Questions on optimization constrained to integer variables.

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2
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1answer
22 views

Model cost for a state change in an integer program

I have a problem involving tool selection I am trying to model right now. (I am fairly new to this). I have a series of manufacturing operations I need to perform for $i \in \{1,\dots,n\}$. Each ...
0
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1answer
44 views

Maximum Coin Changes That Does Not Add To a Dollar

What is the maximal amount of money attained from coins of 1, 5, 10, 25 cent denominations that none of its subset amounts to 100 cents? We can find the solution with exhaustive or naive dynamic ...
0
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1answer
28 views

Sign of a Linear Expression in Mixed Integer Linear Programming

I have a question about a system in mixed integer programming I am trying to solve using GLPK though the question mathematical and is software agnostic. I have converted the whole problem to linear ...
1
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0answers
27 views

Optimizing the product of two integers

How to efficiently solve the following optimization problem? Find $a$ and $b$ which minimize $ c = a * b $ under the constraints $ c \geq C $, $ a \leq A $, $ b \leq B $, where a, b, A, B, C are ...
2
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1answer
68 views

Finding the nonnegative integer exponents that minimize a product

I've been trying to solve a problem which seems to be a multiplicative optimization problem: Given a threshold $T > 0$, and a set of integers $b_1, b_2,\dots, b_n > 0$, find integer ...
8
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4answers
561 views

Are there ways to solve equations with multiple variables?

I am not at a high level in math, so I have a simple question a simple Google search cannot answer, and the other Stack Exchange questions does not either. I thought about this question after reading ...
1
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1answer
73 views

Can this optimization problem be solved?

I am working on an optimization problem but I am not sure if the problem can be formulated as an integer programming problem. Assume the cost minimization problem for a set of subscribers and ...
0
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1answer
32 views

How to find the analytical solution of this optimization problem?

I have an optimization problem of the form $$\begin{align} \text{maximize}\quad&\sum_{i=1}^{k}\sum_{j=1}^{n}w_{ij}x_{ij}\\\text{s.t.}\quad \quad\quad\,\,& \sum_{i=1}^{k}x_{ij}\leq 1,\;\forall ...
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0answers
20 views

Sensitivity analysis on zero-one Knapsack problem

I have a zero-one Knapsack problem as follows: $$\max \sum_i e^i x_i+\sum_j f^j y_j$$ $$\text{s.t.} \quad p\sum_i x_i+c\sum_j y_j \leq b$$ $$x_i, y_j \in\{0,1\} \quad \forall \ i,j$$ where $e$ and $...
0
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1answer
16 views

Formulating a problem involving sets with ILP

Consider set $\mathcal{G} = \{G_1, \ldots, G_K\}$. We are given $\mathcal{A}_i \subset \mathcal{G}$, $i \in \mathcal{N}= \{1,\ldots, N\}$ and for each $\mathcal{A}_i$, there is a corresponding cost ...
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0answers
18 views

Convex hull possesses only integer extreme points

I have the following question. Consider given natural numbers $ 1 \le l_m <\ldots < l_1 < L $. Is it possible to prove that the convex hull of $ \left\lbrace a \in \mathbb{Z}^m_{\ge 0} \, \...
3
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2answers
109 views

Find all solutions to $2 x + 3 y + 4 z = 10$

I do not have a background in math, and am wondering what type of question this is. I looked combinatorics optimization, and the knapsack problem, but found the vocabulary too dense. The problem: ...
1
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0answers
11 views

Why does the Dantzig cut require the constraint data to be integral?

Given the following integer linear program, (ILP) $\min c'x$ subject to $Ax \ge b, x \in \mathbb{N}_0$ where all elements of $A$ and $b$ are integral, and assuming its linear-program relaxation (...
0
votes
1answer
38 views

System of equations with multiple answers, but only natural numbers (and other constraints)

Okay so basically I want the solution with the smallest sum when you add up the integer variable solutions that give a solution to: $975a + 880b + 790c + 585d + 487e + 440f + 292g + 260h + 530i + ...
1
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1answer
18 views

How to determine lowest integer multiple for any given decimal fraction

In the equation a * b = c, given a, how can I find the lowest integer c provided that: <...
3
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2answers
65 views

Packing squares in a rectangle

I need to pack $N$ identical squares arranged in array fashion, $R$ rows of $C$ columns, the last row possibly incomplete, in a $W\times H$ rectangle (reals). The squares must be as large as possible. ...
1
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2answers
29 views

Formulating an optimisation problem into a mixed-integer problem

I'm not sure if I understand this question and was wondering if anyone could provide any insight to an answer. The only thing I can think of adding is a constraint: "x2 = integer", so I'm clearly ...
0
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0answers
23 views

How to linearize a double sum of product of binary variables?

