Questions tagged [linear-programming]

Questions on linear programming, the optimization of a linear function subject to linear constraints.

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Proof of equivalence of definitions for a vertex of a polyhedron

In these lecture slides from Princeton University I found the following definition of a vertex of a (convex) polyhedron (p. 11). A point $x\in\mathbb{R}^n$ is a vertex of a polyhedron $P$ if 1)...
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11 views

Goal linear-programming in Matlab

$$0v_1 + 0.2v_2 + 0.4v_3 + 0.6v_4 + 0.8v_5 + 1v_6 + 0.8v_7 + 0.6v_8 + 0.4v_9 + 0.2v_{10} + 0v_{11} → 0.46$$ subject to $$v_1 + v_2 + ···+ v_{11} = 1$$ $$v_1, v_2, . . ., v_{11} ≥ 0 $$ I couldn't ...
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13 views

Understanding linear optimization better?

I'm taking a linear optimization class, and I'm having a difficult time formulating an 'integer program' from a problem. My main problem is taking given information (often tables), how do I declare ...
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10 views

Using Matlab for linear programming with unknown objective function [closed]

I'm trying to find the optimal scalar (min) $u$ with an optimal vector $x\in \Re^{k}$ that satisfy: $A*x<=u*ones(d,1)$ $u>=-1$ Where A is a constraints matrix $A\in \Re^{d*k}$ and some other ...
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16 views

Hit-'n-Run Monte Carlo on convex polytope

So, I'm currently trying to implement a MCMC to uniformly sampling hyper-points from the polytope defined as $\mathbb{K}=\{x\in\mathbb{R}^{n}\;\;\text{s.t.}\;\; A\,x=b \}$ in the specific case where, ...
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1answer
29 views

Conversion of maximin linear program to matrix form [closed]

Ι can't seem to understand how the author converted the problem into such matrix form. Any help? I think i am missing the point of M and u. Linear Program: $maximize $$\sum_{i=1}^{|S|} min_{a\in(A\...
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1answer
30 views

How to solve linear program $\min \langle c, x \rangle$ using Lagrangian?

Given the following linear programming problem $$ \min \langle c, x \rangle\\ \begin{align} \text{s.t} \,\,\,\,\,\,\,& \sum_{i=1}^{n}x_i=1\\ &x\geq0 \end{align} $$ where $x \in \mathbb{R}^n$. ...
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18 views

Value of $\alpha$,$\beta$ and $\gamma$ in given LPP problem

By simplex method the optimal table of given LPP problem $$Maximize\;\;z=\alpha x_1+3x_2$$ subject to $$\beta x_1+x_2+x_3=8$$ $$2x_1+x_2+x_4=\gamma$$ where $x_1,x_2,x_3,x_4 \geq0$ The ...
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46 views

Proving that a polyhedron contains a line if and only if $Ax=0$ has a non-zero solution.

I would like to check if my proof is correct. I do not have much experience of proving stuff so I would appreciate if anyone could point out and fix some mathematical/proving/wording errors if there ...
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121 views
+100

Custom Nurse Rostering Problem

I've asked this question also on Operations Research Stack Exchange. It's a custom nurse rostering problem: $N$ is a set of nurses; $S$ is the set of shift-type (morning, afternoon, night, rest) $...
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28 views

This question is related to operation research with a model

I have include the question and can somebody able to help this to find and solve this one with a model. It is somewhat difficult to understand the problem so anyone can help me to solve this problem. ...
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1answer
42 views

Determine the vertices of $P^{'}=\{ c^{T}x: x \in P\}$ where $P$ is a polytope

Let $P$ be a polytope and $c \in \mathbb R^{n}$, then determine the vertices of $P^{'}=\{ c^{T}x: x \in P\}$ My idea: Note that since $P$ is a polytope there exists $r > 0$ so that $P\subseteq ...
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18 views

A “lifting property” for linear maps on cubes?

Let $L : \mathbb{R}^n \twoheadrightarrow \mathbb{R}^m$ be a linear map that is surjective. Let $[0,1]^n$ denote the unit cube in $\mathbb{R}^n$, and let $Z := L([0,1]^n) \hspace{5pt}$ ($Z$ is a ...
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1answer
26 views

Operation Research with some recurrence constraints [closed]

I have got this question and i can't think a way to formulate it. So if anyone can help me with this question it will be very helpful Each year, a shoe manufacturing company faces demands (which ...
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1answer
51 views

Problem to find the objective function in this problem

I have the next real problem. I have a set of workers $W = \{w_1,\dots,w_{12}\}$ and I have to find the optimal distribution minimizing the jumps of the workers from one jobsite to another jobsite in ...
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1answer
42 views

Doing a Charnes-Cooper transformation with matrices and an zero-one constraint

I need to solve an assignment problem with the following objective function: $${\max} \frac{\displaystyle\sum_{i=1}^m\sum_{j=1}^n h_{ij}\cdot x_{ij}}{\displaystyle\sum_{i=1}^m\sum_{j=1}^n c_{ij}\cdot ...
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1answer
141 views

Does set remain bounded if these integer constraints are removed?

