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

learn more… | top users | synonyms

3
votes
2answers
68 views

Faster Algorithms for Convex Hulls

I was interested in the following: Given two polyhedra $P_1, P_2$ specified in the form: $$ P_1 = {x : A_1x \le b_1 } $$ $$ P_2 = {x : A_2x \le b_2 } $$ Whereas $ x \in R^n$ and $b_1, b_2$ are ...
3
votes
1answer
307 views

What is the complexity of computing the minimum distance between two convex polyhedra that both have $n$ faces?

EDIT: (in response to what deinst said) sometimes using a sledgehammer for some menial task is rather convenient - especially if it also has the complexity $O(n)$ (which is what my question is about) ...
2
votes
2answers
1k views

How to find out whether linear programming problem is infeasible using simplex algorithm

So in http://econweb.ucsd.edu/~jsobel/172aw02/notes3.pdf, there is a mention about finding out whether linear programming (LP) problem is infeasible by simplex algorithm, but it does not actually go ...
2
votes
1answer
109 views

How to use the simplex method for linear programs?

I believe to be missing something important in the Simplex algorithm, because I can't get it to work. From what I gather, there are three steps per iteration, given a matrix for a linear program in ...
2
votes
0answers
422 views

Comparing two probability distributions

In my research I have to find two discrete probability distributions by solving two separate linear programs. The domain of optimization is the probability space of $m^n$ atomic events, where $n$ is ...
2
votes
4answers
181 views

Dual of a Linear Program

\begin{align} \min_{x} c^Tx \\ s.t.~Ax=b \end{align} Note that here $x$ is unrestricted. I need to prove that the dual of this program is given by \begin{align} \max_{\lambda} \lambda^Tb \\ s.t.~\...
2
votes
1answer
102 views

Find the minimum value of C subject to the given constraints.

C=2x+5y Constraints: x+y>=2 2x-3y<=-6 3x-2y>=6 A-42 B-4 C-49 D-10 I encountered this question while doing the Systems of Linear Equations and Inequalities test at http://www.classzone.com/books/...
2
votes
1answer
137 views

Formulating an LP problem with vectors

We have $m$ vectors $v_1,v_2,\dots,v_m\in\mathbb{R}^n$ and $m$ numbers $t_1,t_2,\dots,t_m\in\mathbb{R}$ and we want to find a vector $y\in\mathbb{R}^n$ such that $$|v_i^Ty-t_i|\leq D$$ for $i=1,\...
1
vote
1answer
121 views

Linear programming duality theorem

As far as I know, there are 2 versions of this theorem: 1) $\max \{xc^T: xA \le b, x \ge 0, x \in R^n\} = \min \{by^T: Ay^T \ge c^T, y \ge 0, y \in R^m\}$ 2) $\max \{xc^T: xA \ge b, x \in R^n\} = \...
1
vote
2answers
2k views

Optimality conditions and Directions in Simplex method

I am trying to understand the optimality conditions in Simplex -method, more in the chat here -- more precisely the terms such as "reduced cost" i.e. $\bar{c}_j=c_j-\bf{c}'_B \bf{B}^{-1} \bf{A}_j$ and ...
1
vote
0answers
452 views

Showing a dual LP solves a primal LP

I originally asked this question: Does solving the LP dual SOLVE the primal LP? It was answered using an example of how the primal and dual solve each other (because of knowledge from strong duality)....
1
vote
2answers
807 views

Invertability of submatrix?

If I have a matrix $A \in R^{(m \times n)}$ with $m \leq n$. All rows in matrix a are linearly independent and therefore $A$ has a full row rank. I can decompose matrix $A$ such that $A = [B|N]$ with $...
1
vote
1answer
358 views

Meaning of the bar over $\bf{c}'$ in $\bf{\bar{c}}'=\bf c' -\bf c'_B \bf B^{-1} \bf A\geq \bf 0$?

I am trying to understand the page 87 Bertimas about Linear Programming. The author uses bolding and bars -- now I am starting to think that the bar means something else to vector, bolding apparently ...
1
vote
1answer
63 views

Solving a linear program thanks to complementary slackness theorem

Using the complementary slackness theorem, say if the following basis optimal: $$x_1*=0=x_5*,x_2*=4/3,x_3*=2/3,x_4*=5/3$$ \begin{cases} \max & 7x_1 &+6x_2&+5x_3&-2x_4&+...
1
vote
1answer
86 views

Efficient (time complexity) algorithm for Linear Programming problems

I have an inequality of the form: $$\sum_{i=1}^n a_i\cdot x_i \ge a_0$$ where $a_i\gt 0$ for all $i$. Subject to this and $x_i\ge 0$ for all $i$, I have to minimize the expression: $$\sum_{i=1}^n ...
1
vote
1answer
42 views

Real linear combinations of intervals

Given intervals at $i\in\{0,1\}$ $I_i=[-a_i,a_i]$ where $0<a_0<a_1<1$ and a third interval $I=[-a,a]$ where $0<a<{1}$, when is there an $\alpha,\beta\in\Bbb R$ such that $\alpha I_0 +\...
1
vote
2answers
53 views

How to find the number of possible solutions of LP problems?

