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

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Linear Programming - deriving the Dual of the Primal

I've the following linear programming problem: This is the LP representation of the uncapacitated facility location problem. This is the dual representation of this problem: My question is how ...
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Linear Programming Neighborhood Problem 2

Suppose we are given a set of 2n integers and we wish to partition into 2 sets $S_1$ and $S_2$ so that |$S_1$|=|$S_2$| and so that sum of number in $S_1$ is as close to sum of numbers in $S_2$.Let ...
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19 views

Linear Programming Neighbourhood problem

In the n-city TSP what is the cardinality of $N_2(t)$ , the neighborhood of tour t determined by 2 change? What is the cardinality of $N_3(t)$?
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175 views

Primal feasible solution implies Dual optimal solution?

Every feasible solution of P puts an upper(or lower, depending on whether it is a maximization or minimization problem) bound on the optimal solution of D(assuming of course that D has a feasible ...
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54 views

Mixed Integer Linear Programming: Construction Rods

I have an interesting problem involving linear programming. The problem is the following, I have 4 different kinds of rods (rod sized found in the local market): 9m rod 11m rod 12m rod 15m rod ...
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18 views

Conditions for a LP to be integral

What are the conditions to be met by a LP that allows to infer that its optimum solution will be integral? Is total unimodularity a necessary and/or sufficient condition? What if all variables are ...
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35 views

LP Word Problem Construction

I am having difficulty constructing the constraints on a word problem as follows: The Brite-lite Company receives an order for 78 floor lamps, 198 dresser lamps, and 214 table lamps from condoski ...
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73 views

Indicator Variable, Mixed Integer Linear Programming

Assume $x$ is a real variable, and $0\leq x \leq1$. Besides, $y$ is a binary random variable. I need a linear program that: if $y$ is $1$: $x>0$, if $y$ is $0$: $x=0$ I know the following ...
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24 views

Solving a linear program using just one call to a procedure that gives a feasible solution.

Suppose we have some procedure $F$ which takes any set of linear constraints and either returns either infeasible or returns a vector satisfying these constraints. If we now take a linear program ...
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46 views

linear programming problem solving

I have written in lp solve and obtain the solution of 0.58 highqualitymeat and 0.41 lowqualitymeat the thing that confuses is doing it through a graph as: let x be high quality meat and y be low ...
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25 views

Linear programming and complementary slackness

If there's a given basic solution $ A = (x,w)$ for the primal problem (where $x$ are decision variables and $w$ are slacks), I can determine the dual variables $y$ and slacks $z$ associated with B.S. ...
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37 views

Maximize $Z=-x+2y$ given $x\geq 3,\ x+y\geq 5,\ x+2y\geq 6,\ y\geq 0$

I am a highschool senior that's new to this topic. So, apologies for my lack of knowledge and misconceptions. The proof of the theory of this chapter is beyond the scope of my textbook, so that may be ...
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26 views

Linear Programming:What combination of two loams to minimize cost

I am fairly new to linear programming so simplification would be helpful.Came across a certain question and unfortunately no answer for it at the back of the book. The question is adopted from a book ...
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2answers
48 views

Prove linear program is unbounded

So I need help on my homework (I feel like a 10 year old). The exercise goes like this: Prove algebraically that the following program is unbounded: Max: $x_1 - x_2$ Constraints: $-2x_1 + x_2 ...
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1answer
242 views

Lemke-Howson pivoting in degenerate bimatrix games

I'm working on an implementation of the Lemke-Howson algorithm for finding Mixed Nash Equilibria of bimatrix games, and I'm running into a roadblock when the algorithm is fed a degenerate game. For ...
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29 views

Feasibility Sets for Integer Program

I have two set of constraints defining feasibility sets $A$ and $A'$ of a mixed integer program. $x_{i}, y_{ij}$ are continuous positive variables, $a_{ij}, b_{ij}, c_{ij}, d_j$ are known ...
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58 views

A linear program for maximizing a fraction

Given $\lambda_1,\ldots,\lambda_n \geq 0$ and an $n\times n$ matrix $A$, I wish to maximize the ratio $$ \frac{\lambda_1x_1 + \cdots + \lambda_nx_n}{x_1+\cdots+x_n}, $$ where $x_1,\ldots,x_n \geq 0$ ...
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37 views

simplex algorithm with tableau

in the picture below you can see the tableau representation of a linear programm with a coefficient $b\in \mathbb{R}$. BV stands for basic variable. For which $b$ is $y_1=\frac{1}{2}, ...
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71 views

how Determine the maximum values of C.

how Determine the maximum values of C. my try is that : To graph the last two bounding lines, I'll want to put the equations into slope–intercept form. The bounding line corresponding to the 3rd ...
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26 views

LP program: does the decision variable coefficients affect the problem?

