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

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2
votes
1answer
151 views

Finding lexicographic maximum point using a linear program

I'm trying to find the lexicographical maximum point of a bounded polyhedron, i.e. I have a set $P = \{x \in \mathbb{R}^n : Ax \leq b\}$ and I'm looking for the lexicographic maximum point of this ...
4
votes
1answer
325 views

Linear programming for combinatorics/graph theory

I just went to a graph theory talk talking about various fractional graph parameters (but focusing on one). These were defined using linear programming. A question was asked, "How can we learn more ...
2
votes
3answers
173 views

Linear inequalities to make a specific solution infeasible

Say we have a binary linear programming problem: \begin{equation*} \begin{aligned} & \underset{\mathbf{x}}{\text{minimize}} & & c\cdot\mathbf{x} \\ & \text{subject to} & ...
3
votes
2answers
174 views

How many ${0, 1}$ solutions would this system of underdetermined linear equations have?

The problem: I have a system of N linear equations, with K unknowns; and K > N. Although the equations are over $\mathbb Z$, the unknowns can only take the values 0 or 1. Here's an example with N=11 ...
1
vote
1answer
242 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 ...
1
vote
0answers
176 views

Sensitivity analysis on non linear problems

First of all, I would like to apologize if this question does not fit into the "soft" category. I am quite a newbie around here, and maybe I can fail to get the feeling of what exactly is a "soft" ...
2
votes
1answer
325 views

Why we call it technological coefficients?

I'm learning linear programming's basic concepts. In following inequality: $$ \begin{align} \text{Minimize }c_1x_1 + c_2x_2 + \cdots+ c_nx_n \\ \\ \text{Subject to }a_{11}x_1 + a_{12}x_2 ...
1
vote
0answers
49 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 ...
1
vote
2answers
569 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' ...
2
votes
1answer
256 views

Column generation algorithm gets stuck — subproblem returns an existing column in master

I have implemented a column generation algorithm to (try to) solve a computationally large transportation routing problem. The gist of the algorithm is the classic column generation scheme: 1) start ...
2
votes
1answer
156 views

Linear Programming - Single Optimal Solution

Is it correct to state that if a linear objective function is not in parallel with any of the constraints, than there is a single optimal solution at some vertex of the polytope?
2
votes
1answer
74 views

Drawing samples from an LP program

Say I have an LP program in standard form: \begin{equation*} \begin{array}{rl} \mathbf{x}^* = \underset{\mathbf{x}}{\text{arg}\;\text{min}} & \mathbf{c}^T\mathbf{x} \\ \mbox{s.t.} ...
2
votes
1answer
230 views

Linear Programming - Getting a vertex of the polytope

I have a standard basic linear programming problem. Is there a polynomial time algorithm that can return a vertex of the polytope that describes the feasible set of solutions. I know that the ...
5
votes
1answer
1k views

simplex method : Entering Variable

In the Simplex method, a variable that enters the basis, cannot depart the basis in the very next iteration. Please explain..why so ?
0
votes
1answer
144 views

Positive semidefinite vector $\bar{x}$ as $\bar{x}>0 :=\bar{x} \lambda \bar{x}^{T}>0$?

$A \lambda A^{T} $ (quadratic form?) is used with matrices to check definiteness. What about with vectors? If I see conditions such as $\bar{x} > 0$, how can I know whether it means $\bar{x}_{i} ...
3
votes
1answer
409 views

How can I get a huge Linear Programming Problem? Any public data set?

