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

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Optimizing over a set of optimization problems

This is my first time asking an optimization question on here, so I am looking forward to see what will happen here. In the lack of a better title, I wrote it as it is. At a high-level, I can perhaps ...
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1answer
10 views

making a function non-linear using a Lagrangian function

How Is this formula a Lagrangian function ? And how can a non-linear element be added to a function using this "Lagrangian function" This is where i got this In order to improve the performance ...
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22 views

Solving Matrix Value and Optimal Strategy (Matrix Games)

How would I solve this matrix game ? I'd like to find the value of the matrix and the optimal strategies for each player. $$ \left[ \begin{array}{cccc} 0 & 3 & -2 & 2 \\ -3 & 0 ...
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1answer
13 views

2 Linear equation problems [on hold]

Write objective, constraints and graph for the following two problems: 1.A test offers 2 types of problems. Type A takes 3 Min to solve and B takes 2. You have 20 min to take the test and can only ...
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1answer
12 views

Duality and Optimality Conditions

I have seen the solution and it involves adding a $x_5$ and $x_6$ to the inequalities. I really do not understand why this happens? I have not seen any questions like this yet. Any pointers would ...
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A question on linear programming

For each $Q \subset \Bbb R^n$, denote $Q^*:=\{z \in \Bbb R^n:zx \leq 1,\;\;\text{for all}\; x \in Q\}$. Let $P:=\{x \in \Bbb R^n: Ax \leq b\}$, for the matrix $A$ and the vector $b$. It is clear that ...
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M-technique (Big M)

I understand how to do normal simplex, but I've seen examples of M-techinque and in the objection function (z) row, M just seems to disappear. As M doesn't appear in the constraints I dont understand ...
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Linear Programming, Optimal Solutions

I posted the whole question to give some context, but my problem lies with (iv). I think you're meant to use a formula for the generalization of the optimal solution, but I'm not really sure what ...
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17 views

Linear programming to find minimal additive and multiplicative factors

Consider samples $\{x_i,y_i\}$ with $x_i\in\mathbb{R}^N$ and $y_i=\pm1$ and additional $z\in\mathbb{R}^N$. Can one use linear programming to find the minimal $m>0$ and minimal $\epsilon>0$ (e.g. ...
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Solving Linear Inequalities Using Primal and Dual LP's

I've been working through my linear programming homework, and I'm having difficulty understanding how and why you would want to use the dual to find the optimal solution to the primal. I'll write up ...
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1answer
48 views

Enquiry to network flow

Could anyone advise me on how to find a feasible flow to the following graph so that the edges $(2,5), (4,5), (6,5),(6,7)$ are saturated? This means, I have to formulate the network flow as a linear ...
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1answer
18 views

condition for having a positive solution to a linear equation.

Let $Y$ be a member of $\mathbb{R}^m$. I need a necessary and sufficient condition on a $n\times m$ binary matrix $A$ for having a solution to the linear equation: $$AX=Y$$ Such that $X_i\geq 0$, ...
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1answer
31 views

Min-Cost-Flow Problem

Given a directed graph $G = (V,E)$ with a cost function $\gamma: E \to \Bbb R_{\geq 0}$ and two vertices $u,v \in V$. How to reduce the problem of finding a directed path from $u$ to $v$ with minimum ...
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19 views

Application of Combinatorics, Logic and computability theory in physical science: Tiling of Wang Tile with proportionality

The original problem of Domino Tiling and Wang Tile has great theoretical interest on computability theory... However, the great emerging problem on application of Wang Tile in material science and ...
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1answer
31 views

Linear programming with non-convex quadratic constraint

Could anyone let me know if the following linear programming problem can be solved in polynomial time or should be NP-hard? $\min c^Tx$ s.t. $x^TQx\geq C^2, x\in [0,1]^n,c\in ...
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Reduced Cost in Network Simplex Algorithm

On page 5 of the slide, [T]he reduced cost of a non-basic arc $(i, j)$ is the sum of the costs of the arcs forming a cycle with $(i, j)$ in the current tree solution. Why is that the case?
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70 views

maximization function of a matrix given a scoring system

This is, from a mathematics standpoint, trivially solvable, but my goal is to solve it with the fewest number of comparisons. I'm hoping to discover that this problem is identical to something in ...
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26 views