I just started reading up on linear programming by myself, and am a bit confused by the decision variable coefficients $c_j$, in the objective function $ \sum_j c_jx_j$. Do they matter? I mean, if ...
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79 views

What gambling/board game or real life thing can (surprisingly) be modelled as a linear programming problem?

So I've taken Linear Programming 101. I've read my textbook, took the test and all that, and - besides all the theory, the nice algebraic interpretations, etc - I've encountered a lot of textbook ...
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duality theory question

Let $A$ be a given $m$ x $n$ matrix, and $c\in R^n$ and $b\in R^m$ be given vectors. Use LP duality theory to show that if the problem $$\min\{x^Tx: Ax=b, x\geq0\}$$ has a finite optimal solution, ...
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43 views

extreme points and representing

$$X=\{(x_1,x_2)^T : x_1-x_2\leq3, 2x_1+x_2\leq 4, x_1\geq -3\}$$ Find all extreme points of $X$, and represent $x^*= {0\choose1}$ as a convex combination of those extreme points. I sketched it out ...
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45 views

L_1 norm optimization as a sequence of linear optimizations?

Does someone know of numerical methods to approximately solve ${\bf x_0} = \min_{\bf x}\{ \left\|\bf Mx - b\right\|_1\}$ by using some sequence of linear optimizations? Links or ideas are both ...
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34 views

Computing a lower bound for the minimal componentwise distance of vertices of polyhedra

Let $A$ be a matrix in $\mathbb{R}^{m \times n}$ and let $P = \{ x \in \mathbb{R}^n \mid Ax \leq b \}$ be a polytope. I want to compute a lower bound on the minimal componentwise distance of two ...
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118 views

Linear programming optimization problem formulation

I need help in formulating an optimization problem. I have a system of equations as follows: $c_1x_1+c_2x_2+c_3x_3=1$ $b_1x_1+b_2x_2+b_3x_3=1$ $a_1x_1+a_2x_2+a_3x_3=1$ In my case the ...
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30 views

Why is a local min also a global min for convex functions?

As the title states, for an unconstrained minimizaton problem, of a convex function, why is it that the local minimum is also the global solution?
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1answer
138 views

How does the Simplex method handle test ratios with zeros?

I've been running into an issue choosing a pivot when there are constraints with an RHS of zero. It appears that sometimes you should include zero test ratios when searching for the minimum test ...
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2answers
71 views

Integer Linear Programming

Without using a computer, I have to solve the following integer linear programming:$$\min \quad x_1+x_2+x_3$$ $$\operatorname{sub} ...
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29 views

If $\phi$ is injective linear map $\mathbb{R}^r \rightarrow \mathbb{R}^s$ then $Im \phi$ is closed in $\mathbb{R}^s$

My optimization theory handbook says that If $\phi$ is injective linear map $\mathbb{R}^r \rightarrow \mathbb{R}^s$ then $Im \phi$ is closed in $\mathbb{R}^s$, where $Im \phi$ denotes image of map ...
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96 views

Prove a property of primal-dual problems

When I was studying the computation aspects of quantile regression, I consulted some linear programming book and found an interesting property as follows: If the primal problem have unbounded ...
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35 views

$ \mathbb{F} = \{x \in \mathbb{R}^n:Ax=b\}= \mathbb{F}_r=\{x \in \mathbb{R}^{n}:A^{(r)}x=b^{(r)}\}$

Show that these two sets are equal. $A$ is an $m\times n$ matrix of rank $r$, $b \in \mathbb{R}^m$. $A^{(r)}$ denotes an $r\times n$ matrix with $r$ linearly independent rows of $A$ and $b^{(r)}$ is ...
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100 views

Systematic Gaussian elimination on a binary matrix?

