ where $A$ has $r=km$ rows and $k=O(1)$. I am trying to write $$ Set=\{x\in\...
0
votes
0answers
84 views

Check VRP instance is feasibility

Beforehand, this is a very long thread, in case you want to know in advance, to see if this thread's interests match with yours, this thread concerns fast ways of determining whether a VRP instance is ...
0
votes
0answers
37 views

Interger programming using gray encoding

Could anyone suggest me a tool or library which takes an integer programming problem written in DOCPLEX or CVXPY as input and outputs the equivalent problem using Gray binary encoding? I am happy to ...