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

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106 views

Question about “linear programming problem” in reference to joint pmf

I'm working on a homework problem and I'm not totally sure what the question is asking... The question reads: "Consider the linear programming problem: maximize $Ax_1+Bx_2$ subject to $x_1+x_2\leq ...
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29 views

Let $P$ be a minimization primal problem $\min c^T x$. Does $P$ and its dual $P^*$ always have the same number of optimal solutions?

Let $P$ be a minimization primal problem $\min c^T x$ and let $P^*$ be its dual. I've been wondering about the following: Suppose $P$ has exactly $n$ optimal solutions. I know that $P^*$ also has ...
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1answer
261 views

Dual problem of a maximization primal problem $P$?

Suppose we have a primal problem $P$ which is stated as a maximization problem $\max c^{T} x$. The dual problem is defined (Introduction to Linear Optimization by Dimitris Bertsimas) only for a ...
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66 views

Normalizing Vectors to get short numbers

$\vec{A}$ is vector agent, $\vec{O}$ is vector Object, $m$ is a constant integer. The following expression is repeated with a different O for every loop cycle: ...
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1answer
106 views

How does the Simplex method of solving LPs use the starting solution?

Say one looks at the LP (in slack form) and sees that assigning $0$s to all the non-basic variables doesn't give a valid solution but some other non-trivial assignment of values to the non-basic ...
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1answer
30 views

Formulating Solution for Branch and Bound

I have a linear programming question which I am setting up for a branch and bound solution. I am having issues with where to begin. The question is asking to find the minimum operating cost to ...
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1answer
76 views

Combining the duality principle and the graphical method

I am trying to minimize this linear program by combining the duality principle and the graphical method: I can't seem to find an example of how to approach this, can anyone show me how I would go ...
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1answer
44 views

How to find a polynomial of order $4$ which minimizes a given condition

How to find a polynomial $P(x)$ of order $4$ such that $\max\{\vert\ln(n)-P(n)\vert : 1\leq n \leq12\}$ is as small as possible? I guessed the solution with linear programming, but I don't know ...
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23 views

Expressing nonlinear problem as LP

I am using GLPK to solve a simple linear problem. Given is a set of distances $d_{ij}$ between nodes of a graph. We want to assign to each edge a velocity $v_{ij}$ such that the average time of ...
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1answer
38 views

unnecessary constraint in optimization problem

I have some optimization problem (optimizing parameter $\alpha$)with those constraints: $$\alpha_i\ge0$$ $$\sum\limits_i \alpha_i y_i =0$$ and a third constraints: $$w-\sum\limits_i \alpha_i y_i x_i = ...
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39 views

Effective convexity criterion for the finite point set in $\mathbb{R}^3$

I need to find effective convexity criterion for the finite point set. Below there is description of what is meant by "effective" criterion. Definition. Let $M = \{A_{1}, \ldots, A_{n}\}$ be the ...
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1answer
32 views

Expressing a Set Using Linear Inequalities

Let $D = {x ∈ R^3: |2x1 − x2 + 3x3 + 1| + |x2 + 2x3 − 2| + |5x2 − 3x3| ≤ 10}$. Express D as the feasible solution set of a linear system of inequalities (meaning, a system of the form $Ax ≤ b$). How ...
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1answer
306 views

On the Proof of Fundamental Theorem of Linear Programming.

Having read the link: Why maximum/minimum of linear programming occurs at a vertex? I understand why the optimal solution of any linear programming problem must be on the corner or lies on a face of ...
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2answers
45 views

Take two pieces of wood one 84 inches the other 74 inches. Need to cut equal amounts of 12.5 inches and 7.75 inches. How to solve?

So the system would look something like this. 74" < 12.5x + 7.75y < 84" 60" < 12.5w + 7.75z < 74" y + z = x + w where x, y, w, z are natural numbers ...
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1answer
53 views

Non linear programming

Could you please help me in solving the problem posted below. A company uses a raw material to produce two types of products. When processed, each unit of raw material yields 2 units of product 1 and ...
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1answer
45 views

Linear programming - geometric change between canonical and standard forms

Suppose that we are given a LP in canonical form, that is in the form $\{x \in \mathbb{R}^d |\ Ax \geq b \}$ and that we want to convert it to an equivalent LP in standard form $\{x \in \mathbb{R}^k \ ...
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1answer
44 views

Proof concerning basic solutions

Prove that every basic solution of $Ax=b$ (where $A$ is a matrix of rank $r$) is set by $r$ linearly independent columns of matrix $A$ (so it is $[A^{k_1}\dots A^{k_r}]\bar{x}=b$ where $A^{k_1},\dots ...
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1answer
156 views

How to solve systems of linear equations of multiple variables (more than 3 to 100s)?

