The linear-programming tag has no wiki summary.

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### Approximate Set Cover Problem by Rounding

Here is the simple algorithm for approximating set cover problem using rounding:
Algorithm 14.1 (Set cover via LP-rounding)
Find an optimal solution to the LP-relaxation.
Pick all sets ...

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### Are there intuitive/classically algorithmic analogues to Semidefinite programs on networks?

Many network optimization algorithms, including shortest path, push-relabel, augmenting path, etc, actually have an interpretation in terms of linear programming.
A famous application of semidefinite ...

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### Need help to find an efficient algorithm for the following problem!

Consider $x$ an n-dimensional vector with $x_i$ is integer in the range $[0 \dots k], k\in N$.
Given $A_{n\times n}$ is the covariance matrix of $x$.
$u$ is a given n-dimensional vector of real ...

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### Lovasz theta function - uses

Lovasz theta function bounds the Shannon capacity of graphs. What are some other uses of the function - especially in asymptotic coding theory and optimization problems?

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### Partially optimal solutions in integer linear programming

Linear programs with a totally unimodular system matrix are known to have an optimal integer point. They are therefore solvable via relaxing the integer constraints to intervals.
An other interesting ...

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### Algorithm for the intersection of a vector subspace with a cone of non-negative vectors

Hi,
I would like to know whether there is some more effective way of how to compute an intersection of a vector subspace of $\mathbb{R}^{n}$ with a cone of vectors with non-negative entries than the ...

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### Guessing game with guess cost

This is a question about Problem 328 on the website Project Euler. A description of the problem is provided in the previous link. I was wondering if there has been any research done on this question. ...

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### Complexity of a weirdo two-dimensional sorting problem

Please forgive me if this is easy for some reason.
Suppose given $S$, a set of $n^2$ points in $\mathbb{R}^2$.
I want to choose a bijective map $f$ from $S$ to the set of lattice points in $\lbrace ...

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### Linear complementarity problem: principal pivoting algorithm

I'm trying to implement the "Dantzig; van de Panne and Whinston" principal pivoting algorithm for solving symmetric positive semi-definite LCPs from "The Linear Complementarity Problem" book ...

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### Mathematical Programming with other Algebras than Linear

Linear Programming is strongly entwined with linear algebra, as are many of its generalizations under the heading of mathematical programming / convex optimization.
What analogies are there for ...

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### Linear programming - uniqueness of optimal solution

Is it possible to build such an objective function for a given set of constraints, so that there will be only one optimal solution?
My general problem is to get any vertex of a polytope formed by a ...

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### Random Sampling a linearly constrained region in n-dimensions…

Hi,
So here is my problem:
Given a nonlinear, discontinous, cost function $f(x_1,x_2,..,x_N)$ along with linear constraints $x_n \ge 0, \forall n$
$x_n \le c_n$
and $\sum_{n=1}^N x_n = 1$ find an ...

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### Number of breakpoints in parametric maximum flow problems

The parametric maximum flow problem can be formulated as
$$f(\lambda) = \min_{x\in\{0,1\}^n} \left( \sum_{i}(a_i + b_i\lambda)x_i + \sum_{i,j}c_{ij}x_ix_j \right),
$$
where all $c_{ij}<0$ (so that ...

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### Linear program to maximize the minimum absolute value of linear functions ?

I'd like to compute
$\max_{x,t} t$ such that $\forall i$, $t < a_i + |x - b_i|$.
where $a_i,\ldots, a_n$ and $b_1,\ldots,b_n$ are fixed and $x \in [0,1]$.
Can this be solved with a linear ...

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### Circumference of convex shapes

Here is a puzzle I found in "Mitteilungen der DMV" (roughly "Letters of the German Society of Mathematicians") issue 19/2011. It was posed by Alfred Schreiber in "Wie man Hasen fangt" (How to catch ...

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### A polytope associated with the Hadamard Transform

In an investigation of whether or not a subset $V$ of "Hamming Space" $M_n = \mathbb{F}_2^n$ is a tile (i.e. whether $M_n$ can be written as a disjoint partition of translates of $V$) in ...

