Questions tagged [integer-programming]
Questions on optimization constrained to integer variables.
1,128 questions
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When to construct LP, IP, or MIP model
After doing a few optimization word problems, I've noticed I'm struggling a bit when trying to determine whether or not to set up the problem as an Integer Program, Linear Program, or Mixed Integer ...
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Need help establishing linear objective and linear constraints
So I have a problem, that is really similar to the assignment problem.
Basically there is a company producing square envelopes. A number of papers should be put into the envelope. Exactly one paper pr ...
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maximum flow algorithm to find a maximum cardinality matching in a bipartite graph.
How could I show how a maximum flow problem can be used to find a maximum cardinality matching in a bipartite graph?
thanks in advance
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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 ...
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Bin packing with load fairness across the bins
The bin pack problem denotes the process of assigning a set of n items into a minimal number of bins of capacity c. It can be simply formulated as an ILP as per the below description:
My question is :...
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Representation of conditions in integer linear programming
How to represent in PLI the fact that inequality (1) or inequality (2) must be satisfied but not both?
$j$ is executed before $k \rightarrow t_{ij} + p_{ij} \leq t_{ik}$ (1)
$j$ is executed after $k \...
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LP relaxation of ILP and the ILP have the same optimal value
I have an ILP (all variables are binary) and on several instances I’ve observed that its optimal value coincides with the LP relaxation optimal value.
The LP relaxation is not integral for fractional ...
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Minimum of objective function over set of integers with constraints?
I have the following objective function:
\begin{equation}
f(i,j,k) = a + b_1 i + b_2 j + b_3 k + \sqrt{(a + b_1 i + b_2 j + b_3 k)^2-d^2}, d\geq 0
\end{equation}
Is it possible to analytically find ...
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The relaxation method in the Linear and Integer Programming, some goniometric stuff
I have a problem in a very nice book by Alexander Schrijver Theory of Linear and Integer Programming in the Relaxation Method.
Can some give me a hint how here on the page $160$ it was obtained the ...
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Sum of integers closest to a given number
I have the following problem at hand:
Given an odd number $n > 7$, find a set of non-negative integers $m_7$, $m_8$, ..., $m_{13}$ and $m_{14}$, such that the sum
$m_7\cdot 7 + m_8\cdot 8 + ... + ...
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How do I solve this integer linear program?
I am trying to wrap my head around this problem:
$$maximize \quad w^t X \quad \text{with}\, w=[a, b, c, d]$$
$$\text{subject to}\quad x_1 a + y_1 b + z_1 c + k_1 d \leq v_1$$
$$\dots$$
$$x_n a + y_n ...
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How do I solve this mixed integer program?
I have a minimisation problem in the following form
$$\textrm{min}: x^TAx$$
constrained by $\sum x_i=N$
where $x$ is a vector containing only 1's and 0's, and $A$ is a square matrix of real numbers.
...
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What is the best way to convert this into a integer linear program and what is the best way to solve such a problem?
I am studying a mixed integer program in the form
$$ \textrm{min}: \sum A x$$ constrained by $\sum x_i = N$
where $x$ is a vector containing only 1's and 0's, N is an integer, and $A$ is a square ...
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Name of the optimization model based on variable type
If the variable is binary 1,0 the model name is a binary integer optimization model
If the variable is any integer, the model name is integer optimization model
What if the variable is any integer ...
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A VRP formulation where customer i needs to be visited before customer j?
The title explains the problem more or less. I am looking for literature for a specific constraint, where a customer $i$ needs to be visited before another costumer $j$ can be visited.
Are there any ...
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Does adding linearly dependent columns to a totally unimodular matrix preserve total unimodilarity?
I was wondering the following. Given a totally unimodular matrix $A$ and a vector $b \in Im(A)$ is then the matrix $[A,b]$ totally unimodular too? My guess is no, since for total unimodilarity every ...
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Methods for binary linear programming
I have an LP problem (linear objective with eq and ineq constraints) in binary variables.
Except for the objective, all the coefficients are integer, mostly in {-1,0,1}. Maybe the objective coeff ...
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Non $0,\pm 1$ integer coefficient polytopes with integral extreme points
Suppose we have a linear system
$$Ax\leq b\quad \text{where}\quad A\in \mathbb{Z}^{m\times n},b\in \mathbb{Z}^m.$$
In integer programming literature, we usually have that $A$ has only $\{0,\pm 1\}$ ...