Model a shortest path linear programming problem

I have a graph with 8 vertex, and i'm supposed to model a linear programming problem which consists in delivering 10 trash containers (1 is in vertex 1, 3 are at vertex 2, 2 are located at vertex 5, ...
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21 views

Minimise a cost of elements into given length - Optimisation - Linear programming

I've been researching optimisation methods used to minimise the cost of elements that could be used in a given length. Here is an example of my problem : For a given length 40 : X1 Length l1 = 13 ...
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2answers
369 views

Linear Programming to find the loan plan to minimize the interest payment

Assume that it is the first of July and you are running a small shop. The sales revenue and the amount of bills you have to pay for the next six months are estimated as following: In short, you ...
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1answer
18 views

Formulation of linear problem

I'd like to ask you how to formulate this problem as linear problem (equations)? Marie wants to buy oranges and apples. She has to buy at least 5 oranges and the number of oranges has to be less ...
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1answer
17 views

Finding matching weight for two differing types of cat food

Attempting to figure out how much cat food to give my cat I came across a problem which I am unsure of any way other than iteration to solve. The problem I have is that I have been advised to feed my ...
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1answer
16 views

Linear programming with reuse of services

I came across some questions of this style and was not sure what the minimization function would be. A hotel requires a known number of hand towels for guests to be given during the week and the ...
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1answer
19 views

Linear Inequalities - Allocation Problem

The problem at hand can be summarized as follows: we have to allocate a ressource to $n$ production units. The allocation to production unit $i$ is $x_i$. Each of the production unit will produce at ...
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1answer
15 views

Where did i go wrong in this linear inequation evaluation?

We are currently studying Linear Programming in school and while going through it i seem to of come across a ridiculous error. Problem is, i can't seem to find it. Essentially there is an equation 8 ...
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Statistical Meaning of LP problem

What is the statistical interpretation of this LP problem for different values of $\mu$? $\min \sum_{j} \left( |x-b_j| + \mu (x-b_j) \right)$ I know that $\min \sum_{j} |x-b_j|$ is the median but ...
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1answer
26 views

orthogonal triangular decomposition and ordinary least squares

I have just come across orthogonal triangular decomposition whilst looking at ordinary least squares regression. I'm not quite sure how this is being used though to find a solution. In my example I ...
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21 views

Connect all nodes of a graph to satisfy demand

I have a non-complete non-oriented graph composed by one Supersource node which produces all the amount of goods the graph needs and n nodes. Every node require a specific amount of goods . What I ...
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1answer
20 views

Gradient Question-Linear Regression

When discussing linear regression, we discuss the error of the out of sample data prediction. That is, $$ E_{\operatorname{out}} = \frac{1}{N} \sum_{n=1}^{N} ...
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1answer
19 views

Is it possible to get the dual solution “quickly” once the optimal primal solution is found?

With the primal objective value, I know the dual objective value. I also know the right hand sides of the original program. However, I don't know the values of the dual variables at the optimal point. ...
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35 views

Examples of non trivial problems in this structure.

I'm looking for examples of non trivial problems that match with the follow structure. Let the function $$g: U \times V \rightarrow \mathbb{R}$$, where $U$ and $V$ are complex vetorial spaces of ...
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1answer
34 views

Minimize the minimum - Linear programming

Consider an optimization problem with variables $x_1, x_2, \dots, x_n \in \mathbb{R}$ (maybe subject to some linear constraints), and linear functions $\{f_i(x_1, \dots, x_n)\}_{1\leq i\leq m}$. We ...
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42 views

Application of the Simplex Method using Gauss-Jordan to solve Transportation Minimization Problem

I am attempting to implement the Simplex method using Gauss-Jordan elimination to solve the transportation problem. This is a minimisation problem, whereby I want to starting from a feasible solution ...
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1answer
28 views

Can the search space of a solvable linear optimization problem be discontinuous?