I'm working on a Parallel Simplex Solver using C and nVidia CUDA for my Bachelor Degree in Computer Science. I've already asked one of my teachers to bring me a super linear problem with thousands ...
3
votes
1answer
100 views

Sensitivity of a solution to an LP Problem to a change in the objective function

Suppose I have a LP problem of the kind $\max f(x) = 2x_1 + c_2x_2$, subject to several restrictions. Suppose I know that the point $(a, b)$ is optimal. How much can $c_2$ change so that $(a, b)$ ...
2
votes
2answers
367 views

Simplex method: Optimality criterion

I have to show that if for a minimization problem, $z_j - c_j <0$, for all non basic variables then it has a unique optimal solution. The proof says "If we start with a feasible point $x$ ...
3
votes
2answers
1k views

Linear programming problem formulation

Stuck in this problem for quite a while. Anyone can offer some help? The problem is as follows: Fred has $5000 to invest over the next five years. At the beginning of each year he can invest money in ...
3
votes
1answer
1k views

Choosing Pivot differently in maximization Simplex- and minimization Simplex method?

In maximization simplex, the pivot is the smallest element in the column divided by the rightmost corresponding number. I am stumbling with the Example 3 here with solution that choose the pivot with ...
5
votes
2answers
1k views

Berlin Airlift Linear Optimization Problem

I am trying to learn more about the Berlin Airlift transport problem. Two links I could find are here: http://drmohdzamani.com/notes/file/Simplex%20Method.pdf ...
6
votes
1answer
173 views

Difficulties in Writing the Dual of a Primal Program

I am a student and I am studying the following problem during my spare time. Your comments and suggestions would be helpful. Given the following primal program: (Decision variables are $\xi_{v}$, ...
4
votes
1answer
103 views

Membership problem for convex cones

Does anyone have a reference for the most efficient or some simple reasonably efficient algorithm for the membership problem for convex cones: Given a finite set of vectors $v_1, ..., v_n$ and a ...
2
votes
2answers
242 views

How to calculate volume given by inequalities?

I need to find the volume of the 3d space that is given by the following conditions: \begin{array}{c} 0 < x_1 < 1\\ 0 < x_2 < 1\\ 0 < x_3 < 1\\ x_1 + x_2 + x_3 < a. ...
9
votes
3answers
5k views

Optimum solution to a Linear programming problem

If we have a feasible space for a given LPP (linear programming problem), how is it that its optimum solution lies on one of the corner points of the graphical solution? (I am here concerned only with ...
3
votes
1answer
163 views

solving linear program with rank constraint?

I have a linear program where the variables are n vectors. Now I'd like to impose an extra constraint that k (k<=n) of the n vectors are linearly independent, or the matrix with the n vectors as ...
1
vote
2answers
220 views

What is an efficient way to get blur from source and blurred images?

I'm doing little program to get blur from source image and blurred image. But I haven't learned so much things about math in school yet. The equation used for blurring the image A into B: ...
0
votes
1answer
87 views

Does max { $w^Tx$ subject to $x$ is a point on a given polyhedron } optimize at an extreme point?

Is it necessary that the linear program max { $w^Tx$ subject to : $x$ is a point on a given polyhedron } attain its maximum at an extreme point of the polyhedron for any arbitrary w ? Let $c$ = ...
2
votes
1answer
190 views

is there a generalization of unimodular matrices for non-square matrices?

Is there a generalization of unimodular matrices for non-square matrices? It is well-known that unimodular matrices guarantee an integral solution for a linear program (if the constraint matrix is ...
3
votes
3answers
190 views

Using Correlation for mouse gesture recognition

I am in need to build a mouse gesture recognition system which will compare given recognition to the the gestures in training data and will say where a given gesture best fits. I am planning to use ...
1
vote
1answer
218 views

Lower bound for the complexity of linear programming

Since it is known that you can sort $n$ numbers by solving a certain kind of linear program - doesn't this imply a lower bound on the complexity of solving linear programs in general via the lower ...
3
votes
1answer
441 views

Finding tight constraints on a linear inequality

I have $a^\intercal M b > 0$, where $\forall a_i > 0$, $\forall b_j > 0$, and M is known. I'd like to find a tight linear constraint on $b$ which is independent of $a$ (other than the ...
2
votes
1answer
251 views

real-time linear programming

I'm going to implement in C a light-weight embedded lp-solver for a production system. I need to be able to sequentially solve a series of (possibly unrelated) linear programming problems with ~6-60 ...
2
votes
1answer
1k views