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Reconfiguration to find other solutions of a Binary Linear Program (NOT ILP)
Assuming we are to optimize 0-1 problem. If we've found the first solutions where multiple solutions might exists. How do we reconfigure the system (maybe through unimodular operations) inorder to ...
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What are the prerequisites to learning integer programming and what textbooks would you recommend to learn it?
There was this course in many undergraduate mathematics programs called integer programming. It included Modelling, Linear Programming Primal, Linear Programming Duality, Dual Simplex Algorithm and ...
user1077349
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Efficient way of solving a matrix equation with integer solution
The FLAC audio codec is able to losslessly compress audio by removing redundancy from an audio signal with (forward) linear prediction and coding the residual of this prediction with a rice code. The ...
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Cutting bars problem with integer programming.
How to model the Linear Programming for the problem below with the most complete + reasonable constraints.
A production facility has 2 types of reinforcement bars 6m, 8m long (unlimited quantity). ...
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How to formulate the following constraint in OPL (or mathematical program)?
(Originally posted here https://stackoverflow.com/q/72687231/10291218)
Suppose I have an integer array A of size n with two ...
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What are data matrices of a SDP problem?
I'm reading the paper 'New bounds for the max-$k$-cut and chromatic number of a graph' by van Dam and Sotirov. On page 221, it says:
"It is well known that one can restrict optimization of a SDP ...
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if else statement linear programming
I am trying to work out an if else statement for the following problem, which should be mathematically linear programmed:
when both item 1 and item 2 are picked, both their costs are reduced with 20%. ...
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How to prove that solution have at most $m$ fractional coordinates
I started to study a bit of mixed linear programming, and I am facing the following exercise that after quite some time I don't know how to approach:
Let $A\in\mathbb{R}^{m,n}$, $b\in\mathbb{R}^{m}$, ...
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Find the minimum integer satisfying an inequality
Let $\alpha \in (0,1)$ and $\beta \in (0,1)$. I want to compute the smallest integer $n > 0$ such that:
$$
1 - \alpha^n - [1 - \alpha]^n \geq \beta.
$$
For example, with $\alpha = 0.75$ and $\beta =...
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How to solve an Integer Programming problem using Gomory's Cutting Plane method, without using the Dual?
How to solve an Integer Programming problem using Gomory's Cutting Plane method, without using the Dual? This is a concept question.
Im not opposed to using the dual in practice. Im just curious why ...
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Polynomial Curve Fit without floating point
big math dummy here hoping to get some advice. I'm working on a closed loop servo system that requires a curve fit on some feedback. The controller for this system is $16$-bit. With the help of excel ...
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Integer Linear Programming - Capacitated Vehicle Routing Problem
I am dealing with a CVRP with multiple vehicles. I am struggling to come up with a formula for the constraint that each node with a non zero demand must be visited by one vehicle, once.
I am trying to ...
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Mysterious distributed optimization problem
Problem. Let $x = (x_1,\dots,x_N) \in K^{N}$, i.e., each element $x_i$ can take at most $K$ discrete values. Let $x_{(i)}$, for $i \in \{1,\dots,I \}$ possible overlapping subsets of $x$. For example, ...
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On the non-sufficiency of total unimodularity of the constraint matrix in the definition of an integer polytope
Crossposted at Operations Research SE
Is there an example of an $m\times n$ integer matrix $A$ and an integer vector $b\in \mathbb {Z}^{m}$ such that the polyhedron $P := \{ x\in \mathbb {R}^{n} \mid ...
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Select a subset to Minimize a continuous unimodal function
I want to find an approximation algorithm for the following problem.
$\qquad$ Find a $S\subseteq N$ such that $\rho(S) = \frac{\sum_{i\in S}\ V_i}{(1+\sum_{i\in S}\ V_i)(4+\sum_{i\in S}\ V_i)}$ is ...
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The optimal value function over all doubly stochastic matrices
Let $I = J = \{1,\dots,n\}$. Define set $X \subset \Bbb R^{I \times J}$ as all $n \times n$ doubly stochastic matrices $x = (x_{ij})$ satisfying
$$\sum_{j=1}^n x_{ij} = 1, \quad \forall i $$
$$\sum_{i=...
