Questions on optimization constrained to integer variables.

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How to minimize $\min_k k \frac{b^k/n}{\lfloor b^k/n \rfloor}$

This problem looks familiar, but I don't remember its solution: $$ \min_k \ \ \frac{b^k/n}{\lfloor b^k/n \rfloor}k $$ subject to $$ b^k \ge n \\ b,n,k \in \mathbb{N} $$ Does it have a name? What's ...
4
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154 views

On the integer feasibility of polytopes defined by idempotent integer matrices

EDIT: I realized that while writing this question, I was reasoning about orthogonal projections. Thus, I forgot to transpose when forming the projection on to the space orthogonal to the image of $P$. ...
3
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111 views

Binary optimization

Let me first make my background clear. I am a PhD student with not much knowledge in optimization but I need to do some optimization as a part of my research work. My problem is as follows: There are ...
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29 views

Find bounded integers $x, y$ minimizing $| t - x * y |$

How do I find the integers $x$ and $y$ minimizing $| t - x \cdot y |$ with $1 \leq x < N$ and $1 \leq y < M$ ? Background: A clock signal is divided by two hardware prescalers (with a limited ...
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57 views

Quadratic Integer Programming

Would anyone mind helping me solve this problem $$ \min\space f(x) = \frac12 x^\mathrm TQx + bx + c \qquad \text{s.t. } \sum_i x_i=\lambda $$ where $x$ is a vector whose entries are positive ...
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87 views

Clarification of variable values in Arithmetic Coding algorithm

I have been trying to follow this video to implement my own Arithmetic Coding algorithm in Java. I am having a bit of trouble figuring out what some of the variables in the video should be. For ...
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115 views

An optimization problem involving Latin Squares

Let $C$ be a given $n \times n$ matrix of real numbers and let $p$ be a given $n$ vector of non-negative numbers such that wlog $\sum_i p_i = 1$ and wlog the $p_i$ are non-increasing. I'll write ...
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60 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} 1 ...
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37 views

Prove this statement about inequalities

Can someone help to prove this. For $n$ and $\{a_{11},\dots,a_{nn}\}$, if we know that $a_{ij}$ is either $0$ or $1$ or $-1$, and further assume that the following inequality system on $\{b_n|b_n\in ...
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44 views

Minimising waste in a cutting problem.

I have three possible board sizes: $8$, $10$ and $12$ feet long. I want to make some number of cuts to these, say, $3, 2,1,1,1,6,5,3,4,2,1$ feet cuts and I want to minimize waste. I've done a quick ...
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39 views

Program for Handling Huge Primes

I am trying to run a program with really large primes (around the $10^{20}$th prime), but Mathematica seems to only be able to handle around the first $10^{12}$ primes. Is there any software that can ...
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65 views

Help solving this linear (?) programming problem with odd integer constraints.

I would like some help writing the following linear (integer? quadratic?) programming problem in matrix form including the application of the constraints. I am drawing a dashed line around the ...
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44 views

How to minimise an objective function which is not a direct function of the decision variable?

I have a problem with partitioning a water network by closing some pipes. I use some graph theory techniques to find some candidate pipes to close; but to select which pipes among them to close (my ...
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23 views

Highest (lowest) index of positive time-indexed variable

I have a simple problem involving a variable $x_{it}$ representing the amount of a resource allotted to a task $i$ in time $t$. The quantity of the (renewable) resource is constrained at a value $R$ ...
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75 views

Is 0-1 integer programming always NP-hard?

I have the following problem. Maximize $\sum\limits_{m=1}^M\sum\limits_{n=1}^N x_{mn}$ subject to: $\sum\limits_{\substack{m^\prime=1\\ m^\prime \neq m}}^M\sum\limits_{\substack{n^\prime=1\\ ...
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64 views

Integral Farkas Lemma

The context of this question is commutative algebra, however the question itself is more related to convex geometry. All necessary information is given. In the proof of Lemma 3.1.1 in the book ...
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30 views

integer programming with bounded dimension

We know that integer programming with bounded dimension or fixed number of variables can be solved in polynomial time by Lenstra's result(from results of the LLL algorithm). After heavy foraging i ...
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17 views

Internals of a MIP Solver

I would like to learn about the internals of a Mixed Integer Programming (MIP) solver. Which concepts shall I read about? Are there a couple of standard books which can be a good start?
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162 views

Constraints of a linear programming problem

QUESTION Sandy Arledge is the program scheduling manager for WCBN‐TV. Sandy would like to plan the schedule of television shows for next Wednesday evening. Of the nine possible one‐half hour ...
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43 views

How to express coprimality as a constraint in an optimization problem over integers?

