Questions on optimization constrained to integer variables.

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60 views

How to think about minors of the rectangular matrix in the context of a system of Diophantine linear equations

My question is related to my previous question How to prove existence of solutions to the system of Diophantine linear equations. In particular, to the theorem which I've used to prove some subset of ...
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1answer
81 views

Can you generate math problems that are solveable?

If you take Linear Programming, it problems are formulated like this: You know that Cabinet X costs 10 cents per unit, requires 6 square feet of floor space, and holds 8 cubic feet of files. ...
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2answers
48 views

Fog of War Influence Map

I'm in the process of making an RTS game and I've ran into a problem I could use help with. I want to create a fog of war system that reveals an area around a unit that belongs to you like how it's ...
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1answer
28 views

Replacing a max constraint in a binary program

For $x \in \{0,1\}$, I want to express $x = 1 \Leftrightarrow \exists k: y_k = 2$ where $y_k \in \{0,1,2\}$, i.e. $x \leq 0.5\max_k\{y_k\}$ using binary decision variables but I can't figure out how ...
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1answer
34 views

Integer linear programming problem

There is an integer linear programming problem with this constraint: $$\left|X1 - X2\right|= 5\text{ or }10\text{ or }20$$ How it can be solved with adding auxiliary variable $y$? Main problem is ...
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1answer
89 views

Mixed Integer Linear Programming: Construction Rods

I have an interesting problem involving linear programming. The problem is the following, I have 4 different kinds of rods (rod sized found in the local market): 9m rod 11m rod 12m rod 15m rod ...
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0answers
247 views

Indicator Variable, Mixed Integer Linear Programming

Assume $x$ is a real variable, and $0\leq x \leq1$. Besides, $y$ is a binary random variable. I need a linear program that: if $y$ is $1$: $x>0$, if $y$ is $0$: $x=0$ I know the following ...
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70 views

Solution techniques for optimization problems

I am very new to solving such optimization problems. Following is the problem, I need to know the various methods (preferably advanced machine learning techniques) that I can use to solve this. ...
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2answers
88 views

Integer Linear Programming

Without using a computer, I have to solve the following integer linear programming:$$\min \quad x_1+x_2+x_3$$ $$\operatorname{sub} ...
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1answer
86 views

What kind of algorithm might solve this type of optimization problem?

I am trading futures contracts in baskets at ratios that I compute by some method. Suppose there are $n$ contracts in a basket, and the ratio is given by $\mathbf{r}\in \mathbb{Z}^n$, so that the ...
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4answers
64 views

Can I find solutions to $a^4 + a^2 + a = b^2 + b$, $a,b \in \mathbb{Z}$ and $ 1 < a < b$?

I was wondering if anyone could point me in the correct direction for either finding a solution to my problem or proving that it does not exist. $$a^4 + a^2 + a = b^2 + b \;\text{ for }\; a,b \in ...
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178 views

Wrong ILP solution with LPSolve (simple example)

I added the following example into LPSolve and found a strange issue. I don't want S1 and S2 to overlap within certain margins. ...
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1answer
207 views

Show both relaxations of boolean LP give equal lower bounds

Given the boolean LP: $$\text{Minimize}\;\; c^Tx$$ $$\text{Subject to}\;\; Ax \leq b$$ $$\hspace{57mm} x_i(1-x_i)=0\;\; i=1,...,n$$ Show that the LP relaxation: $$\text{Minimize}\;\; c^Tx$$ ...
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1answer
55 views

How is D.Knuth MMIX instruction MXOR useful for finite field multiplication?

Knuths TAOCP, Fascicle 1, exercise 37 (page 26) - http://mmix.cs.hm.edu/doc/fasc1.pdf: Explain how to use MXOR for arithmetic in a field of 256 elements; each element of the field should be ...
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0answers
52 views

How do you calculate Up and Down Penalties on a Branch and Bound algorithm of a MILP?

My notes really don't explain this clearly at all, so I have no idea what to do. If I have the following MILP: In which I've been told to solve it using: (a) Rule 1 (choose the variable with the ...
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0answers
27 views

What decides the structure of the dual variables taken in designing min-max type combinatorial optimization algorithms?

