Questions on optimization constrained to integer variables.

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Why should the matrix $A$ in an ILP be integer?

Almost everywhere I read about integer linear programming (ILP), I found that the matrix has to be integer (by definition). More precisely, an ILP is defined as follows: An ILP in canonical form is ...
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49 views

Prerequisite reading for Concrete Mathematics? [closed]

I'm a freshman computer science major who has just started reading Concrete Mathematics, mathematics for computer science. Is there any prerequisite reading or learning I should do before embarking on ...
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31 views

Transportation: Minimizing Cost

I am trying to solve this problem, but I have had no luck. I have tried to set this up in MS Excel, so I could use Solver to find the solution, but I don't really know how to form this problem. As far ...
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44 views

Binary Integer Programming

I need to form teams. There are 8 projects and 60 students. Each project has different requirements. For example, out of 5 total requirements, project 1 has 2 requirements: must have a programmer and ...
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37 views

Eliminating non-integer solutions to $ab / (2\sqrt{ab} + a + b)$

I am writing a program to output all $a,b \in \mathbb{N}$, where $a \le b \le n$ (for a given $n \in \mathbb{N}$), such that $$ \frac{ab}{2\sqrt{ab}+a+b}=c\in \mathbb{N} $$ For example, $a=9$, ...
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18 views

Finding grid nodes a line passes through

For 2D grid pathfinding, I want to do a quick broadphase to check if there is a direct path from the start to the target by conceptually checking all nodes touching the line segment formed by ...
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13 views

CPLEX solver on a standard vertex coloring problem

I have a very straightforward ILP model of vertex coloring that I'm trying to solve with CPLEX. With a binary variable $x_{vc}$ for every $(v,c) \in \{ V \times C\}$ there is a constraint $$ \sum_{c ...
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7 views

Gomory's cut typical running time until the constraint is fractional

I was considering the following problem. Say we are given an linear programming problem $$ \max c^Tx $$ $$ Ax \le b $$ $$ x \ge 0$$ Where instead I consider $i^{th}$ the optimal solution $X_i$ of ...
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NP-hardness of solving congruence equations in several variables

You are given the following equation modulo $N$ (where the $\beta_i$'s are given integers modulo $N$, and the $x_i$'s are unknown integers modulo $N$): $$\beta_1x_1 = \beta_2 x_2 = \ldots = \beta_l ...
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42 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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40 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. ...
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21 views

(Proving NP Hardness) Maximizing ratio of polynomials

I have a function $\prod_{i = 1}^N \alpha_i$ $\alpha_i = \displaystyle \frac{(\sum_{j = 1}^{M} R_j x_j I_i(j))^2}{\sum_{j = 1}^{M} R_j x_j}$ The variables are $x_j$s, and $R_j$s are some positive ...
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27 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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52 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. ...
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53 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 ...
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70 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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20 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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19 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 ...
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65 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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77 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} ...
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30 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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18 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 ...
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56 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 ...
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30 views

minimising multivariate quadratic function over integer variables

I have a quadratic function $x_1^2+x_2^2-(u_1x_1+u_2x_2)^2$ which I need to minimise over integer $x_1$ and $x_2$; also, the coefficients $u_1,u_2<1$. In other word, assuming coefficients ...
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36 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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10 views

optimization of formulas involving binomial coefficients

I encountered such a problem. We need to find the min value and max value of $f(x,y)$. $x$ and $y$ are integers $\in[0,n]\times[0,n]$ and $(x,y)\neq (0,0)$ or $(n,n)$. $$ ...
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183 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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17 views

Linear Integer Optimzation Problem (scheduling problem)

Does any of you know how to get this done?
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48 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 ...
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60 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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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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64 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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39 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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19 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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24 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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51 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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71 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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62 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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71 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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80 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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8 views

Number of satisfied constraints in unsolved ILP

I have ILP constraints as follow: $\forall l \in L_i \exists t \in T^l_i : \sum_j w_j(t) \geq r_i^l(t)$ For unsolved case, how can I compute the number satisfied $l$? Does it depend on the solver I ...
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42 views

A good enough solution for the 0-1 multiple knapsack problem

Can you please explain, or point to an easy explanation of a good enough solution to the 0-1 multiple knapsack problem? This is a single constraint problem so only one-dimensional weights are to be ...
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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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50 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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99 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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38 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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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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14 views

Help on Binary Integer Programming

I am about to do a paper about distributing subjects to students with the constraints on: number of subjects that the student need, number of students whom these subjects will be distributed to, and ...
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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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16 views

Linearization bounds on 0-1 quadratic problems

What are the best linearization methods for approximating the following constrained 0-1 Quadratic problem, where $Q \in \mathbb{R}^{n\times n}$ and $k$ is an integer $1\leq k \leq n$ $$ \max ...