I am currently working with an optimization problem that is defined over a a set of $D$-dimensional integer vectors where each component is bounded by $M$. Let us refer to this optimization problem ...
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38 views

Regression/compressive sensing with non-linear constrains where the coefficients are assumed to be integer or binary {0,1}

The following regression problem $$ \mathbf{y} = \mathbf{A}\mathbf{x} $$ where $\mathbf{y}$ is a $N\times 1$ column real vector, $\mathbf{A}$ is a $N\times M$ real matrix where each column ...
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38 views

How to find best set of customers

I've got a problem with getting optimized solution from this task: We have n customers, which buy articles from m suppliers. There are much more customers and suppliers on the market. We've got sales ...
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169 views

Strange but practical Bin packing problem

I am trying to solve the following MILP through LP solve. A link for the original problem is here I am re-iterating the problem as follows: I am trying to write an application that generates drawing ...
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65 views

Speeding up solution of a binary integer program

To solve the problem of making a "good" schedule for a tournament between N teams, using memories from my (long gone) student days, I expressed it as a binary integer program. With the current set of ...
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34 views

Program to determine the relationship of one variable to several possible variables

Suppose I have a system with several variables a, b, c, d, and x. I am trying to solve for the unknown x. I don't know exactly which of those variables x is dependent on, or exactly how the function ...
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95 views

Why these two problems lead to same answers?

Suppose these two problems: Problem 1: $$\min_{X,P} \quad\max_{1\leq l\leq L-1} \quad {|\sum_{1\leq i\leq N_p}^{N_p}x_ie^{\frac{2\pi l}{N}p_i}| \over {\sum_{i=1}^{N_p} x_i^2}} \quad \equiv \quad ...
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329 views

How does Microsoft Excel Solver's Simplex algorithm deal with integers?

I was wondering how the Simplex algorithm in Excel's Solver deals with integers. From what I understand, the Simplex algorithm is meant to be used for linear programming/optimizations only. Yet, Excel ...
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29 views

If the following LP has an integral solution

I know the constraints matrix A of a linear program "Min cx such that Ax>=b" is totally unimodular. So, the program has integral solutions for integral vector b. If this is also the case for the ...
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93 views

Determining if data can be fit by a continuous piecewise integer-valued polynomial

This question concerns the sequence of integers which form the solution to a particular computational problem. See the bottom for the full formulation; basically, for some value n, $G(n)$ is the ...
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78 views

Problem formulation for maximizing the number of smaller rectangle inside larger rectangle

I stumble upon a problem which i would like to pose it as "Optimization Problem". Given the dimension of larger and smaller rectangle, i would like to find the maximum number of smaller rectangle ...
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215 views

sum of maxima vs the maximum of the sum

Consider the following integer program $$ \begin{align} \max &\sum\nolimits_{i,j} U_i(j)\cdot x_{i,j}\\ \text{subject to}& \sum_{i}x_{i,j}\cdot f\left(i,j\right)\leqslant c_j,& ...
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252 views

Book recommendation on Applied Integer Programming/Combinatorial Optimization/OR

Having some very basic and theoretical knowledge about these topics from my study, I'm looking for a book (or other good sources) that explains the stuff from a practical point of view. On the one ...
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162 views

Optimization via Simulation

I want to minimize and objective function $\hat{B_i}$ $i\in l$, which can be computed by a matlab code (assume $\operatorname{findB}(a, b, c)$ returns $B$. I have the following optimization problem: ...
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148 views

Modeling propositional formulas in integer programming

Say I have an binary integer programming problem: \begin{equation*} \begin{aligned} & \underset{\mathbf{x,y}}{\text{minimize}} & & f_0(\mathbf{x,y}) \\ & \text{subject to} & ...
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17 views

