Questions tagged [binary-programming]
An optimization problem in which the decision variables are binary.
231 questions
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Semidefinite relaxation for a binary quadratic program with an $\ell_1$ penalty term
I am considering the following regularized binary quadratic optimization problem with a sparsity penalty
$$ \min_{{\bf x} \in \{\pm 1\}^n} \; {\bf x}^\top {\bf C} \, {\bf x} + \| {\bf A} {\bf x} - {\...
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Seeking advice on how to "see" this derivation
I'm struggling to model this constraint for a problem:
$$x_C^4 = 1 \implies (x_A^4 + x_B^4 \geq 1 \land x_A^1 + x_B^1 = 0) \;\lor\; x_A^2x_B^3 = 1 \;\lor\;x_A^3x_B^2=1.$$
where all variables are ...
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Solve system of linear diophantine equations such that solution vectors only have entries $\in \{0,1\}$
I want to find all solutions $\mathbf{x}$ of the system of linear diophantine equations
$$
\mathbf{A} \mathbf{x} = \mathbf{b}
$$
where $\mathbf{A}$ is a $m \times n$ matrix such that $A_{ij} \in \...
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Relaxing a binary constraint in an optimal control problem
Crossposted at Operations Research SE
I am attempting to optimize the operation of an electrical system that produces some amount of thermal power $P_t$ and keeps a temperature $x_t$ within a certain ...
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Generating relations from linear equation system with binary coefficients
Given a binary coefficient matrix $A \in \{0,1\}^{m\times n}$, $n>m$, and a real-valued vector $b\in \mathbb{R}_+^m$, the corresponding linear equation system is underdetermined and does not admit ...
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How to tackle this optimization with binary variables?
So, I have a (practical) optimization problem in which (somewhat large, say $1500$) $N$ binary variables need to be found.
$
\min_{a \in \{0,1\} ^N} \sum_i \left(\frac{a_i w_i}{\sum_j a_j w_j}\right)^...
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Is there an elegant general method for solving linear multiplicative system of equations in modulo 2? Here is an interesting example problem.
Here is the following problem:
I have solved the system of equations with simply using brute force but I feel there must be a ...
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Efficient Algorithm for Binary Tensor Decomposition?
I've been working with high-dimensional binary tensors (e.g., tensors with entries that are only 0s and 1s) and I'm looking for an efficient way to decompose them into rank-1 components. The tensors I'...
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Calculation of special subsets in high-dimensional binary matrices
I need to solve a rather specific problem related to binary matrices. The task is to count the number of specific "combinations", where "combination" means the following:
this is ...
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Minimizing $\|Ax-b\|$ given that x can only take values 0 or 1
earlier I stumbled upon a question about finding a vector x that minimises $\|Ax-b\|$ where A is a known matrix and b is a known vector. However, I was wondering whether this can be achieved under the ...
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How to make this conversion from a binary integer linear program to a quadratic program?
I saw a conversion from a binary integer linear program (BLP) to a quadratic program (QP) in this link https://qr.ae/psu9Wr. I will repeat the problem below. The original problem is
\begin{align}
\...
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Convert 0-1 integer linear program to quadratic form.
I am searching for a general conversion from 0-1 integer linear programs to (integer) quadratic programs. And I see this answer using a general example. https://qr.ae/psu9Wr. I checked the optimality ...
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Explanation of multiple constraints from one rule [closed]
I'm trying to understand this case study:
https://github.com/DorisRipley/Art-Exhibition-Optimization-A-BIP-Modeling-Approach/blob/main/Art%20Exhibition%20Optimization.pdf
and I'm having trouble with ...
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Why this ILP and LP are equivalent?
Let's consider a competition with $n$ questions. Each question has a price $p_i$ and a score $v_i$. To advance to the next round of the competition, we need to accumulate a minimum score of $D$. We ...
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Dual of LP representation of graph coloring
I have found a representation of the graph coloring problem as an ILP. Given a graph $G = (V, E)$.
Let $C$ represent the set of colors. Let $w_c$ be a binary variable that is $1$ if the color $c$ is ...
