Questions on optimization constrained to integer variables.

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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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$a^2-b^2 = k$, $ab = l$ 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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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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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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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
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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
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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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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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1answer
50 views

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 ...
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38 views

Almost linear programming problem

I have a problem that is almost the typical in linear programming, but not quite. All variables take real non-negative values. Certain simple linear inequalities and equalities should hold. But what ...
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60 views

Minimize the following function with integer values with the given constraint. [closed]

$$f(b_1,b_2,\ldots,b_m)=(b_1)^2+(b_2)^2+\cdots+(b_m)^2$$ such that $$ b_1+b_2+\cdots+b_m=l$$ m is fixed and all values are positive integers including zero. We want to minimize this function with ...
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1answer
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Disjunction of conjunction in linear programming

I'm trying to get my model working with less variable/constraints possible. I want the binary variable $R$ to store the result of this Boolean expression: R = (a1 and b1) or (a2 and b2) or (a3 and ...
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3answers
239 views

Combinatorial optimization - Bijections between duplicated numbers

English is not my native language: sorry for my mistakes. Thank you in advance for your answers. Two Bijections and an Array... Here is a 2D array (in this particular example: rows: 1 to 4; ...
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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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Integer programming feasibility is NP-what

What is the complexity class of the general problem of integer programming feasibility? The sources I've looked at are, in my opinion, very confusing. Some say NP-hard, some say NP-complete. Some ...
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2answers
271 views

Optimization problem: Maximize the sum of minimum.

Given positive integers $L$ and a set of non-negative integers $N$. Find maximum of: $$\large \sum_{i = 1}^{4L}\ N_i\cdot(\min(\vert i - c\vert, 4L - \vert i - c\vert))$$ with $c \in \{1, 2,\dots ...
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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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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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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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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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1answer
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XORing consecutive integers has an interesting property. Does anyone know why?

I hesitated to post on StackOverflow but I think the problem has little to do with programming and more to do with mathematics. So, here it is: I wanted to compute the function $ f(n) = 0 \oplus 1 ...
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1answer
51 views

Lagrangean Relaxation of quadratic assignment problem to yield $n$ knapsack problems?

Consider the assignment problem: $$ Z = \min \sum_i\sum_j\sum_k c_{jk}\cdot x_{ij}\cdot x_{ik} $$ s.t. $$ \sum_i x_{ij} = 1 \quad\forall j $$ $$ a \leq \sum_j x_{ij} \leq b \quad\forall i $$ $$ ...
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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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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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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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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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1answer
27 views

How is the upper bound of a minimisation IP determined during branch-and-bound?

When using the branch-and-bound algorithm to solve an Integer Programming (IP) problem, the entire enumeration tree doesn't need to be evaluated and that's where the speed-up is achieved. Suppose the ...
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1answer
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Why optimization problems cannot be solved by simple derivative?

Let $f(\cdot)$ be a linear function. $f:\mathbb{R}^n\rightarrow\mathbb{R}$ $\;\quad\;\mathbf{x}\;\rightarrow f(\mathbf{x})$. Let $\mathbf{A}$ be a matrix in $\mathbb{R}^{m\times ...
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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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1answer
47 views

Symbol or notation for quotient operator

I'm trying to describe an algorithm in pseudocode where I've used the integer division operator. In VB.NET, the language I'm using, the operator used is "\", but I don't know if this is unambiguous to ...
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Minimize error function with integer constraints

Much time has passed since I studied any form of math so I wanted to take this cheap shot of asking someone else to think for me. I need to write some software that, for any given real number ...
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1answer
56 views

primal and dual lp optimal?

I have a simple assignment problem. I have four tasks that can be assigned to two persons. It is possible that not every task is assigned to a person due to capacity limitations. Each task requires: ...
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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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how to impose binarity constraint in a vector

This is part of a homework problem. In an optimization problem, I need to have a K dimensional vector S, such that each entry of the vector is either 0 or 1, and $l_1$ norm of S is <= K. I can't ...
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Given the sum of a geometric progression and the number of terms, can we recover the progression?

Consider a set of numbers which are in geometric progression: $n, nd, nd^2, \ldots ,nd^{M-1}$ Their sum is $S=\frac{n(d^M-1)}{d-1}$. Now if we know the values of $S$ and $M$, can we find values of ...
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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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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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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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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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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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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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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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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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1answer
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Non 0-1 integer programming

Many interesting combinatorial problems - graph coloring, maximal matching, set cover, and knapsack among others - can be reformulated as integer linear programs. One thing that all of these ...
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How to enforce a constraint that a decision variable can only take 1 of $k$ integer values?

How would you enforce the constraint that $x$, a decision variable, can only take values -3, 7, or 19? I think I probably need to introduce a binary variable here but not sure where to start. Thanks. ...
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1answer
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Designing an algorithm to determine if a linear combination of k-1 sets is contained in the k-th set .

I am trying to solve the following problem - given $k$ sets : $A_1,A_2,...,A_k$ containing $O(n)$ integers each I need to design an algorithm that will determine if there is such a group of elements ...
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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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Mixed integer programming with quadratic obj and quadratic constraints?

I was trying to use cplex for matlab to solve my optimization problem. However, It seemed to me that cplex was only able to solve PURE integer programming problem with quadratic objective function and ...
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1answer
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organizing rectangles on top of each other

We have some rectangles that should be organized in a number of columns. Each column height should be in the range of $[H, H+d]$ in which $d$ is a small number relative to the height of the ...
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1answer
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Global consistency of constraints in a MIP program

How does a Mixed Integer Programming (MIP) solver ensure global consistency of constraints while adding an additional constraint (during branch and bound). A naive method would be to add the ...