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

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1answer
25 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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1answer
21 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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1answer
30 views

Is the polar of the polar set the original set?

For each $Q \subset \Bbb R^n$, denote $Q^*:=\{z \in \Bbb R^n:z\cdot x \leq 1,\;\;\text{for all}\; x \in Q\}$. Let $P:=\{x \in \Bbb R^n: Ax \leq b\}$, for the matrix $A$ and the vector $b$. It is ...
1
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0answers
39 views

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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0answers
27 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. ...
1
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1answer
54 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
24 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
49 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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0answers
36 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
58 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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0answers
22 views

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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78 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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0answers
45 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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3answers
466 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 ...
0
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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
19 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 ...
0
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1answer
27 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 ...
0
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1answer
28 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
19 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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0answers
20 views

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 ...
0
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1answer
42 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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0answers
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
22 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
27 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. ...
2
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0answers
47 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
56 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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1answer
33 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 ...
1
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1answer
76 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 ...
2
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1answer
38 views

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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0answers
39 views

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
76 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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0answers
50 views

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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0answers
50 views

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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0answers
58 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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0answers
29 views

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 ...
0
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1answer
54 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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0answers
35 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 ...
4
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1answer
90 views

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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0answers
53 views

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 ...
3
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2answers
105 views

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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0answers
32 views

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 ...
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0answers
93 views

Adjacent basic solutions and adjacent bases

I'm reading chapter 2, "The geometry of linear programming", in Bertsimas & Tsitsiklis's "Introduction to Linear Optimization" (Athena Scientific, 1997). I'm having some difficulty with the ...
0
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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: ...
2
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1answer
89 views

Linear Programming Transformations

What is the process of performing a transformation from a given problem to another linear programming problem such that the transformed problem has an optimal solution iff the initial problem has a ...
0
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1answer
55 views

Recommendations of programming language / software for computer-assisted mathematics

I have always used R and Python for statistical analyses and object-oriented programming. Now, I have to perform relatively demanding (long to perform) mathematical analyses such as derivations and ...
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2answers
35 views

Enquiry to Linear Programming.

I came across this Theorem on Optimum solution to a Linear programming problem: " If $S$ is the feasible region of some linear program with objective function $ z=c^{T}\textbf{x}$ then 1) $z$ ...
0
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1answer
33 views

Polyhedron's Representations and spanning the Euclidian space

Let's say you have to different representations of the same polyhedron $P\neq \emptyset$: $$P=\{x\in \mathbb{R}^n\;|\;h_i^Tx\leq c_i, i=1,...,k \} =\{x\in \mathbb{R}^n\;|\;g_j^Tx\leq d_i, j=1,...,l ...
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2answers
30 views

Reduce occurrence of $x$, retain definition at $x=0$

I need to apply a gamma curve to render an output variable $(x)$, to make better use of screen real estate, and it has me scratching my head with what is probably a simple math question. For $0 <= ...
1
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1answer
41 views

Big M constraint question

I have a question regarding using Big M constraints to solve the following problem: Given: $a, b \ge 0$ and integers. $$2a + 5b \le 17\\ a + b \le 5\\ 3a + 6b \le 20$$ For at least two of the ...
3
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1answer
278 views

optimal basis and optimal solution

Determine which of the following is true: a) Consider a maximization LP in SEF. Suppose $x$ is a basic feasible solution for which all nonbasic variables have strictly negative reduced costs. Then ...