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Binary solution to least squares linear regression [closed]
I am looking for a closed form solution $x^*$, binary vector, to
$$\arg\min_{x}(\|M x + b\|_2),$$
restricted to $x \in \{ 0,1 \}^n$. Here $b \in \mathbb{R}^{m}, M \in \mathbb{R}^{m \times n}$ are ...
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How to linearize or reformulate an implication constraint that implies that a decision variable belong to an interval?
I am an electrical engineer who is working in computer network and I need to model my delay with respect to a binary variable $x$ as folow
$\left\{ {\begin{array}{*{20}{c}}
{x = 1 \Rightarrow \left( {...
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Solving an SAT problem with objective
I have 8192 bits, denoted $b_0, b_1, ..., b_{8191}$. The bits are subject to some XOR constraints (e.g. $b_0 \oplus b_3 \oplus b_{42} \oplus \cdots \oplus b_{8191} = 1$). The objective function to be ...
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Formulation of an integer linear programming problem
I want to formulate the following problem in an integer linear programming problem:
We have $n$ elements $m_1,\dots, m_n$ elements with $m_i = (m_i^1, \dots, m_i^p) \in \mathbb{R}_{\geq 0}^p$ for all $...
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Constraint Integer Linear Programming
I have an integer linear programming problem where i want to maximize over $\{0,1\}^n$, so i have the problem $$\max_{x \in \mathbb{R}^n}c^Tx, \text{ subject to } x_i \in \{0,1\} \text{ for all } i ...
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Counting the number of binary solutions to system of equations by finding the coefficient of a term in a generating function
I am trying to solve the number of binary solutions to a system of linear equations, the same as in this question: number of binary solutions under linear restrictions.
Shortly:
Consider $ x1,…,x_n ∈ ...
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Approximating the Max Cut Solution of a Graph by the Graph Laplacian
Given a weighted graph with $n$ vertices and weights $w_{ij}\geq 0$, the max-cut problem is equivalent to
$$
\max_{x \in \mathbb{R}^n} \sum_{i,j} w_{ij} (1-x_i x_j) \quad \mbox{s.t.} \quad x_i \in \{-...
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How to reformulate or linearize the phrase "become redundant" or "not needed"?
I am an electrical engineer and currently I have to deal with an optimization problem with a very specific requirement:
$\begin{array}{*{20}{c}}
{\mathop {Min}\limits_x }&{f\left( x \right)}\\
{{...
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Condition of constraint in BILP
I have a set of $n$ activities $a_i \in A$. Each activity $a_i$ has a set of child activities $C_i \subset A$, an associated cost $k_i$, and a start time $t_i$.
In the BILP model I use, I consider a ...
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Modeling AND of implication in integer/binary linear programming
Problem statement
Let $\beta \in \{0, 1\}$ for brevity. A set of $K$ numbers $M_k$, represented as individual bits $B_{ik} \in β $, must be distributed to a set of $ J \le K$ pairs $F_j = (c_{ij} \in ...
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Binary program that maximizes ratio of quadratic forms
I'd like to solve the following optimization problem. Given $\mathbf a, \mathbf b \in (0, \infty)^n$, find $\mathbf x \in \{0, 1\}^n$ which maximizes
$$ f (\mathbf x) = \frac{\left( \sum\limits_{i=1}^...
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Model legal shift constraints
I need your help.
My decision variable $b_{fds}$ indicates whether a fireman $f$ works shift $s$ on day $d$. I need two constraints:
a) No more than 5 consecutive working
days
b) At least 2 ...
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How to linearize a weighted average using Pyomo?
I'm learning about linear optimization problems using Pyomo. At this time I'm looking to practice and ended up getting stuck in a constraint that contains the weighted average formula. Using this ...
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Defining binary variable
I am currently working on my modeling skill and I wanted to try to find a linear constraint, that models a binary variable in a specific way. The new binary variable $\gamma_i$ should take the value $...
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How to ensure an increase happens a certain interval after the last decrease in a variable?
I have three binary variables x, y, and z each indicating an increase, decrease, and stable values of another variable P. Now the issue is I want to formulate a constraint that makes sure that a y can ...
