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

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23 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 ...
0
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2answers
35 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 ...
2
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1answer
175 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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1answer
38 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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33 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}, ...
2
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1answer
47 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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1answer
23 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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1answer
67 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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0answers
141 views

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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41 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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0answers
30 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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0answers
105 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 ...
0
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1answer
25 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?
2
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1answer
69 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 ...
2
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2answers
62 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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2answers
28 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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1answer
66 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 ...
0
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1answer
34 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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58 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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0answers
95 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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0answers
15 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 ...
2
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1answer
32 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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2answers
42 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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1answer
49 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 ...
1
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1answer
39 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 ...
0
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0answers
52 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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0answers
24 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 ...
0
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1answer
66 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 ...
2
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1answer
30 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 ...
2
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2answers
86 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 ...
4
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1answer
68 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 ...
0
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1answer
25 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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0answers
92 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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1answer
33 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 ...
0
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1answer
31 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 ...
0
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1answer
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 ...
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0answers
172 views

Linear program for way optimization with unusual constraints

I've just finished the lecture "Introduction to Optimization", so I know something about linear programming, and came across the following game on lumosity.com: The goal of the game is to pick up ...
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0answers
19 views

refomulation of an optimization problem

I have written a program for optimizing a set of generators. And I need to reformulate this problem, to include additional generators and constraints. I have hourly price and cost data and need to ...
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166 views

Positive/Negative Definite Bordered Hessian?

I understand how to check a function for concavity and convexity using the Hessian matrix and the rules for the determinants of the leading principal minors. I understand if these rules are violated, ...
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0answers
19 views

maintaining monotonicity in an optimization problem

I have written a program for optimizing a set of generators. I have hourly price and cost data and need to figure out when a generator should run or just stay off. I describe the problem in more ...
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0answers
27 views

Why does the simplexmethod 'break up' - unbounded, LP program, very basic problem

I've calculated a very, very basic LP problem: with >= "greater or alike" and <= "smaller or alike" max x + 2y 4x + 3y >= 12 x <= 4 ...
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2answers
66 views

How to find on each face of a polyhedron one point?

We have a polyhedron in $\mathbb R^n$ generated by the intersection of a collection of finete hyperplanes or the convex hull of the set of vertices. My question is: Is there any algorithm for ...
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1answer
83 views

Show both relaxations of boolean LP give equal lower bounds

Given the boolean LP: $$\text{Minimize}\;\; c^Tx$$ $$\text{Subject to}\;\; Ax \leq b$$ $$\hspace{57mm} x_i(1-x_i)=0\;\; i=1,...,n$$ Show that the LP relaxation: $$\text{Minimize}\;\; c^Tx$$ ...
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1answer
17 views

Finding the intersection between 2 lines using matrices

My professor uploaded some notes, and there's a step in his explanation of a Linear Programming Problem which I do not understand. He takes 2 lines and converts them into matrices in order to find the ...
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1answer
32 views

Primal of Dual of LP problem

Given that the following relation holds: $$\begin{align*} &\textbf{Primal problem} \\ &\max Z = c^Tx \\ &s.t. \\ &Ax \leq b \\ & x \geq 0\end{align*}$$ $\Longrightarrow$ ...
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22 views

Duality of Linear Program

Formulate the dual of the following linear problem: Min $\sum$ $\sum$ $c_{ij}$ $x_{ij}$ s.to $\sum$ $x_{ij}$ - $\sum$ $x_{ji}$ = 0, $\hspace{5mm}$ $x_{ij}$ >= $l_{ij}$ and $x_{ij}$ $\leq$ $u_{ij}$ ...
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0answers
13 views

Showing that every extreme point of the set of solutions of the standard form of constraints of any L.p.p. is a basic feasible solution

Let $\vec y$ be an extreme point of the convex set of solutions of $A \vec x=\vec b $ where only the solutions of $\vec x(\in \mathbb R^n)$ with all components non-negative are taken ; then I want ...
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2answers
60 views

Represent if-else or OR condition in a linear equation (optimisation with simplex algorithm)

I would like to write some linear equations and inequations to state that the sum of all possive x - C is smaller than L. As my ...