Finding the optimal solution to an ILP, when feasibility is not ultimately required

I have the following problem: I would like to solve an ILP with binary variables, i.e. I have a set of possible items, each having properties like "size" "weight" "value" "age" and so on, in total, ...
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24 views

Finding integer vectors in the column space of a matrix

Consider a given set $S \subset Z$. $S$ is a finite set. Matrix $A \in S^{N \times M}$ is also given. Does there exist an algorithm to find all the vectors belonging to the space Col$(A)\cap S^N$ ...
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19 views

Name search for special Linear Mixed Integer Programm

I am looking for a name for the following question in literature! (and if you know it, then it would be great) I couldn't find it and due to wide audience here, presumably you know more. Thank you ...
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18 views

Good MIP formulation of a timetabling problem

I am trying to formulate a university timetabling problem as a mixed-integer program. The choice variables are binary variables of the form $x(c,s,r)$ which is $1$ if a class of course $c$ is held in ...
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32 views

Finding the largest $3^k$ number less than the natural number

Given the natural number $N$ in binary representation (computer memory). How to obtain the representation of the $N\rightarrow\sum^{M}_{k=0}(a_k\cdot(3^k+1))$ form or, at least, how to find the $M$? ...
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2 views

Multiplying separately the different units,(other multiplication methods)

Is there any technique that performs multiplications taking into account the units separately? For example when you have to multiply 15*13 how can you process the tens separately? In general are there ...
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31 views

Mixed Interprogramm remodeling

for example i have the following problem min z 5 x_1a + 6 x_1b - 3 x_2a + 0 x_2b <= z -3 x_1a + 0 x_1b - 1 x_2a + 2 x_2b <= z x_1a + x_1b = 1 (Constraint say of this group only one variable ...
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27 views

Why is integer programming in fixed dimension easier than in general?

When the dimension is an a priori fixed constant, then integer programming feasibility (the existence of an integer point in a polyhedron) can be decided in polynomial time. If the dimension is not ...
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38 views

Calculate $\lceil \frac{n}{log_2k} \rceil; n \geqslant 1, k \geqslant 2$ with only integer functions

How to calculate following expression with only integer fuctions? $$\lceil \frac{n}{log_2k} \rceil; n \geqslant 1, k \geqslant 2$$ I mean with using of only integer division, integer log with base ...
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54 views

Optimization - Integer programming Problem

The city of Shamut has called for bids for construction of its new town hall. The call for bids lists five parts of the total job: F - Foundation S - Structure P - Plumbing and heating E - ...
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47 views

Find all $a_i$ such that $(x_{a_1} - x_{a_2} + x_{a_3}) +\ldots + x_{a_{3k}}$ min

Given $n$ numbers $x_1, x_2, \ldots,x_n \in \mathbb{Z}$ and an integer $k \le\frac n 3$. Find $a_i$ $(i = \overline{1,2,3,\dots,3k}),\ 0 < a_i < a_{i+1} \le n$ such that: $$M = (x_{a_1} - ...
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28 views

Algorithm Request, choosing rows from a sparse table of integers to sum to a minimum row value

I'm writing some software, and one part of the software needs to be able to solve this problem as well as possible. Consider a table of integers and goal, for example: $$T = \begin{array} ...
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86 views

Minimum cost problem

I have been given $n$ points on a $2d$ plane. In terms of their $(x,y)$ coordinates. Now suppose I have to set, say firms, at these positions and the cost for building the first one is zero. For every ...
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21 views

Binary IP - Binari-zed result of negative value to 0, and positive value to 1

I have a set of variables $T_i = \{22, 23.6, 24, 24.2, 25\}$ and a constant value $C=24$. Given $$ a =T_i - C $$ I'd like to turn the result of subtraction a to binary value $\{0,1\}$, such that if ...
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47 views

Integer programming with pairwise relaxations: optimality?

In David Sontag's thesis [1] (page 11, 3rd paragraph from the end), it is mentioned that "Most previous linear programming approaches to approximate inference optimize over the pairwise LP ...
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52 views

Minimizing an expression with linear constraints

Given a system of under-constrained (i.e. infinite solutions) linear equations (all values will be integers, all coefficients will be 0, 1, or -1), I want to pick values for the variables to minimize ...