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

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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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32 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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14 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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32 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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79 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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1answer
122 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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22 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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29 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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25 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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139 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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25 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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24 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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1answer
43 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
34 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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1answer
58 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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42 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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37 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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28 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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23 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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52 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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25 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
44 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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2answers
992 views

How can not-equals be expressed as an inequality for a linear programming model

I have this linear programming model I'm building but one of the constraints needs to specify that the solution's basic variables need to all be different from one another. This is an integer linear ...
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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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39 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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67 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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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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13 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

Objective Value of LP as a function of RHS of Constraints

I saw the following statement in a paper, but am having trouble finding a reference for it. Consider the optimization problem $y = \max_x c^\top x$ subject to $Ax = b$ and $x \ge 0$. Then, written as ...
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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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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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65 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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89 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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34 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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1answer
48 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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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
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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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1answer
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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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1answer
22 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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77 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
27 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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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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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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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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116 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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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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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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Linear Optimization: Objective function value, basic feasible solutions and reduced cost

For the system $$Ax=b, x \geq 0$$ for $A \in \mathbb{R}^{m \times n}$, $m \leq n$, we call a set $B \subseteq \{1, \dotsc, n\}$, $|B|=m$ a basis for $A$, if $A_B$ is invertible, where $A_B \in ...
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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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Find all optimal solutions by Simplex

Let "stable operation" be an operation on a simplex tableau such that the entering variable has a reduced cost of 0. Recall that a pivoting operation will not change the objective value if either the ...