This was a question asked during an interview for programming job. And the bottom line was to write an alogrithm to solve such equations. As much as it numbed my neurons - it really provoked me. I had ...
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1answer
402 views

Converting a Linear Program to Canonical Form

A linear program is said to be in canonical form if it has the following format: Maximize $c^Tx$ subject to $Ax ≤ b$, $x ≥ 0$ where $c$ and $x$ are n-dimensional real vectors, $A$ is an $m × n$ matrix ...
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1answer
44 views

Simplex method and basic solutions

I have put this into the form $0.5x_1 + 0.25x_2 + x_3=6$ $-x_1 - 3x_2 + x_4=-2$ $x_1 + x_2 = 10$ Is this correct? If so, how do I find a basic solution so that I can begin the simplex algorithm? ...
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0answers
64 views

Mixed-Integer Linear Programing : get the maximum constant associated to a non null variable

Does anyone know a way get the maximum constant associated to a non null variable using Mixed-Integer Linear Programing ? I would like to get the variable $a$ in this description : $$ i = 1,\ldots,m ...
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1answer
50 views

What optimization problem is this?

Minimize $$\sum_{i=1}^{m}w_i x_i$$ with $w_i \in \mathbb{Z}_{\ge0}$, and $x_i \in \{0, 1\}$ subject to a set of $n$ conditions of the form $$\sum_{i\in S_k} x_i \equiv c_k \pmod{2}$$ for $S_k ...
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1answer
13 views

Formula for rate that changes when negative

Is it possible to reduce this code to a single formula, rather than check if x is negative? ...
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1answer
53 views

Linear Programming Inventory

A company is opening a new franchise and wants to try minimizing their quarterly cost using linear programming. Each of their workers gets paid 500 per quarter and works 3 contiguous quarters per ...
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1answer
277 views

Linear programming and the simplex method

I am trying to solve this system of equations. My approach would be to introduce slack variables and then somehow use the simplex algorithm to solve this. Can anyone show me how this is done?
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1answer
170 views

books on the application of linear algebra on statistics/finance/machine learning

I am reading "linear algebra done right" by Axler and like it a lot. One thing though, in the end I would like to put these theory to use and as a math textbook it doesn't cover much application. ...
3
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1answer
71 views

Finiteness of the Supremum of Inner Product of Two Finite Sum Positive Sequence

Let $$A = \Big\{(a_1,a_2,\dots)\ \Big|\ a_i\ge 0, \sum_{i=1}^\infty a_i=1\Big\},$$ $$v(x)=\sup\left(\bigg\{\sum_{i=1}^\infty a_ib_i\ \bigg|\ (a_i)_{i=1}^\infty,\, (b_i)_{i=1}^\infty \in ...
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1answer
36 views

Optimization options to select multiple items with different features and values

I'm trying to identify which approach would work best to select a set of elements that have different features that minimise a certain value. To be more specific, I might have a group of elements with ...
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0answers
28 views

Linear Optimization: What is the difference between these two theorems?

I attend a lecture about linear optimization where we had the following two Theorems. But I somehow cannot spot the difference: Theorem 1: Let $P$ be a polyhedron with an extreme Point and $c \in ...
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1answer
64 views

Taking the dual of this non-standard linear program

I am just beginning to learn linear programming have a question about taking the dual of a non-standard LP specifically the one below: $\min M\\ 2x_1 + 3x_2 + 4x_3 \leq M \\ 2x_1 - x_2 + x_3 \leq M\\ ...
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0answers
236 views

Proof of Simplex Method, Adjacent CPF Solutions

I was looking at justification as to why the simplex method runs and the basic arguments seem to rely on the follow: i)The optimal solution occurs at some vertex of the feasible region (CPF points) ...
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0answers
54 views

What is a closed chain (or circuit) that is used in solving a transportation problem (a special type of linear programming problem)?

What is a closed chain (or circuit) that is used in solving a transportation problem (a special type of linear programming problem)? I'm having some problems with it. Please clarify it. I have posted ...
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1answer
45 views

Conversion of a general linear program into a standard linear program

I am trying to teach myself the basics of optimization of linear programmes, for example the following question: How do I tackle such a question?
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128 views

Convert a problem o minimizing a function to linear programming problem in standard form

I have to 1) convert a problem o minimizing a function to linear programming problem in standard form. It is something new to me. Can somebody explain it to me? $$\min(\mathbb{R}^2\ni(x,y)\rightarrow ...
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1answer
62 views

Optimization problem in the standard form

Let $x\rightarrow x^{T}c$ be an objective function of an optimization problem in the standard form, for which the optimal solution doesn't exist. Does then exist an optimal solution to $x\rightarrow ...
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1answer
152 views

Dual simplex doubt (unrestricted)

I have this two problems and i only want to find the dual form: $\begin{gather} max\hspace{.1cm}z =5x_1+6x_2\\ s.t\hspace{.1cm}x_1+2x_2=5\\ -x_1+5x_2 \ge 3\\ x_2 \ge 0\\ x_1\hspace{.1cm} ...
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1answer
315 views

Linear Programming : Is there any other way to solve than graphs?