I am trying to understand the mathematics behind the lights out puzzle (http://mathworld.wolfram.com/LightsOutPuzzle.html). There's a very helpful webpage at ...
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131 views

Converting nonlinear program into linear program

Consider the following nonlinear optimization problem \begin{align} \min \quad c^Tx &+ f(d^Tx)\\ \text{s.t.} \quad Ax &\geq b\\ x &\geq 0 \end{align} where $$ f(y) = \begin{cases} -y+2 ...
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16 views

A basic solution to a linear program

I know that with an objective function of two variables, the basic solutions to a linear programming problem are the points where the constraints intersect in an xy-graph. But, if we are given an ...
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33 views

Constructing a newton sequence

How may I construct the newton sequence for the following: $(1) f(x_1,x_2) = x_1^4 + 2x_1^2x_2^2 + x_2^4$ with $x_0 = (1,1)$ and $x_0 = (1,0)$ $(2) f(t) = t^4 - 32t^2$ and $t_0 = 1$ To find ...
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46 views

Transpose notation question

In a course I am undertaking, I was exposed to the following notation for Taylor's theorem: Fix $x^∗ , x ∈ \mathbb{R}^n$ and assume that $f : \mathbb{R}^n → \mathbb{R}$ has continuous first and ...
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62 views

Convolution Properties

I have a quick question about certain algebraic properties of convolution. If I have 3 functions $f(x)$, $g(x)$ and $h(x)$, is the following true? $\Big[ f(x) . g(x)\Big] \circ h(x) = \Big[f(x) \circ ...
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1answer
41 views

Lowest upper bound for linear system?

Assume you have an under-determined linear system $AX=B$ where you have more variables than constraints. It is also known that $X>0$ (element-wise). How can you determine the (scalar) lowest ...
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69 views

Linear Programming problem - Packing Trucks Objective Function

I'm learning about linear programming and I want to see if it's applicable to a problem I'm trying to solve (one that's probably been solved many times before). I'm having trouble writing a good ...
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29 views

Linear System with non zeros count constraint

I trying to solve a simple linear system: $Ax=b$ But with constraints like: $\sum{x_i}=S$, Usually S = 1. $L \le x \le U$, Lower & Upper bounds (usually $0 \le x \le 1$) And "Maximum count of ...
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1answer
97 views

Prove that optimal solution is an extreme point in LPP.

While proving this I have proved that Optimal solution cannot lie inside the feasible set and that each supporting hyperplane to a set bounded from below (which is the case as in LPP we can always set ...
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1answer
31 views

What's a basic solution, and how do we find them?

I've just started learning linear programming, and for some reason, have run into a question about something that isn't mentioned in the first chapter (and we're supposed to answer these questions ...
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168 views

When change making problem has an optimal greedy solution?

A well-known Change-making problem, which asks how can a given amount of money be made with the least number of coins of given denominations for some sets of coins (50c, 25c, 10c, 5c, 1c) will ...
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73 views

Maximum / Minimum Question with 3 Variables?

I seem to be stuck in this problem, would need your help! Question: Assume I have : 147 of x, 174 of y, 238 of z A different amount of x, y ...
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1answer
31 views

Nonnegative solution to underdetermined linear system

I would like to show that the underdetermined system $Ax=b,\; x\ge 0$, with $b$ being a positive vector and $A$ being a binary matrix, has at least one solution. I've seen several other related ...
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96 views

Conversion into linear program

I have an optimisation problem with decision variables that are multiplied with another (a weighted average is calculated). I'd like to convert it into a linear program. I found this link that ...
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48 views

Maximizing the number of zero entries in a linear combination of matrices

I was wondering if there exists an algorithmic way of solving the following problem. Let's say you have a bunch of square $N\times N$ matrices (call them $M_i$), and you want to form a linear ...
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40 views

Lexicographic Pivoting Rule: Why does that work?

On purpose, I do not write details about it, because I am interested in the intuition and not the details. But if someone asks for it, I can do that. So my question: Could anyone help me with the ...
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30 views

Prove that $b_{\perp}^{T}b_{\parallel}=0$

If $A \in \mathbb{R}^{mxn}$ then the unique expansion of every $b \in \mathbb{R^{m}}$ is $b =b_{\perp}+b_{\parallel} $. Prove that $b_{\perp}^{T}b_{\parallel}$. Comment: Saying that they are ...