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

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minimum possible value of a linear function of n variables

Suppose $x_1,x_2,\ldots,x_n$ are unknowons satisfying the constraint $a_1x_1 + \cdots + a_nx_n ≥ b$, where $a_1, \ldots , a_n, b ≥ 0$. Then the minimum possible value of the expression $c_1x_1 + ...
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84 views

Warm start of simplex algorithm after update of constraint matrix

Assume we found an optimal solution $\mathbf{x}_1$ of the linear program \begin{gather} \max \mathbf{n}^T\mathbf{x}\mbox{ s.t. }A\mathbf{x} \leq \mathbf{b}\tag{1} \end{gather} using the simplex ...
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1answer
1k views

How to test if a feasible solution is optimal - Complementary Slackness Theorem - Linear Programming

I have this linear program $$\begin{cases} \text{max }z=&5x_1+7x_2-3x_3\\ &2x_1+4x_2-2x_3&\le8\\ &-x_1+x_2+2x_3&\le10\\ &x_1+2x_2-x_3&\le6\\ &x_1,\,x_2,\,x_3\ge0 ...
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1answer
849 views

Reduced cost in the Phase II of the two-phase Simplex?

My lecture slides outline how the two-phase simplex works: this table shows the end result of the phase I for the standard-form problem and the auxliary table of the phase I here. I understood until ...
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3answers
200 views

Is an unit-cube polyhedron? What about other platonic solids?

Definitions According to my linear programming course and screenshot here (Finnish), a polyhedron is such that it can be constrained by a finite amount of inequalities such that $$P=\{\bar x\in ...
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2answers
326 views

How can I infer a result using primal feasibility, dual feasibility, and complementary slackness?

I am trying to find the minimum of $-x_1$ with restrictions $\bar g\leq\bar 0$ so that $$\bar g=\begin{pmatrix} (x_1+2)^2+(x_2-4)^2-20\\ (x_1+2)^2+x_2^2-20\\ -x_1\end{pmatrix}\leq ...
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1answer
74 views

LP problem involving producing assemblies

I have to construct an LP problem based on the ff scenario that might be similar to a scenario in another question (in the sense that I felt the need to use $mod$): The productivities are ...
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1answer
70 views

Express the constraint “$x = 0$ or $y = 0$” in linear programming

How to express the constraint "$x = 0$ or $y = 0$" in a linear program? Is it possible at all?
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47 views

Travelling salesman problem as an integer linear program

So the travelling salesman problem is a problem wherein a salesman has to travel through all cities in a way that the total travelling distance is minimal. You can rewrite this as an integer linear ...
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119 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 ...
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326 views

Computationally proving a linear programming solution is unique?

I have a simple linear programming problem min $c^{T}x$ subject to $Ax\leq b$. That gives me the solution I am looking for when solving in maple. My only problem is that I do not know how to check, ...
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517 views

Is the inverse of an invertible totally unimodular matrix also totally unimodular?

My question is learned from here. Let me restate it as follows: A unimodular matrix $M$ is a square integer matrix having determinant $+1$ or $−1$. A totally unimodular matrix (TU matrix) is a matrix ...
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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 ...
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302 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) ...
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1answer
114 views

How can I determine B-inverse from an optimal tableau of a LP?

(This is NOT a homework question, I am reviewing for my upcoming exam) Given this linear program: and this optimal tableau: I am attempting to determine $B$ inverse using the table above. From ...
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1answer
136 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 ...
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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 ...
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1answer
94 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 ...
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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 ...
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1answer
169 views

Previous table of simplex algorithm

I have this problem of simplex method which shows cycle . Problem : Maximize $Z=10x_1-57x_2-9x_3-24x_4$ Constraint to : $ 0.5x_1-5.5x_2-2.5x_3+9x_4<=0 $, $ ...
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0answers
419 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 ...
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4answers
176 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 \\ ...
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1answer
60 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 ...
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1answer
79 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 ...
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2answers
52 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 ...
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1answer
204 views

Tutorial for Simplex Method with No Slack Variables

I found a nice tutorial here http://www.math.ucla.edu/~tom/LP.pdf for applying the Simplex Method to problems of the form: maximize $c^T x$ with the constraints $Ax\leq b$, $x_i \geq 0$. It suggests ...
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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 ...
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1answer
140 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}. ...
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1answer
90 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 ...
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67 views

computing dual LP in graph matchings

I'm having a trouble converting the following LP to a dual LP. Help on some starting steps would be greatly appreciated!
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238 views

simplified Linear programming model with time constraints

Based on my other question, here is a simpler hypothetical exercise that isolates that time constraint issue all together: A farmer has 1000 ha forest that is already at a mature age assumed to be ...
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0answers
445 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 ...
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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\} = ...
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1answer
356 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 ...
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2answers
797 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 ...
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172 views

1s surpassing 0s in binary strings of odd length

Let $A(k)$ be the number of distinct binary strings of length $2k+1,$ for which the number of $1$s surpasses the number of $0$s for the first time at digit number $2k +1$, i.e., in the final digit in ...
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2answers
662 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 ...
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1answer
373 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. ...
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1answer
383 views

Sudoku mathematically, MILP?

My homework contains a word (freely-translated) "target-function" that I should generate somehow for 9x9 sudoku solver with some MILP problem. But I am bit lost what they mean. I have sofar described ...
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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 ...
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2answers
965 views

Optimizing with Absolute Value Objective Function

max : $w = |q^T y|$ subject to $A y \leq b$ $y \geq 0$ Please describe how one could solve the non-linear programming prob. above by using linear programming methods. I tried changing $y$ to $y' ...
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1answer
279 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 ...
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31 views

Formulating the Dual of a linear program

I have a linear program: Maximize 18x + 12y subject to: x+y <= 20 x <= 12 y <= 16 x,y >=0 I have found ...
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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 ...
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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 ...
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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 ...
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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 ...
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89 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 ...
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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 ...
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1answer
4k views

Finding the number of basic/zero variables at an optimal corner point in linear programming

Draw a graph of the following problem $$\begin{align}4x+3y &\leq 180 \\ 7x+4y &\leq 280 \\ y &\leq 40 \\ x &\geq 0 \\ y &\geq 0\end{align}$$ a) If the problem is to ...