In my highschool curriculum there's a a chapter on Linear Programming Problems. In the chapter there are bunch of unproved statements and mechanical ways to solve linear problem. But my question is- ...
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1answer
58 views

Linear Programming and Geometry Question

I have a question that involves some linear programming and linear algebra, and I really don't have a clue how to approach this question. Could someone give me some hints and ideas as to how to attack ...
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1answer
177 views

Linear Programming and Standard Form

In order to find the dual of a primal linear program, do I always have to convert it to the standard form first? For example, if I have the following LP, would the dual also be a min since the LP in ...
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2answers
102 views

References for Linear Algebra needed for Differential Equations and Linear Programming

I am in need of learning the Linear Algebraic theory behind the following Applied disciplines. Could someone please recommend Linear Algebra books for: Differential Equations: Specifically learning ...
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0answers
53 views

Write down a linear programming problem

I want to replicate a linear programming problem.I have the following information, for the background." A fuzzy regression analysis with only one independent variable X results in the following ...
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2answers
69 views

Is $[u_1,u_2]$ an edge of the polytope $conv(F)$?

Here's the problem. I have a finite set of vectors $F \subset \mathbb{Z}_{\geq 0}^d$. I define $P$ to be the convex polytope $conv(F)$ i.e. the convex hull of $F$. Given $u_1, u_2 \in F$, is the ...
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1answer
60 views

Given a set of integers find two disjoint subsets $I$ and $J$ so $|I|+|J|=k$ and $\sum\limits_{i \in I}x_i = \sum\limits_{j \in J}x_j = n^2$

Given a set of integers $1 \le x_1,x_2\dots,x_n \le n^2$ and a number $k \le n$. describe an algorithm that will determine if there exists two disjoned subsets $I$ and $J$ such that $|I|+|J|=k$ and ...
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1answer
289 views

Show that every extreme point in Q is either an extreme point of P or a convex combination of two adjacent extreme points of P

P is a bounded polyhedron in $\mathbb{R}^n$, $a$ a vector in $\mathbb{R}^n$, and $b$ some scalar. Define $$Q = {x \in P | a'x = b}$$. Show that every extreme point in Q is either an extreme point of P ...
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1answer
51 views

Find Maximum of Lower Envelope

Okay, I'm not really sure whether the title is good. Consider \begin{align*} \min\{ 5x_1 + \frac{5}{2}x_2 + \frac{5}{3}x_3 + \frac{5}{4}x_4, \\ x_1 + \frac{6}{2}x_2 + \frac{6}{3}x_3 + ...
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1answer
3k views

simplex M-method minimization problem

Solve using the simplex method. Identify the solution of the dual in the final simplex tableau Minimize: $$z=12x_{1}+4x_{2}+2x_{3}$$ **Constraints:**$$ x_{i}\ge 0$$ $$-6x_{1}+3x_{2}\ge 9$$ ...
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1answer
70 views

Feasible Condition with a single constraint

A linear program with a single constraint minimize $z = c_{1}x_{1} + c_{2}x_{2} +· · ·+c_{n}x_{n}$ subject to $a_{1}x_{1} + a_{2}x_{2} +· · ·+a_{n}x_{n} ≤ b$, $x_{1}, x_{2}, . . . , x_{n} ≥ 0.$ (a) ...
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1answer
174 views

Easy elementary proof of Farkas Lemma?

Is there any elementary proof of Farkas lemma which does not use convex analysis and hyperplane separation theorem? What about special case below: If the Matrix $A$ is invertible, then there is ...
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1answer
148 views

Set of optimal solutions for a linear programs

Consider the linear program: minimize $z = x_{1} - x_{2}$, $x_{1}, x_{2}\geq 0$ subject to: $-x_{1} + x_{2}\leq 1$ , $x_{1} - 2x_{2}\leq 2$ Derive an ...
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0answers
79 views

How to justify that a basic feasible solution to a Linear Program corresponds to an extreme point of the feasible region?

Say we have an LP Problem in standard form. That is, $$\text{Maximise} \;\; C^T X $$ $$ \text{subject to:} \;\;\; AX = B --(1) \;\;\;\; \text{where $A$ is an $m \times n$ matrix }$$ I read ...