# Questions tagged [mixed-integer-programming]

A mixed-integer programming (MIP) problem is a linear program where some of the decision variables are constrained to take integer values.

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### Relaxation of the binary optimization problem doesn't make sense in terms of the original problem - how do I repose the model to make sense?

Suppose I have a set of binary variables $x\in \{0,1\}^N$, where in the context of my optimization problem these binary parameters mean to include the parameter in a model or not include it, where ...
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### Minmax combinatorial optimization with target set size constraint

I have a combinatorial optimization problem whose objective is in the "min-max" form. Suppose that there is a row stochastic matrix $P=[p_1^T,\cdots,p_n^T]^T\in\mathbb{R}^{n\times n}$ whose ...
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### Reformulate indicator function using mixed integer program

Consider the following constraint: $$\sum_i I(a_i x \leq b) \leq m.$$ Can we reformulate this constraint using big-M constraint? A similar question can be found in Involving indicator function as a ...
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### Changing a model in to a mixed integer linear programing model

Trying to learn about integer programming in quarantine and I've come across a problem that stumped me. I searched the net but couldn't see anything similar and would appreciate another set of eyes on ...
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### How to enforce graph continuity constraint?

I am working on a single source multiple destinations problem formulation using MILP. The network I have designed is as shown in the figure below Graph Network. The objective function is to minimize ...
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### Assignment problem with batching costs

I am studying an assignment problem with batching costs, and I would like to know if there is a standard name or algorithm for this problem. I know this problem can be formulated as mixed-integer ...
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### What will be an efficient joint clustering solution to this problem?

There is a system with $S$ transmitters/service points, $U$ subscribers/users. The link quality/gain between any transmitter indexed by $s$ and a user indexed with $u$ is denoted by $h_{u,s}$. Any ...
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### Proving that an inequality is a split inequality (in cutting plane, Integer Programming)

As I know, to prove if a given inequality is a split inequality or not, we need to find $\pi$ and $\pi_0$ such that $πx ≤ π_0$ and $πx ≥ π_0 +1$ and the given inequality must be valid for both $Π_1$ ...
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### Making a variable in a linear program be equal to a ramp function

In a linear program, I have a variable $y$ that must obey $y=\max(x-a,0)$ where $x$ is a linear combination of the other variables and $a$ is a constant. $x$ and $y$ and $a$ are always non-negative. ...
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### MILP formulation of constraint

I'm very new to MILP, and I'm testing python-mip. I'm trying to model a simple power distribution system with some constraints, but I'm stuck.... I want to find a valid power balance between a device, ...
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### 'Weird' Indexing In The MILP Literature

Fairly new to MILP. I posted this question as an issue on the Gurobi examples repo too, but it might be more appropriate to ask it here. NOTE: Since I "need at least 10 reputation to post images&...
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### Solving a system of piecewise linear equations with multiple variables in minimum function

General problem I am facing a problem from the transportation field. For a given network, the maximum flow $q$ on a link is either the sum of the flows from upstream links, entering at the upstream ...
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### Column Generation?

I implemented a method to solve a large integer program but am not sure how its called. The idea is the same as column generation, just less explicit. Take the following problem as a simple example (...
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### Valid inequality with chvatal gomory for problem with binary variables

I am not able to find a valid inequality with chvatal gomory for my problem that increases the value of the relaxed problem. My problem: Determine the optimal allocation of n workers to m machines in ...
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### Integer solutions to a large sparse equation

How might one find integer solutions to: $$\textrm{diag}(x)Ax = Bx$$ $A$ and $B$ are integer-valued, square, non-invertible, sparse, and large. The values of $x$ should be $0$ or $1$ (ideally; ...
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### Mathematical formulation of an optimization problem

I have the following optimization problem at hand. There are N integer numbers, $a_{i},a_{i+1},\dots,a_{n}$ where $a_{i} > 0$. We need to partition these numbers ...
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### Any nice/good way to allow an “or” option in a linear program?

I am looking for a way to express an "or" option in a system of linear inequalities for a linear program I am working on. I will explain what I mean precisely: Lets say I have a set of ...
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### If Else Statement in Linear Program, making it Mixed Integer Program [closed]

How to write the following if-else condition in Linear Programming or in Mixed Integer Program? If a <= b then c = a else ( a > b) then c = b a, b, c are all variables. Is it possible to rewrite ...
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### How to write hard ILP problem?

I want to write/create hard ILP problems. When I say hard I mean the question should be solvable in at least 20 minutes with non-commercial ILP solvers. Can someone show me a direction or know how to ...
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### linearizing non-strict inequality If.. else conditional constraint that includes decision variables

I would like to linearize a conditional constraint as follow: 0 <= x1 <= 1. If 0 <= x1 < 1, then x2 = 1, else x2 = 0. I find it very difficult because it is not a typical If...then ...
Given is the following problem: There are $N$ items that have to be processed in given intervals over a year. For example, item $i_1$ is processed about every 14 days, item $i_2$ is processed monthly. ...