Questions on optimization constrained to integer variables.

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1answer
294 views

How to linearize this constraint a summation of a product of a integer with a binary

I have to linearize the following constraint, $$ \sum_{i \in V_C} \sum_{j \in V} \sum_{k \in K} y_{ik} \cdot x_{ijk\ell} \leq I_\ell \qquad \forall \ell \in V_D $$ where $y$ is a integer variable ...
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1answer
61 views

Operations Resarch Optimal Scheduling

Consider the following problem: A car manufacturing company needs to transport car frames, which are $10$ cubic units each, and wheels, which are $2$ cubic units each, across the Atlantic ocean. ...
0
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1answer
45 views

minimising quadratic function subject to integer solutions

I would appreciate if one could help me to solve this problem. I have a bivariate quadratic function: $$ f(a_1,a_2)=(1-u_1^2)a_1^2 +(1-u_2^2)a_2^2 -2u_1u_2a_1a_2 $$ where $u_1^2+u_2^2=1$ and $a_1$ ...
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1answer
279 views

Solving an integer linear programming problem without a graph

I am new to linear prorgramming and so far I have been solving LP problems with the help of a graph solution. However, when there are more than 2 variables obviously I can't plot them on the graph. ...
2
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1answer
73 views

Chocolatier sampler boxes problem: applying goal programming and mixed-integer programing to optimally compromise goals.

QUESTION: A boutique chocolatier is planning to make a number of sampler boxes, each containing $36$ chocolates. (Therefore the total number of chocolates should be divisible by $36$.) The ...
2
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1answer
96 views

Help with Math software (macaulay 2)

I just started working with Macaulay 2 and need some help. I need to find the number of solutions of a system of equations. I am having difficulty imputing this into the software so please be specific ...
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0answers
33 views

Binary depending on the sign of another variable

I'm writing a mixed integer linear problem, where I have an indicator function in the objective function counting the instances of negative values of a decision variable. I thought of defining a ...
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0answers
23 views

Book recommendation on integer programming ? (in order to solve a set cover problem)

I'm trying to solve a set cover problem. To put it shortly, my problem is about covering a $N \times M$ grid, by using various rectangles which have associated cost depending on their shape and ...
1
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1answer
299 views

How do I convert a constraint with a product of two integer variables to a linear constraint?

I have a constraint of the form: $$\theta \leq a_1x_1 + a_2x_2 + a_3x_1x_2$$ where, $x_1$ and $x_2$ are integer variables with ranges $x_1 \in \{0, m\}$ and $x_2 \in \{0, n\}$. I would want to ...
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1answer
92 views

How to solve the coupled integer programming problem?

I have the following integer linear programming problem: $$\begin{equation*} \begin{aligned} & \underset{x}{\text{maximize}} && \sum_{k=1}^K\sum_{t=1}^Tx_{kt} \\ & \text{subject to} &...
0
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1answer
65 views

Solution of the LP relaxation - always round to the nearest integer?

If an optimal solution to the LP relaxation of an IP is not integer, can we always get a feasible IP solution by rounding it to the nearest integer? Or can we generalize this process by saying, if we ...
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0answers
24 views

Is there any general algorithm to solve such a 3D cutting problem?

Suppose a cuboid $\mathbb{A}$ has $L$,$M$ and $N$ as its length, width and height respectively, where $L\ge{M}\ge{N}>0$; Now we want to cut $\mathbb{A}$ into smaller cuboids with length $x$, width $...
2
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1answer
62 views

On/off variables in MILPs with infinite bounds

I have an LP defined by $$A x = b$$ $$0 \leq x \leq U$$ and would like to extend it to an MILP through introduction of binary on/off variables $z$ such that $$z_i = 0 \implies x_i = 0.$$ This ...
-1
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1answer
203 views

Linear Programming constraint equivalent of conditional

I would like to use the following conditional in my linear program: if(A == 1) then B = C + 1 A = binary, B and C are continuous. In the else case, any relation ...
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0answers
193 views

Global and local maxima in a weighted sum of logarithms of linear functionals?

Is is possible to describe, and locate efficiently, the maxima of the function below in the parameters $\mathbf{x}$ $$\sum_{i} p_i \log( N + \sum_j x_j[B_j +(A_j-B_j)\delta_{ij} + min(A_j,B_j) ]) \...
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0answers
21 views

Linear Integer Optimzation Problem (scheduling problem)

Does any of you know how to get this done?
5
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1answer
57 views

Is the optimal solution of a strictly convex function over $\mathbb{Z}^d$ a rounded version of its optimal solution over $\mathbb{R}^d$

Consider a strictly convex function $f: \mathbb{R}^d \rightarrow \mathbb{R}$. Let $x^* = \min_{\mathbb{R}^d} f(x)$ denote the (unique) minimum of this function over $\mathbb{R}^d$. Similarly, let $x_{...
0
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1answer
124 views

convertion into integer linear program

I am trying to model the Ising spin state problem into Integer linear program and find the optimal ground state using lp_solve. (This is just a miniature version of Ising state problem) $$ maximise: \...
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1answer
64 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
82 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 ...
0
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1answer
29 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 ...
0
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1answer
37 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 ...
0
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1answer
90 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
286 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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1answer
72 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. ...
2
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2answers
92 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} :\begin{cases}x_1\le9\\x_2\le7\\x_3\le5\\3x_1+6x_2+8x_3=80\\x_1,x_2,...
2
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1answer
90 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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1answer
193 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
223 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: $$\text{Minimize}\;\; c^Tx$$ $$\...
0
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1answer
62 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
53 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 ...
2
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0answers
28 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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0answers
54 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}. \...
1
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1answer
35 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 $i$...
3
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3answers
481 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 ...
2
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1answer
62 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 ...
0
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1answer
171 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 ...
0
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1answer
223 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 ...
0
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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 ...
0
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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 $\mathbb{F}_2^...
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2answers
155 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: ...
2
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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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0answers
364 views

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 ...
0
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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?
0
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1answer
38 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. ...
2
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3answers
993 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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2answers
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 ...
1
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1answer
78 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 $...