Question: Let $P$ be a nonempty polyhedron in $\mathbb{R}^n$, and let $l_i, u_i \in \mathbb{R}$ for all $i \in I$, where $I \subseteq \{1,\dots,n\}$. I'm looking at a problem where the feasible ...
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1answer
94 views

Find pairs of linearly associated linear combinations

I am trying to figure out if the following problem can be formulated using linear or quadratic programming: consider a set of $n$ normalized vectors $V = \{V_1,..,V_n\}$ with $V_i \in \mathbb{R}^d$. ...
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2answers
117 views

A property of a linear image of the cube

Let $[0,1]^3$ denote the unit cube in $\mathbb{R}^3$. Let $L : \mathbb{R}^3 \to \mathbb{R}^2$ be a surjective linear map, and let $H := L([0,1]^3)$ (which is generically a hexagon). Can you provide a ...
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1answer
48 views

Unboundedness of an LP

Let $(P)$ be an LP of the form $$\min \{ c^tx\}, \text{ with } Ax \le b,x \ge 0.$$ Show that if $(P)$ is unbounded then there is a vector $d$ such that $Ad=0,d \ge 0$ and $c^td<0$ holds. ...
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1answer
29 views

A condition for an LP to be unbounded

Let $(P)$ be an LP of the form $$min \ c^tx \text{ with } Ax \le b,x \ge 0.$$ Show that if there is a vector $d$ such that $Ad=0,d \ge 0$ and $c^td<0$ $(P)$ is unbounded . Could you help ...
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1answer
35 views

When does there exists $x\geq,\neq 0$ s.t. $(\sum_{i}A_{i}\alpha_{i})x\neq 0$ $\forall$ $\alpha_{1},…,\alpha_{k}\geq 0$ s.t. $\sum_{i}\alpha_{i}=1$

Let $A_{1},...,A_{k}$ be $n\times n$ matrices. When does there exist a vector $x\geq,\neq 0$ such that $(\sum_{i}A_{i}\alpha_{i})x\neq 0$ for all $\alpha_{1},...,\alpha_{k}\geq 0$ such that $\sum_{i}\...
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53 views

Why does standard quadratic programming contain $\dfrac{1}{2}$ in the objective function?

Does the $\dfrac{1}{2}$ provide any of computation convenience?
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12 views

Algorithm to solve 'user optimum' 'polygamic' stable marriage problem: Optimally assign travellers to shared rides.

I am looking for inspiration to reformulate the classical assignment problem into something behaviourally richer (closer to Nash equilibrium or or stable marriage problem). I find it tricky to ...
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1answer
61 views

Solving for $x_i$ and $y_i$ in this system: $(\mu-x_1)y_1+(x_2-\mu)y_2=d$, $x_1y_1+x_2y_2=\mu$, $y_1+y_2=1$

Suppose we have a system of equations: \begin{align} (\mu-x_1)y_1+(x_2-\mu)y_2&=d \\ x_1y_1+x_2y_2&=\mu\\ y_1 + y_2 &=1\\ \end{align} The decision variables are $x_1$, $x_2$, $y_1$, $...
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1answer
33 views

Propositions about Multiple Solutions in LP

I'd like to check if the following statements are true about optimizing a linear objective function $f$ over a convex space. For the purpose of the problem, the space over which we are optimizing is ...
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1answer
32 views

Linear programming: reduce a constraint that includes a minimum

I have an almost linear programme. However one of the constraints has a form $z = min(x,y)$ (all the other things are linear in the model). Is there a way to substitute this with something (or ...
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1answer
36 views

Converting inequalities into equalities by adding more variables

I was trying to solve a rather large system of equalities and inequalities and was stuck, until I realized that converting the inequalities into equalities by adding more variables showed that zero-...
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1answer
32 views

Distribute N balls into M bins

Problem 1 :- There are N bins and M balls. Each Ball have a score. Problem is to distribute M balls into N bins such that - 1. Average score of balls in Bin 1 = Average score of balls in bin 2 and ...
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24 views

Resource allocations

There is a certain factory and there is time for which it is necessary to consume a certain amount ($ X $) of a resource. At each specific period of this time (and the time is discrete), a factory can ...
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1answer
89 views

Convert a Non-Linear Constraint of Integer and continuous variables to a Linear Constraint for programming

I have nonlinear constraints: \begin{equation} \sum_{i} \dfrac{X_{ij}^{t}}{r_{ij}} \le T_{disp1} * w_{t} * NN_{j}^{t} * \overline{\mu}_{j} \quad \forall j \ ,\ t \end{equation} \begin{equation} O_{...
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1answer
27 views

Optimal control problem in non-standard form with many integral constraints

I am trying to solve the following optimization program and I am hoping for some help as I am stuck: $\min_{R_{t}\in[0,1]}\,\,\int_{0}^{\tau}e^{-t}R_{t}\,dt$ s.t. $e^{-t}R_{t}-\int_{t}^{\tau}e^{-s}...
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2answers
54 views

Prove that a polyhedron in the $\mathcal{H}$-representation is bounded

Given a polyhedron $P$ specified by a set of linear constraints $P=\{x \in \mathbb{R}^n \mid Ax \le b \}$, what are the conditions on the matrix $A$ such that $P$ is bounded? I have the following ...
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39 views

How to optimize in an assignment problem when the objective function is a quotient?