I have a double summation of the form $$ x_{kn}\sum_{k'\in K}\sum_{n'\in N} x_{k'n'} A_{k'n}\leq B_{kn},\quad\forall\; k\in K,n\in N $$ where $x_{kn}$ is a binary variable. How to linearize this ...
0
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1answer
26 views

When modeling a multi-objective problem, is there a simple way of choosing to fully minimize one function, then to go on and minimize the second?

I am modelling a problem where I have two objectives. My goal is to fully minimize the first objective function, then choose among the solutions that fully minimized the first objective function to ...
0
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0answers
14 views

Solving Binary Linear Programming Problem Using KKT

Execuse me, I know that if I searched a lot I could find the answer, However I have already did my research and I am running out of time. I need the detailed solution of the following linear problem (...
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2answers
31 views

Binary optimization problem

I am facing the following problem: Let P be a fixed m x n finite matrix and D be a matrix of ones and zeros with the same dimensionality as P plus the following constraints: sum of row entries <=...
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2answers
40 views

Trying to sell the most batches of animals using linear programming

I'm trying to sell the most batches of animals... Let's say I have 200 dogs, 100 cats, and 100 ferrets. ...
1
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1answer
39 views

How to convert a non-linear constraint to a linear constraint for integer programming?

I have non-linear scheduling model and I want to convert it to a linear model. But I have no idea about how can I do it. The nonlinear constraint is: For each $i, i'\in I$ and $j, j' \in J$ and $q, ...
0
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1answer
49 views

How to linearize the following constraint on abs terms with coefficients of mixed signs

I am implementing an optimization program on 2 variables with a constraint of the form: 2*|x1| + 3*|x2| <= 2.25 * (|x1| + |x2|) Given that the effective coefficients on the two abs terms are + ...
0
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1answer
20 views

Why this scheduling MIP model is not working?

I have an integer programming model for Parallel Machine Scheduling. The parallel machine scheduling problem have $i$ jobs, $j$ process and $k$ number of machines. Each processes has to be done in ...
0
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0answers
13 views

Existence of solutions for a scaled integer linear inequality

Assume that I know there exist non-negative integer solutions to a linear system of integer equations (with coefficients from $\{-1,0,1\}$ and non-negative constant terms in my case). Is there any ...
2
votes
1answer
87 views

Integer Programming (formulating a problem)

The Record-a-Song Company has contracted with a rising star to record eight songs. The durations of the songs are 8, 3, 5, 5, 9, 6, 7, and 12 minutes, respectively. Record-a-Song uses a two-sided ...
3
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1answer
31 views

Transforming a $0$-$1$ knapsack problem into the standard form

I have the following $0$-$1$knapsack problem: $$\begin{align*} &\mathrm{Max} : \quad z= 3x_1 -4x_2+5x_3+7x_4-6x_5+x_6\\ & \mathrm{subject\ to}: -2x_1 +x_2 +10x_3 +3x_4 -5x_5+12x_6 \leq 4 \...
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0answers
9 views

Correct definition of the co-occurrence graph of a pseudo-Boolean function

In section 4.6 of Pseudo-Boolean Optimization, Boros and Hammer have defined the co-occurrence graph of a pseudo-Boolean function as follows. If a pseudo-Boolean function $f : \mathbb{B}^n \mapsto ...
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1answer
15 views

Tree-width of a quadratic pseudo-Boolean function

A pseudo-Boolean function $f : \mathbb{B}^n \mapsto \mathbb{R}$ is of the following form. $$ f \left(x_1, \ldots, x_n\right) = \sum_{S\subseteq V} c_S \prod_{j \in S} x_j $$ Here $c_S \in \mathbb{R}$,...
0
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1answer
15 views

Mixed integer linear programming - making a variable have influence on another variable only when it is equal to zero

Having an integer variable $D \in \{-k,-k+1,...,-1,0,1,...,k-1,k\}$, how do I make it affect another variable only when $D = 0$? Specifically, I have a binary variable $U$ in my model. I want it to ...
1
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1answer
15 views

Mixed Integer linear programming - absolute value of a variable not involved n the objective function

I'm looking to find the absolute value of the expression s-t. I have begun by introducing the following constraints: Where A is the absolute value. Unfortunately, A is not involved in the objective ...
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0answers
27 views

Kinds of logic and constraint programming

I am currently solving combinatorial optimisation problems using integer linear programs (ILP), and I would like to try something different (constraint satisfaction, logic programming, ...). I tried ...
2
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0answers
18 views

Does it make sense to compare sets (polygons) with different dimensions?