Background Say you have a traditional linear-optimization problem, there is a linear cost function, $\vec{c}\cdot\vec{x}$ and a set of linear constraints, $A_1\vec{x} \geq b_1 $ $A_2\vec{x} \leq ...
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linear programing

suppose that the time(hours) required to produce desks, chairs and shelves through two processes are shown in the table below: process desks chairs shelves pressing 8 ...
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1answer
32 views

How to convert a linear optimization problem into a normal form?

The following linear optimization problem is given: $$ \begin{eqnarray} x_1 + 2x_2 -7x_3 \leq 1\\ |3x_1-5x_2-20| \leq 4 \\ x \geq 0 \\ 6x_1+5x_2-3x_3 \rightarrow min \end{eqnarray} $$ And it is my ...
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Get reduced costs from simplex tableau

This is probably a dumb question... but I'm trying to find how to calculate the reduced cost for a particular variable based on the information in a simplex tableau after I've minimized a linear ...
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Maximize minimum optimization using linear integer programming

I am trying to solve a maximize minimum optimization. I have four different items that each of them has 10 values of Rates and for each value it has a corresponding weight. Then I have a free table ...
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3answers
49 views

Linear Programming and differentiation, why can't we differentiate to find the optimum solution?

I do understand that differentiating a linear function (for a maximization) subject to some linear restriction (such as the problem $p=ax+by$ s.t. $cx+dy \leq m$) won't necessarily give me the right ...
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Online convex programming: Projection followed by normalization

I have the following projected gradient descent online linear programming problem which has been well studied in www.cs.cmu.edu/~maz/publications/techconvex.pdf‎ $\mathbf{y}_{t+1}=\mathbf{w}_t - ...
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Assigning jobs to minimize cost - Linear programming

I'm stuck trying to solve this linear programming question. You want to make a website with a list of features F, which are n elements long. Each feature has a corresponding value for how long it'll ...
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24 views

Linear Programming Problem(Algebra of Simplex Method)

May I know if my proofs to the following claims are correct? Please advise. Thank you. 1.) Reduced cost corresponding to basic variables are zero. Proof: Consider the standard L.P. : max ...
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Prove that a point is optimal in LP-problem

I have the following LP-problem: Minimize $B_1^t Y_1 + B_2^t Y_2 + B_3^t Y_3$ subject to $$ (C_1,C_2,I) \begin{pmatrix} Y_1 \\ Y_2 \\ Y_3 \end{pmatrix}\geq 2 \text{ and } Y\geq 0 $$ where $B_1$ is ...
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1answer
27 views

Existence of Integer solution for a set of linear equations

Can anyone give a proof sketch of the following claim: If a system of homogeneous linear equations with integer coefficients has a positive real solution, then it also has a positive integer solution? ...
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33 views

When to stop the trolley

First, I'm not good at math. Am not good at making formulas but I will try to explain as much as I could. Second, we are developing a software/application for wake boarding. We are using trolley to ...
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Writing a linear program in standard form

Usually I have been asked to write problems in standard form that have inequalities involved. However, this problem has none and I was wondering if anyone had insight on how to go about solving it. ...
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Linear Programming with Matrix Game

It seems from an easy google of "learning linear programming" that a common way of learning it is to work with Matrices that represent "games" for two players. Here is one I have stumbled across. We ...
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30 views

Linear programming with quadratic constraints

I want to solve a problem of this form: $max_{y,k} \,\,\, w^\top y + C 1^\top k$ s.t. $k y^\top B^\top = I $ $A^\top y \geq b$ is there an algorithm that can solve such a problem? Is there an ...
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2answers
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Can a non-extreme point be an optimal solution of a Linear Programming problem?

Consider a linear programming problem. Is it possible for an optimal solution to exist, but not at an extreme point? According to Bertsimas & Tsitsikalis ("Introduction to Linear Optimization", ...
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on stochastic matrices

I have some questions on stochastic matrices in Discrete Mathematics as follows. The set $P_n$ of all $n \times n$ doubly stochastic matrices is a polytope in $\mathbb R^{n^2}$? If $A$ be a vertex ...