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Modelling a shift change
I have the following problem. I am currently modeling shift schedules. There is the variable $x_{itk}$ which tells whether the cashier $i$ completes the shift $k$ on day $t$. Now I want to model the ...
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Problems to find a suitable formulation for a constraint
I have the following problem. I would like to model the following relationship. I have three binary variables $x_1$, $x_2$ and $x_3$. These become either 0 or 1. I need a constraint for a mathemstical ...
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How to do i prove that this inequality is valid in the given set?
How do I prove that $$z_1 + x_{12} + x_{22} + x_{13} + x_{23} + z_4 \geq 2$$
is a valid inequality for my constraints:
\begin{align}
x_{11} + x_{12} + x_{13} &\geq 1 \tag1\label1\\
x_{12} + x_{13} ...
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Placing number blocks so that the resulting matrix is symmetric
There are some number blocks given as follows:
The aim is placing these blocks in such a way that the resulting $4\times 4$ matrix is symmetrical. Blocks cannot be rotated, they must be used as given....
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Does a (5,3,4)-code exist?
I am a bit confused on whether a binary (5,3,4)-code exists.
As far as I am aware, this code exists if and only if a binary (4,3,3)-code exists according to Theorem 2.7 in Raymond Hill's book "A ...
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How to model this constraint linearly in binary integer linear programming?
I have a directed acyclic graph, and two binary decision variables:
$a_{ij}$, which is equal to one when the corresponding edge between the nodes $i$ and $j$ of the graph is selected, and zero ...
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Linear programming combination of variables
I am trying to formulate the following linear programming problem.
My inputs are the following:
A set of $N$ tables $\Pi_1, \dots, \Pi_N$
A cost budget $G$
I have the following decision variables:
$...
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Linearizing product of binary variables
How would I linearize the following expression
$$ z = (1-x)y $$
where $x,y \in \{0,1\}$? Ideally, I would want to formulate this as a system of linear inequalities.
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Conditional constraints in Integer Linear Programming
I think it's rather a simple question. I'm trying to construct a reduction from graph problem to ILP. When I have variables $x_1, x_2, \dots ,x_n \in \{0, 1\}$ for every vertex, can I create ...
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Linearize Min Max Index in List as Constraint
I'm trying to solve an optimization problem by creating an optimization model (which I shall solve using CBC solver) and I need to linearize it. Please help me to reformulate it :
Given Data : A1, A2, ...
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Triangles in a graph via LP
I have a linear program and I can't formulate the objective function and constraints.
For a graph $G = (V, E)$ we may select a set $S$ of vertices of $V$. Each vertex carries a cost $c_v > 0$ if it ...
1
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1
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Minimizing a quadratic function with binary variables and a totally unimodular constraint
Let $q=q(x_1,...,x_n)$ be a quadratic polynomial. I want to solve the following optimization problem:
$$\min_{Ax = b, x\in \{0,1\}^n}(q)$$
where $A$ is totally unimodular. Is there some neat algorithm ...
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Nonlinear discrete and continuous optimization problem
I'm trying to minimize a cost function that is made up of dependent binary variables and continuous variables.
For example the cost function could look like:
$F(x_{0}, x_{1}, x_{2}, r_{0}, r_{1}) = 0....
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How can I apply the McCormick Envelopes to the product of two binary variables?
I've seen the McCormick envelopes applied many times to the product of two continuous variables, but I can't seem to find when both of them are binaries. Also, I applied the restrictions as described ...
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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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About the greedy solution for a specific linear-fractional programming problem
Let $a_i>0, b_i>0, \forall i $.
The optimizing problem is
$$\max_{x_i}(\frac{\sum_{i=1}^{n}x_i a_i b_i}{\sum_{i=1}^{n}x_i b_i})$$
with constrains:
\begin{align}
x_i \in \{0,1\}\\
\sum_{i=1}^{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.
...
3
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1
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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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How do I transform the following set of conditions into inequalities?
I've been working on a mixed integer linear program for quite a while now and I need to set up constraints involving binary variables. I just can't find the correct answer to the following problem. ...
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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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