Mathematics Stack Exchange
Mathematics Stack Exchange

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.

In mixed-integer programming one seeks to find the best (optimal) solution to a linear programming problem. In constrast to ordinary linear programming, however, some of the variables are allowed to take discrete values. Mixed-integer programming has many applications in practice and is often used to model "yes"/"no"-decisions via binary variables. Even though discrete decision variables allow a user to define better and often more realistic models, their introduction to a linear programming problem comes with a high computational cost.

456 questions
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How to write if else statement in Linear programming?

How to write the following if-else condition in Linear Programming? If $a > b$ then $c = d$ else $c = e$ $d$, $e$ are variables. How can we write a linear program without multiplying d and e with binary variables? But we can use binary…
Vinay
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Shortest palindromic Egyptian representation for reciprocal integers

Consider the problem of representing the reciprocal of an integer as an Egyptian fraction where all the denominators are palindromes. i.e. write $$ \frac{1}{n} = \sum_{i} \frac{1}{a_i} $$ where $a_i$ is a palindrome (repeating $a_i$ is allowed). …
Peder
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Column generation and reduced costs

Suppose I have a Master Problem (MP) with several inequality constraints for the decision variables, e.g. $$\min c^Tx \quad \text{s.t.} \quad Ax \leq b, \quad \Vert x\Vert_1 \leq r, \quad x\geq 0.$$ Suppose the columns of $A$ are the vectors of the…
M....
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Problem-Heavy References in Linear/Integer/etc. Programming and Operations Research

I am reaching out for problem-heavy references in Linear / Integer / Mixed-Integer (MIP) / Non-Linear / Network Programming and Operations Research (and Linear Algebra as it pertains to the programming and OR subjects). I find it easier to learn…
JJMae
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Minimizing the difference of convex functions in MIP

I am working on the following mixed-integer program (MIP): $\min_{\mathbf{x}, \mathbf{y}} \ f(x_{1}) - f(x_{2}) + c_{x}^{\top}\mathbf{x} + c_{y}^{\top}\mathbf{y} \\ \text{s.t.} \\ A\mathbf{x} + B\mathbf{y} \geq b ,\\ 1 \geq x_{1} \geq x_{2} \geq…
cbs
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Linearize optimization problem with absolute value

Is there any method to linearize the following optimization problem? \begin{align} \min_{x,y} &~~ c~[x; y] \\ \text{s.t.} &~~ \sum x\leq \alpha_1 \\ &~~ \sum |y|\leq \alpha_2 \\ &~~ \sum y= 0 \\ &~~ x+|y| \leq 1 \\ &~~ (x,y)\in \{0,1\} \times…
Reda
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How to prove that 1/3 is the optimal solution for the muffin problem with 5 students and 7 muffins?

The Muffin Problem Definition Let there be $m$ muffins and $s$ students. The problem is to divide the muffins into pieces where every student gets exactly $\frac m s$ muffin, such that the size of the smallest piece you created while cutting the…
PHPirate
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1 answer

Linear Programming: "at most k out of n variables nonzero" constraint

I have a Mixed-Integer Program that contains (among other things) $n$ variables $v_1, \dots v_n$ (continuous or integer doesn't matter, in $[0, M)$ for some $M$). I want to formulate the constraint that at most $k < n$ of these variables may become…
Lukas Barth
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Column generation when number of rows depend on number of columns.

Say I have the following optimization problem: $$ \begin{align} \textrm{minimize } & \sum_{p\in P}{c_p \lambda_p} \\ \textrm{s.t. } & \lambda_{p_1} + \lambda_{p_2} \leq 1, \forall p_1,p_2 \in P \\ & \lambda_p \in \{0,1\}, \forall p \in…
mvc
  • 179
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Several questions on Column Generation: degeneracy, columns in/out and non-integer solutions

I am currently using Column Generation accompanied with Dantzig-Wolfe decomposition to solve MILP. I have several questions to ask. At the beginning, the objective value of RMP does not improve even if columns with negative RC (my problem is a…
user3458994
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1 answer

Issue with integer subproblem in Dantzig-Wolfe decomposition

I am dealing with an integer model, and I'm using Dantzig-Wolfe (DW) algorithm that I have developed in Matlab to decompose the model into a master and sub-problems. I put the integer constraints in the sub-problems. The algorithm works fine;…
Mahdi Rouholamini
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How many variables and constraints can modern mixed integer programming solvers handle?

I know it depends on the specific problem instance, but approximately how large of an MIP problem can we write and still be guaranteed that an optimization solver will find a solution? For instance, if we have a mixed integer program with 1,000…
graphtheory123
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Modeling contiguity of machine processing in a flow shop environment via a MIP

I'm working on a Mixed-Integer-Programing (MIP) formulation for a flow shop scheduling problem. One of the requirements/wishes is that for each machine $i$, processing should be contiguous, or at least contiguous to a certain extent. What I mean…
Rutger
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Select five vectors that upon undergoing elementwise multiplication are most similar to another vector

I have a sparse $60000\times10000$ matrix where each element is either a $1$ or $0$ as follows. $$M=\begin{bmatrix}1 & 0 & 1 & \cdots & 1 \\1 & 1 & 0 & \cdots & 1 \\0 & 1 & 0 & \cdots & 1 \\\vdots & \vdots & \vdots & \ddots & \vdots \\0 & 0 & 0 &…
Alex Pharaon
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2 answers

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 how to approach it. Turn the given model in to…
Songaro
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