Let us assume that we have a linear optimization problem (LP) that has multiple optimal solutions. I would like to know if there is a solver or an algorithm that can provide the number of optimal ...
1
vote
2answers
663 views

Sign restriction on the Lagrange multiplier? Why?

Say we are given a linear program where the goal is to minimize $c^Tx$ with the constraints $Ax\ge b$. Why is there a sign restriction on the Lagrange multiplier associated with the active constraints ...
1
vote
1answer
376 views

Underlying assumption in a Primal/Dual table

I just read in one of the questions answered by @MikeSpivey that the following table is provided in Sierksma's Linear and Integer Programming: Theory and Practice, Volume 1, page 144. ...
1
vote
0answers
55 views

A question about $n\times n$ matrix [duplicate]

Possible Duplicate: For every matrix $A\in M_{2}( \mathbb{C}) $ there's $X\in M_{2}( \mathbb{C})$ such that $X^2=A$? Square root of a matrix Let $A$ be $n\times n$ matrix on $\mathbb C $...
1
vote
2answers
69 views

Find the optimal solution without going through the ERO's

All I got is that $$12y_1 + 7y_2 + 10y_3 = 2(0) + 4(10.4) + 3(0) + 1(0.4)$$ and $y_2 = 0$ because $x_6$ is in basis. How do I find $y_1$ and $y_3$ without going through the simplex method? I took ...
1
vote
1answer
92 views

Degeneracy in Simplex Algorithm

According to my understanding, Degeneracy in a linear optimization problem, occurs when the same extreme point of a bounded feasible region $X$ can be represented by more than one basis, that is not ...
1
vote
1answer
143 views

Proving that Unit Intersection is NP-complete

I am extremely stuck on how to go about this problem and any help would be so appreciated. We are told to consider the following combinatorial problem: Unit Intersection: Let X = {1, 2,...,n}. ...
1
vote
1answer
282 views

Questions about weak duality theorem

Following are some corollaries regarding the weak duality theorem. Consider a constrained problem, $\min_{x \in X} f(x),$ subject to $g(x) \leq 0$ and $h(x) =0$. Its dual problem is $\sup_{u \geq 0,...
0
votes
0answers
28 views

Use graphical analysis to solve a parameterised LP problem

For $c < 0$, we have no feasible solutions and hence no optimal solutions. For $c=0$, our only feasible solution is $z=0$ obtained by $(0,0)$. For $c > 0$, well... I graphed the constraints ...
0
votes
1answer
101 views

How do I convert this into a linear programming problem?

A farmer is planning to raise wheat and barley. Each acre of wheat yields a profit of \$50 and each acre of barley yields a profit of \$70. To sow the crop, two machines, a tractor and tiller, are ...
0
votes
1answer
36 views

maximize 3-variable linear function [version 1.0]

This problem came up when I was trying to solve a bigger, probabilistic problem. So at the end it boils down to this: how can we maximize the function $f(x_2,x_3,x_4) = \frac{18}{100}x_2 + \frac{...
0
votes
0answers
23 views

Find if an integer can be written as a sum of n positive integers with only k unique integers k<=10

Given a set of $n$ positive integers $A$ ($n \leq 10^4$), where the number of unique integers is $k$ ($k \leq 10$), I need to find if a positive integer $W$ can be written as a sum of a subset of the ...
0
votes
1answer
70 views

Derive this variant of Farkas' lemma, through another variant of Farkas' lemma.

Derive the following variant of Farkas' lemma: For each $mxn$ matrix A and vector $b\in\mathbb{R^m}$ one of the following statements is true: $\exists x\in\mathbb{R^n}$ such that $Ax=b$ $\exists y\...
0
votes
1answer
120 views

Linear programming. Find the maximum number of vertex disjoint paths in a directed graph.

How I can write like an objective function subject to its corresponding restriccions the next problem? (max "...") subject to ($\sum "..." - \sum "..."=0$ $\forall$ "...") I have a directed graph ...
0
votes
1answer
74 views

Optimum exists but not extreme point in Standard Form LP problem?

Standard form problem $$\min \bar c^T \bar x \text{ so that } A \bar x=\bar b, \bar x\geq \bar 0$$ I am thinking the point II (Finnish) i.e. optimum exists but it is not extreme point, why it ...
0
votes
1answer
32 views

Integer Points in Simplex

Let $$A_w(d,q):=\left\{{\bf k} \in \mathbb{N}_0^d: \sum_{j=1}^d w_j k_j \leq q\right\}$$ denote the number of non-negative integer points in the $\ell_1$-ellipse with semi-axes of length $\frac{q}{...
0
votes
2answers
46 views

LP: An algorithm to decide whether a polyhedron is a subst of another polyhedron

I've encountered the following question which I am unable to solve: $$ P = \{\vec x | A\vec x \geq \vec a\} \\ Q = \{\vec x | B\vec x \geq \vec b\}\\ P, Q \subseteq R^n $$ Find an algorithm to ...
0
votes
2answers
862 views

GameTheory, Solve for optimal strategies by solving a system of linear equations

In a book on game theory I saw the following example of a game, a modified version of Roshambo (or Rock-paper-scissors), which is described by the following payoff-matrix: $$ \begin{array}{c|c|c} ...