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Convergence Analysis of A Vector Sequence with Discretization Recurrences (with Toy Examples)
I'm confused by how to analyse if a vector sequence is convergent or not. Here I first post the original problem as follows:
(Although this post is long, the problem meaning is easy to understand but ...
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Linear program with exponential decay between variables
I'm trying to create a linear program to solve a scheduling problem, below is a description of the problem, I'll try my best to keep it short but comprehensive.
The core of the problem is that a daily ...
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How good is an optimal solution of the Lagrangian relaxation of an integer linear program?
From what I learned, the Lagrangian relaxation of an integer program is used to find a bound. Is the solution to the relaxed problem considered to be a good approximate solution of the original ...
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The number of feasible subsets of the knapsack problem - the combinatorial explosion
I am reading the book "Integer Programming" by Wolsey (1998). In 1.4, the author is counting the number of the feasible subsets of a knapsack problem. The formulation is
$\max \sum_{j=1}^n ...
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Formulate constraints to an Integer programming: How to algebraically formulate a geometric constraint that the colored grids must form a rectangle?
I am stuck in a constraint formulation of a discrete optimization problem. Consider a board of Cartesian grids (M rows by N columns). We are going to color some grids among them. There is a geometric ...
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Transfer a nonlieanear function to a linear function
I'm using Java to solve a maximization problem in Cplex. My objective function is quite complex. In a nutshell, there are two parts, A and B. Both of them contain variables.
My goal is to maximize A/B,...
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Integer eigenvectors of a rational matrix
I have a $2n\times 2n$ rational matrix $A$ for which I know can be diagonalized in the form
$$A=MDM^{-1}$$
where $D$ is a $2n\times 2n$ matrix consisting of half eigenvalues $1$ and half eigenvalues $...
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What is the linear description of a transformation of Birkhoff polytope?
Let $I, J$ be finite sets and $|I|=|J|=n$, Let $F$ be a Birkhoff polytope formed by the convex hull of $n\times n$ doubly stochastic matrices:
$$F=\{R^{I\times J}_+: \sum_j x( i,j)=1,\forall i\in I,
...
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Primal Dual algorithm for set cover
I am having trouble understanding the approximation algorithm for set cover
using primal dual.
The entire approximation algorithm in a nutshell.
A set cover problem is given
Form the integer linear ...
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Combining multiple binary variable combinations in sum statement
I have a question regarding how to formulate a constraint of an MILP.
I have 2 platforms p and v(p) which are neighbours. Depending on the state of both of these platforms a specific value is chosen. ...
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Consecutive ones property fails to agree partition condition
I have the matrix below:
$A=\begin{pmatrix}
0&1&0&0&0 \\
0&1&1&1&1\\
1&0&1&1&1\\
1&0&0&1&0\\
1&0&0&0&0\\
\end{pmatrix}.$
...
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Given that the positive integers $x>1$ and $y$ satisfy $2007x-21y=1923$. Find sum of digits of minimum value of $2x+3y$.
Given that the positive integers $x>1$ and $y$ satisfy $2007x-21y=1923$. Find sum of digits of minimum value of $2x+3y$.
Here we have to solve for two variables using only one equation. How is ...
user1017704
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Edge Interdiction Clique Problem seperation of inequalities
I am currently working on the paper A branch-and-cut algrithm for the Edge Inderdiction Clique Problem.
Basically, the problem asks to find a subset of at most $k$ edges to remove from a graph $G$, so ...
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Linear minimization over two integers
Given $x\in]0,1[$, let the function $f:\mathbb{N}^+\times\mathbb{N}\to\mathbb{R}$ be defined by
$$ f(p,q) := x p - q $$
Is there an analytic formula for the minimum of $f$ under the constraint
$$ f(p,...
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Number of possible combinations that satisfy the constraints
Let $x_{1}, x_{2}, x_{3} \in \mathbb{Z}^{++}$ (i.e., strictly positive integers).
Suppose the following (in)equalities are given:
\begin{align}
x_{1} &\geq x^{\min}_{1} \tag1\\
x_{2} &\geq x^{\...
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how to define the dual of a primal assignment-like program with Hadamard product as a cost function?
I need to define the dual of the assignment-like problem where the cost function is defined as the Hadamard product of two matrices $C=[c_{ij}]$ and $X=[x_{ij}]$ as follows:
\begin{align}
\text{...
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