# Questions tagged [integer-programming]

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

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### 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 ...
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### How this problem can be defined as MultiObjective optimisation

I need to optimize the end-to-end latency of a multi-component application. Assuming that the application has 10 components, component 1-5 is hosted by device 1, and device 2 is hosting the other 5 ...
133 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 $1\leq j\leq k$....
132 views

### Switching of decision variables to be larger than or equal to a decision variable according to an indicator variable value

I would like to seek some advice on modeling the following: I have two integer decisions variables, $x, x'$, that are either equal or greater than zero and either of them is greater than or equal to a ...
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### Switching of decision variables to be equal to a certain decision variable according to a binary (indicator) variable

I would like to seek some advice on modeling the following: I have two integer decisions variables, $x, x'$, that are either equal or greater than zero and either of them is to be equated to a third ...
137 views

### Strong MIP formulations for a large-scale mixed-integer nonlinear feasibility problem

I'm trying to construct a strong MIP formulation for the following integer nonlinear feasibility problem. Informally: We have a $m \times n$ decision matrix of binary variables Each row of the matrix ...
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### Formulating the conditional constraint

I want to develop a model extension of capacitated location problem. The variables are a binary $x_i$ and a continuous $Q_i$. The following condition must be satisfied: if $x_i = 0$, $Q_i$ must be ...
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### Integer decision variables as index

The following problem has only two integer variables; however, they appear in the index of the parameters. Appreciate it if anyone has any efficient idea to transform it into a canonical integer ...
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### 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 ...
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### 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 ...
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### 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 ...
33 views

### Min-cost flow with per-edge flow conservation

I am trying to solve a linear program that is identical to a min-cost flow problem, except for a difference in the flow-conservation constraint. Instead of the summed outgoing flow equaling the summed ...
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### Faster implementation of “or” constraints in ILP

I have implemented a set of "or" constraints in my ILP using binary decision variables (as in this method). It works fine for smaller problems, but when I try to increase the number of ...
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### Linear objective function with non-linear constraints

I would like to choose a set of $\beta_j$s that maximizes a simple linear objective function of the type $$\underset{\beta_j}{\operatorname{max}}\sum_{j=1}^{J}X_j\beta_j \\$$ subject to the ...
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### Gurobi stuck at start - 16GB RAM exhausted

I'm solving a large instance using Gurobi 7.0, 64-bit version on machine with 16GB of RAM. Gurobi is stuck at the start and all the memory is exhausted. Here is the output: ...
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### How is Big M calculated?

Because of excessive pollution on the Momiss River, the state of Momiss is going to build pollution control stations. Three sites (1, 2, and 3) are under consideration. Momiss is interested in ...
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 ...