In the context of integer programming, I am considering 3 different linear models for a given problem. The goal is to determine which formulation is the tightest, that is, the one that gives the least ...
0
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1answer
22 views

What is the class of this Integer programming prob.

I have an optimization problem which seems to be non-linear because of the constraints (right?): $max (\sum U_i\times x_i)\\ \sum x_i\times y_i\times r_i\leq R\\ \sum y_i=1\\ \sum x_i=1\\ x_i, y_i\...
0
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1answer
23 views

Lower bound for third order polynomial over integers

I have the following polynomial: $P(a,b,c,d) = -2 a + 3 a b + 3 a^2 b + 6 a c + 6 b c + 2 d + 3 a d - 3 a^2 d - 3 b d - 6 a b d - 3 d^2 + 3 b d^2 + d^3 \;,$ where $a$, $b$, $c$, $d$ are integers $\...
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3answers
59 views

Integer solutions to $xyz = w^2(x+y+z)$

I'm looking for a way to enumerate all positive integer solutions of the equation $xyz = w^2(x+y+z)$ where $w \le W$ and $1 \le x \le y \le z$. Could anyone provide a hint at how to approach this?
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0answers
12 views

Pseudo-Boolean functions restricted to integers

The Pseudo-Boolean functions are of the following form. $$ f : \mathbb{B}^n \to \mathbb{R} $$ I would like to know if there is a special sub-category of $$ f : \mathbb{B}^n \to \mathbb{Z} $$ with ...
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0answers
8 views

What is the best function to increment a cycling counter over a finite (binary) integer representation?

What is the best function to increment a cycling counter over a finite (binary) integer representation? Edit 1 : i found this https://en.wikipedia.org/wiki/Ring_counter#Johnson_counter Edit 2 : ...
0
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0answers
24 views

Linear integer programming

I am trying to find the optimal solution for the following linear integer programming: \begin{eqnarray} &&\underset{x_i, \forall i} {\text{maximize}} ~ \sum_{i=1}^N x_i a_i \\ && \...
0
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0answers
28 views

What are the steps to optimize simple Sphere function

I am new to optimization. How can one optimize the simple test function, Sphere function available on https://en.wikipedia.org/wiki/Test_functions_for_optimization?
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0answers
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Quadratization of $5 x_1 x_2 − 7 x_1 x_2 x_3 x_4 + 2 x_1 x_2 x_3 x_5$ using Rosenberg's algorithm

In section 4.4 of Pseudo-Boolean Optimization by Boros et al., the authors have reproduced Rosenberg's quadratization algorithm as follows. Then they have given an example of implementing the ...
0
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1answer
32 views

Quantifier in Integer Programming/Logic

It is common to write constraints with something like $x_s \leq y \quad \forall s \in X$ In Integer Programming, where x_s and y is a variable. However my tutor said that this not so absolutly ...
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0answers
31 views

Integer programming with linear constraint

I am trying to find the optimal solution for the following problem \begin{eqnarray} &&\underset{x_i, ~y_i ~\forall i} {\text{maximize}} ~ \sum_{i=1}^N x_i f_i(y_i) \\ && \text{s.t.}...
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0answers
57 views

Directed Weighted Graph with no cycles - LP

I have directed weighted graph. I have to find a set of edges with minimal sum of their weights that without the set graph becomes acyclic. I can call lp solver multiple times. I'm kind off lost on ...
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0answers
3 views

How do we accomplish the subtraction of two infinities in a PWL Approximation?

I am trying to implement a piecewise-linear function of an M/M/1 Queueing system in an ILP to approximate the delay values. I have expressed my PWL constraint as follows: $\alpha_{i}+ \beta_{i}u_{n_s} ...
3
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0answers
56 views

Candy Crush as an integer programming problem

I'm trying to model the basic version of a match-three game, where the player (has a maximum number of swaps) must swap any two adjacent gems (no diagonals) in an 8x8 grid of gems in order to match ...
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0answers
34 views

Model linearly: Determine amount of units for production

A company produces 2 products in a week. Let $x_i$ denote the number of units of product $i$ to produce. Each product requires liters of Chemical X to make. Info is given below: \begin{array}...
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0answers
40 views

Weights in goal programming

I'm not quite convinced about assigning weights in goal programming. Here is an example formulation problem. What I tried: Let $x_j$ be the number of minutes for ad $j = R, T$ We want to ...
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0answers
87 views

What programs or websites solve linear integer or goal programming problems?

I don't think I can use Excel. My solver doesn't work so I can't even use Excel for regular linear programming. Something like this but for integer or goal programming. This seems to allow integer ...