There are a bunch of combinatorial optimization problems like min cost flows and min weight perfect matchings that invoke duality and complimentary slackness to improve the primal feasible solution. ...
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53 views

Solution of a linearly constrained quadratic programming problem

What is the solution of the following optimization problem: \begin{align} &\min{\mathbf{p}^\mathrm{T} \mathbf{B} \mathbf{p}}\\ &\text{subject to}: \mathbf{0}\leq{\mathbf{p}}\leq \mathbf{1}. ...
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1answer
33 views

Job scheduling to minimise squared completion times using mixed 0-1 quadratic program

I have come across an Optimization question as follows: There are $n$ jobs that have to be processed on a machine. The machine can process only one job at a time. The time taken to process job ...
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3answers
444 views

How to divide natural number N into M nearly equal summands?

How to divide natural number N into M nearly equal summands? For example, to divide 20 by 13, in geometric representation, I should get How to generate the sequence above? What is the name of ...
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1answer
59 views

Description of a constraint for a mixed integer program.

Suppose we have 100 items that are labelled from the set $P = \{A, B, C, D, E\}$. My constraints are as follows: I want to choose exactly seven items. The choice should have at least one item of ...
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1answer
162 views

matlab MINLP optimization with ga

I have written a program for optimizing a set of generators. I have hourly price and cost data and need to figure out when a generator should run or just stay off. There are additional constraints but ...
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1answer
201 views

integer linear programing in matlab with the symbolic toolbox

I am writing a program to optimize a set of generators. I have hourly data and but dont want to necessarily optimize the whole time series. For a similar problem in the past I used the symbolic ...
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1answer
106 views

A binary min-max optimization problem

I encountered a very special optimization problem for a practical application. We have a variable $$\mathbf{s}=(s_1,s_2,s_3, s_4)^T$$, where $s_i$ can only take $1$ or $-1$, and we also have a ...
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1answer
81 views

Which algorithms are commonly used to solve this kind of Binary Integer Programming problem?

I want to solve the problem of minimizing $$\mathbf{c}^T\mathbf{x}$$ subject to the condition that $$A\mathbf{x} = \mathbf{b}\text{,}$$ where $\mathbf{b},\mathbf{c}$ are given vectors in ...
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2answers
140 views

How to find smallest integer which is greater than N positive primes

I know this can't be computed exactly, but I just need a rough estimate. I know one can compute a rough estimate of the number of primes less than N using the famous formula: ...
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2answers
56 views

Escaping from a point in linear programming

Is there a trick for explaining the following constraint as a set of linear (in)equalities? $$ \sum_{i=1}^n|x_i-a_i|>0, $$ where $a_i$'s are real constants.
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Determining information in minimum trials (combinatorics problem)

A student has to pass a exam, with $k2^{k-1}$ questions to be answered by yes or no, on a subject he knows nothing about. The student is allowed to pass mock exams who have the same questions as the ...
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1answer
29 views

One solution of a diophantine system

How to find one solution of $Ax = b$, where $A$ is a $(m, n)$ matrix and $x$ a vector of size $(n, 1)$. $A$, $x$ and $b$ are matrices of integers entries. How to check whether is a solution exists?
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34 views

Determine largest possible position for integers

I'm sorry if I'm not using the proper terminology but here's my question. When two numbers are multiplied, the position of the largest number can only be at the position of the sum of their operands. ...
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3answers
818 views

Is there any methods to solve for integer solution of a quadratic equation like $ax^2 + bx + c = 0$

Is there any method to solve for integer solution of a quadratic equation like following: $$ax^2 + bx + c = 0$$ where $a, b, c \in \mathbb{Z}$ If not is it possible for the Special case: ? $$x^2 -x ...
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58 views

Combinations of fruits and their “nutrients”

As a computer scientist and not a mathematician, I know not some of the formal language to describe my problem, so I'll present it in a word problem form. Maybe someone can help me hone my search and ...
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1answer
75 views

If-Then Constraint: If X < 4 Then T = 4 -X

All the generic If-then constraints do not seem to be gaining me any insight into this. I would like to form a mixed-integer program with Lingo which can minimize cost given that a series of: When ...
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1answer
42 views

Finding a sequences so the image approximates integers

$x$ is fixed in $[0,1]$, $a_n,b_n,c_n$ are integers and not all of them are $0$ $y(a_n,b_n,c_n) = a_n x^3 + b_n x^2 + c_n x = Y_n$ Find an algorithm to go from $(a_n,b_n,c_n)$ to ...
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1answer
87 views

Existence of graphs when given the degrees of all vertices

My question is: How to decide whether a graph is exist when given the degree sequence of all vertices? This question can be easily reduced to the {0,1}-solutions of integer linear equation ...
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0answers
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Mixed-Integer Linear Programing : get the maximum constant associated to a non null variable

Does anyone know a way get the maximum constant associated to a non null variable using Mixed-Integer Linear Programing ? I would like to get the variable $a$ in this description : $$ i = 1,\ldots,m ...
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1answer
20 views

Some operation like determinant

we have determinant operation that is like below: $ det(\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{bmatrix}) $= $ (-1)^{1+1}a(ei-fh)+ ...
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2answers
86 views

How to find nonnegative solutions of a linear system?

