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

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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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69 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
51 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
58 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
335 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
190 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. ...
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83 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
40 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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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
87 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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298 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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55 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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0answers
140 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
63 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
192 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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437 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
64 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
235 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
108 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
56 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
73 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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61 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
353 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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53 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
88 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
219 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
167 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
96 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 ...
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1answer
149 views

how to use linear programming for Heaviside Step function and L1 norm?

I want to find a hyperplane that can divide my sets of points into 2 groups that have nearly equal size. If the hyperplane is $w$, there is a scalar offset $b$. I have $N$ points that are ...
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1answer
166 views

Linear programming: Basic solution

Let $P = \{ x \in R^n : Ax \leq b, x\geq 0\}$ and $Q= \{ (x, r)' \in R^{n+m}: Ax + r= b, x \geq 0, r\geq 0\}.$ Show that $$x \text{ is basic solution of } P \iff (x, b-Ax) \text{ is basic solution of ...
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1answer
139 views

Do lagrangian multipliers converge to dual variables in LPs?

Can anybody clarify the following to me? Consider an LP, say a maximization problem, with solution x* and optimal value Z*. Its dual will have optimal value W*=Z* (by strong duality) and optimal ...
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120 views

Let $P$ be a polyhedron. Prove, $P$ has at least one extreme point $\iff$ $P$ does not contain a line, by using a lemma.

Let $P$ be a polyhedron. Prove, $P$ has at least one extreme point $\iff$ $P$ does not contain a line, by using a lemma. I've a Lemma saying: Suppose $P=P(V,E)$ where $V,E \in \mathbb R^n$ are ...
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1answer
163 views

What constraints are needed to to switch on a binary variable in a linear programming model?

I am trying to create a binary variable ($z$) within a linear model which 'switches' on only when another variable ($d$) becomes greater than zero. I have had some success using the following two ...
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18 views

Find relationships between events

I have a set of Events $(E_i)_i$ which have a probability $(P_i)_i$. I am able to write each event as a sum of distinct events that form a partition of the space. My goal is to find all the ...
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1answer
102 views

Solving LP with two $L_1$ inequality constraints

Is there a "fast" way to solve the following LP formulation with the following constraints: $$ \max_{\mathbf{f}} \mathbf{f}'.\mathbf{g} \\ \mathbf{1}'\mathbf{f}=1\\ \|\mathbf{f}-\mathbf{h}\|_1\le ...
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29 views

Let $P \subseteq R^n$ be a polyhedron. Why does $\{ x + \alpha d \mid \alpha > 0\} \subseteq P$ for some $x \in P$ imply $d$ is a recession direction?

Suppose we have a polyhedron $P \subseteq R^n$ and let $d \in P$ be a recession direction, that is $\{ x + \alpha d \mid \alpha > 0\} \subseteq P$ for all $x \in P$. Why does $\{ x + \alpha d \mid ...
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1answer
47 views

Need help formulating a Traveling Salesman (Deliveryman) Variation

I am working with the following TSP variation: A deliveryman has to deliver orders of items to several different locations around a city. The orders are: ...
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1answer
38 views

Write the convex hull of the points $(0,0), (1,1), (2,0) \in \mathbb R^2$ as the set of solutions to a finite number of inequalities.

Write the convex hull of the points $(0,0), (1,1), (2,0) \in \mathbb R^2$ as the set of solutions to a finite number of inequalities. The convex hull of $x_1, \ldots, x_n \in \mathbb R^n$ is defined ...
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1answer
105 views

Finding Optimal Value with Linear Programming Subroutine

I am stuck on this question: Consider the problem: $\underset{x}{\text{minimize}}{\frac{c^{T}x+d}{f^{T}x+g}}$ $\text{s.t. } Ax\leq b$ $f^{T}x+g>0$ Suppose we have prior knowledge that the ...
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0answers
163 views

Executing Branch and Bound using a Simplex tableau

I'm studying the branch and bound method and how it is used in conjunction with a simplex tableau. The issue I'm struggling with is how you incorporate the branches in your tableau to find out ...
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1answer
45 views

what is the dual of the following linear program over a convex set?

Let $\mathbf{x}=[x_0,x_1,\dots,x_N]^T$ be a $(N+1)\times 1$vector. Let $\mathcal{S}$ be a bounded, compact convex set in strictly positive quadrant of $\mathbb{R}^{N+1}$. Consider the following ...
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0answers
149 views

Linear programming - Textbook recommendations

Next term, I will attend a course on linear programming. Due to the assignments, we will have to write many thorough proofs. I anticipate that we will be supposed to cope with in-depth background ...
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159 views

Closed-form solution of the following LP problem

I am considering the following LP problem: $$ \begin{array}{cl} \text{maximize} & c^Tx\\ \text{subject to} & a^Tx\geq0 \\ & 0\leq x\leq x^\max \end{array} $$ where ...
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2answers
49 views

Maximum cardinality affinely independent subset of $\mathbb{R}^n$

Let $S \subset \mathbb{R}^n$ such that $S$ is affinely independent. Then $$|S| \le n + 1.$$ Why? (e.g. does anybody know some place where this is proven?)
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44 views

Linear programming, how to do the opposite transformation?

This is from a pdf file I found linked to from this site: They first define the primal dual problem like this: Then they have another representation of (p), and show that it's dual is: Now here ...
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1answer
47 views

Affine and Linear programming

Can someone give a simple explanation as to why the feasible region of a set of linear program/equations is affine?
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58 views

Every polyhedron $P \ne \mathbb{K}^n$ equals an intersection of finitely many half spaces.

Currently, I am reading some lecture notes on linear optimisation. I cannot see why the following (seemingly trivial) proposition holds. (How could I understand/proove it?) Every polyhedron $P \ne ...