Simplex Method row operations help?

before programming an algorithm which implements the simplex method, I thought I'd solve an issue before the actual programming work begins. For some reason, I can NEVER get the correct answer. I've ...
2
votes
1answer
670 views

Sufficient Conditions for a Bounded Feasible Region in the Linear Programming Problem

I am working on a problem where it would be nice to prove that the feasible region of a LP problem is bounded, but where it is not necessary to solve any particular problem. In particular, given an ...
2
votes
1answer
586 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 ...
4
votes
1answer
356 views

Confused about linear programming exercise solution in my textbook

please see this simple linear programming exercise and its solution from my textbook. The task is to convert the prose and matrix to a formal linear programming problem. My answer matched theirs ...
1
vote
0answers
148 views

l1-metric and cut metric equivalence

I would like to show that the following two statements are equivalent. Let (A, d) be an n-point metric space. And B set of $\binom{n}{2}$ pairs of points of A. $\exists t \geq 1$, integer m, and ...
1
vote
1answer
482 views

What is the restriction matrix used for in the stepping stone method?

Let's say that we want to solve a classic transportation problem without capacities using the stepping stone method. (Problem definition: A bipartite graph with supply nodes a1...m, demand nodes ...
1
vote
1answer
216 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 ...
1
vote
1answer
457 views

Weak duality theorem and false corollary

Let $A\in\mathbb{R}^{m\times n}, \ c\in \mathbb{R}^n, \ b\in\mathbb{R}^m$ and consider the linear program $$\max \{ c^Tx : Ax\le b\} \ (1)$$ Its dual is $$\min \{ b^Ty : A^Ty=c, \ y\ge 0\} \ (2)$$ The ...
1
vote
1answer
656 views

How to set up a linear programming problem?

I'm not 100% sure if I set up the following problem right. Once I have the problem set up I know how to solve it. (this is a homework problem) The problem goes like this:"A company can use plastic, ...
3
votes
1answer
1k views

Understanding proof of Farkas Lemma

I've attached an image of my book (Theorem 4.4.1 is at the bottom of the image). I need help understanding what this book is saying. In the first sentence on p.113: "If (I) holds, then the ...
0
votes
1answer
113 views

binary variable question

I need to make a constraint for the following condition: Student 1 can only be on the team if students 2, 3, 4, and 5 are also on the team. I'm not sure how to model this using equations. The ...
1
vote
2answers
1k views

Help with binary variable

I need to make a constraint for the following condition: Among students 1, 2, 3, and 4, at least two of them must be on the team, if there are any on the team at all. I have defined Y1, Y2, Y3, and ...
6
votes
2answers
4k views

Primal and dual solution to linear programming

Lets say we are given a primal linear programming problem: $\begin{array}{ccc} \text{minimize } & c^{T}x & &\\ \text{subject to: } & Ax & \ge & b \\ & x & \ge & ...
2
votes
2answers
3k views

Can a non-degenerate LP have multiple optimal solutions?

In linear programming, an LP can have multiple optimal solutions if it contains degenerate vertices, i.e. where one of the base-variables is 0. Can an LP also have multiple optimal solutions if it ...
1
vote
1answer
114 views

Basic linear problem solving

I have some LP problem and I'm willing to solve it (this is an exercise from some optimization-related book). Now, Mathematica tells me that the problem is unbounded and I want to make a generic ...
0
votes
1answer
44 views

How can I adapt my problem so that it is amenable to the simplex algorithm?

According to the Wikipedia article, the Simplex algorithm depends on constraining all the unknowns to be >= 0. I have a problem where one of my variables is highly likely to be negative in many ...
3
votes
0answers
211 views

Bender's Decomposition for Mixed Integer Programs

Say I have 2 LPs, LP_1 and LP_2 which have real and integer variables and a staircase structure (i.e. the solution and feasible region of LP_2 depends on the solution of LP_1). $LP_1$ has the form ...