I have a $M$ equation and $N$ variables like this : $ \begin{bmatrix} 3 & 0 & 1 & 0 & -1 & -3 & 2\\ 1 & 2 & 0 & 4 & 0 & 0 & -1\\ 1 & 1 & 0 ...
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0answers
149 views

Algorithm to reduce expressions to canonical form

I'm writing a small computer algebra system that only knows rational numbers and all expressions that you can get from them by using basic arithmetic operations and powers. So the expressions are ...
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0answers
42 views

fill up a square with rectanagles

I have a square 46*46, I want to fill this up with three types of rectangles of size 29*20, 21*5, 10*4 such that the free space can be minimized? any rectangles can't be overlapped and they can be ...
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1answer
59 views

Integer Optimization

I have an integer optimization problem that I've been pondering for the last several days. Here's an abbreviated version: I have several wav song files with variable sizes (601201 kilobytes for ...
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1answer
44 views

Number of Integer solutions for this optimization problem

What is the number of integer solutions to the problem $$\sum_{i=1}^{i=k}x_i = n$$ subject to $\forall_i\ \ x_i \ge 0 $ note This should hold for both cases $k < n$ and $k \ge n$
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Find relationships between events

I have a set of Events $(E_i)_i$ which have a probability $(P_i)_i$. I am able to write each event as a sum of distinct events that form a partition of the space. My goal is to find all the ...
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2answers
53 views

$a^2-b^2 = k$, $ab = l$ for fixed integer $k,l$ when $a,b$ are both integers

Let us fix integers $k,l$. Let all numbers be integers. Now we want integer $a,b$ to satisfy: $$a^2-b^2 = k, \,\,\,2ab = l.$$ We want to maximize the number of possible $(a,b)$. In order to do ...
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1answer
67 views

How to minimize the square of $\sum b_i x_i$ where each $b_i$ can be either $0$ or $1$?

Would you please help me solve the following problem where $b_{ij}$ is my decision variable that must be determined and all other parameters are known. $$\min \left(\sum_{i=1}^n b_i x_i\right)^2$$ ...
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1answer
42 views

Can there be a unique natural number vector solution to $Ax =b$ where $A$ is not a specific type of square matrix?

Let $A$ be $(n-1) \times n$ matrix that is of the following form: $$\left( \begin{array}{ccc} n-1 & 1 & 0 &.... & ....\\ 0 & n-2 & 2 & .... & ....\\ 0 & 0 & n-3 ...
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2answers
37 views

Determine page number for question based on count of questions and number of questions per page

I'm trying to build a simple quiz, and I want to split up the questions into pages. However, the number of questions and pages can change based on a few parameters, most important is a variable number ...
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2answers
222 views

Free solvers in C/C++ for convex integer programming

I need to solve the following integer program: $\text{minimize } \sum_{i=1}^n(a_{i0} x_i + \sum_{k=1}^3 a_{ik}w_i^k + \sum_{j=1}^m d_{ij}y_{ij})$ $\text{subject to}$ $$ \sum_{i=1}^n y_{ij}=1, \quad ...
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1answer
45 views

Question about making a custom 'formula'

I am working on some program for myself, and I am stuck on one thing, which I am not sure how to make exactly, I do know how to calculate it manually on paper, but I don't know any kind of formula or ...
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84 views

Traverse resultant 2D array after integer partition

I have used the solution of integer partitioning using dynamic programming explained in this post and in this article. Following is the resultant matrix when $N$ is equal to $6$: $$\begin{bmatrix} ...
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Shortest path problem: dual formulation and proof of total unimodularity

The IP formulation of the shortest path problem looks as follows: \begin{align*} \min & \sum_{u,v \in A} c_{uv} x_{uv}\\ \text{s.t } & \sum_{v \in V^{+}(s)} x_{sv} - \sum_{v \in ...