I know that if I had an assignment problem with an objective function that looked like this $${\max} \sum_{i=1}^m\sum_{j=1}^n c_{ij}\cdot x_{ij},$$ I could use linear programming to solve it. But ...
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27 views

Is there a standard way to see if the solution set to a systems of linear equations crosses a given region?

I'm interested in the following question: Determine if there exists a solution to linear system $A\vec{x} = \vec{b}$ that lies in region $R$. Is there a general procedure for doing so? Currently, ...
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31 views

Recursive solution of linear optimization with incrementally added dimensions and constraints

I'm interested in understanding whether there is a nice recursive-type relationship between solutions of linear optimizations problems of lower and higher dimensions. E.g., Fix $a,b,c,d,e,f\in \...
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1answer
31 views

Nursery wants to maximize the total profit

A nursery planted deciduous and evergreen shrubs in an area of $30,000 \,\rm{m}^2$. An evergreen shrub requires $1 \,\rm{m}^2$ and a deciduous $2 \,\rm{m}^2$. The two types of shrubs have different ...
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1answer
64 views

When does a given system of linear inequalities form a bounded convex polytope?

We know that a Closed Convex Polytope may be regarded as the set of solutions to the system of linear inequalities: $$\begin{array}{ccc}{a_{11} x_{1} +a_{12} x_{2}+\cdots+a_{1 n} x_{n}}\leq b_{1} \\ {...
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31 views

Theta-Ratio of a Simplex Method for a degenerate solution, are they always equal?

Are the $\theta$-ratios of two degenerate solutions always equal? So as to say: If we know two unique points yield the same objective value, must their $\theta$-ratios always be equal? For two ...
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18 views

Special Case of the Two Phase-Method

I'm sorry this question must be slightly vague. In the two-phase method, my general understanding is that you try to exit your Aritifical Variables to make your Auxillary problem to $0$. But what ...
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1answer
50 views

Prove that exactly one of the systems has a solution

Let $A$ be an $m$x$n$ matrix, $B$ $l$x$n$ matrix and $c\in R^n$. Prove that exactly one of the systems has a solution: i)$$Ax\leq0,\:\: Bx=0,\:\: c^Tx>0,\:\: x\in R^n$$ ii)$$A^Tp+B^Tq=c,\:\:p\geq0,...
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1answer
28 views

Linear Programming Price Change After X Units Sold

I have a question regarding writing some formulas for LP. How would you code the price change after X number of units sold. So lest say the base price for the first X units sold is £5 and then there ...
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1answer
55 views

Linear Programming problem of accepting reservations on different fare classes to maximize revenue

this question has my stuck, I am unsure on how to incorporate the some of the information into constraints and without them the answer seems a bit silly. Below is the question, please let me know how ...
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0answers
48 views

Merging two line equations having similar angle into a new one [duplicate]

I would like to merge two lines having a y = mx + b equation with very close angles. I am simply comparing their angle difference, then I create a new line passing ...
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9 views

what pivot should I choose when introducing new constraint trying to apply the Dual Simplex Method and all $b_i$ are positive?

There is an LP. It is already given that $x_1 = 0, x_2 = 1$ is the optimal solution. First I find the corresponding simplex tableaux. Then what I don't get is how to apply the dual simplex method when ...
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0answers
27 views

Local branching in Benders Decomposition

I am trying to understand how local branching is used in Benders Decomposition. I was wondering if someone could kindly explain me how exactly local branching works. If my understanding is correct, ...
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1answer
71 views

A question on algebraic inequalities [closed]

Given \begin{align} X+Y&\leq C_1\\ Y+Z&\leq C_2\\ Z+X&\leq C_3 \end{align} Find the maximum of $X+Y+Z$, where $X, Y, Z$ are non-negative integers and $C_1, C_2, C_3$ are positive ...
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1answer
24 views

Asked to find the dual of a given primal problem. (Is my solution wrong? Solutions included)

I'm not understanding how there can be two separate solutions to this problem. I've doubled checked and followed all the steps but I'm assuming my answer is wrong but similar? Sorry, I don't have ...
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0answers
24 views

In a Linear Programming Problem, how is Degeneracy affected by the number of variables and constraints?

In an LPP, with m constraints and n variables. How does the number of constraints and variables affect the degeneracy of the system?
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1answer
43 views

Need help formulating this linear program [closed]

A company's pension fund manager must invest a maximum of $300,000 in bonds and stocks in order to obtain the highest possible return on investment. However, in order to obtain a risk-adjusted ...