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### Infinite Linear Programming

I'm trying to prove optimality for a continuous linear program. That is, I have a linear program with an uncountable number of variables and constraints. I'm not sure how to demonstrate feasibility ...

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### Detection of Redundant Constraints

Suppose I pose the following query to a constraint logic programming
system:
?- Y <= 6 - X, Y <= (- 4) + 4 * X, Y <= 4 + X / 3.
Are there systems that would recognize the last inequality as
...

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### Continuous Transportation Problem

Hi all, I'm trying to formulate an infinite linear program to prove optimality (via duality) for the Continuous Transportation Problem, e.g. the Kantorovich-Wasserstein distance. This is the ...

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### Maximization of a matrix product by iterative methods

This might not be very difficult, but I think I may have gotten a little confused.
Suppose we are given a matrix A, and would like to find the vector x of modulus 1 which maximises the product xt A x ...

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### solving multiple linear programming problems with the same set of constraints

Hi,
I need to solve a set of linear programs of the form:
Problem $i$: $\quad \max c_i \cdot x$ s.t. $ A x \leq b$.
The $c_i$'s are different vectors so each problem has a different objective ...

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### The cone of positive semidefinite matrices is self-dual? (reference needed)

I'm seeking a reference for the following fact.
The cone of positive semidefinite matrices is self-dual (a.k.a. self-polar).
This result is relatively easy to prove, has been known for a long ...

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### Model for shipping widgets in an optimal way

I am a programmer and have the following requirement.
We are trying to figure out the optimal way to ship widgets. Below is the scenario:
We need to ship 1,000,000 widgets
We have two different ...

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### Set Cover:Greedy vs LP

Hi
Both, the greedy and the LP approach for Set Cover give a O(log n) approximation. Is there some inherent difference on the two approximation approaches?
thanks

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### existence of l1 embedding using LP feasibility

hello
Let (A, d) be an n-point metric space
for $t \geq 1$,the task it to find an integer $m$ and an embedding $f : A \rightarrow R^m$ s.t.
$\forall x,y \in A$ : $d(x,y) \leq d_1(f(x), f(y)) \leq ...

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### Split sum into equal terms

Given a sum of $l$ integers $r_1+...+r_k+...+r_l$ and an integer $t$.
Find indices
$1 < p_1 <...< p_h <...< p_{t-1} < l$
such that in sum
...

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### Efficient algorithm for finding the minima of a piecewise linear function

Consider real numbers $a_i$ and $b_i$ for $i=1\dots n$ and define a function by
$f(x) = \max_i ( a_i + b_i x )$
We desire to find $\min_x f(x)$. Obviously this occurs at an intersection of two ...

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### When is polytope compatible with network flow?

A linear program is the problem of optimizing an linear objective function within some polytope $A$ over $\mathbf R^n$. My question is motivated by the question of when a linear programming problem ...

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### Maximum flow with negative capacities?

I'm trying to compute an (s-t) maximum flow through a network which includes a number of arc pairs ((u,v), (v,u)) that have equal, negative capacities (weights). I'm not aware of any efficient ...

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### efficient way to compute the inversion of the following matrix

Hi, there
I have looked it up in the current textbook. The conventional numerical method to compute the inversion of an $n \times n$ matrix requires $O(n^3)$. However, for the following special ...

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### Degenerate case of linear programming duality?

Let's say we have a maximization linear program that looks like this: maximize $\vec{c}\vec{x}$, subject to $\matrix{A}\vec{x} \leq 0$, $\vec{x} \geq 0$. If we take the dual, we have "minimize ...

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### Minimum tiling of a rectangle by squares

Given the $n\times m$ rectangle, I want to compute the minimum number of integer-sided squares needed to tile it (possibly of different sizes).
Is there an efficient way to calculate this?

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### Linear Programming Cost Function [closed]

I need to add the following to my LP problem:
If the amount of workers hired in period $t$ ($H_t$) is higher than 25, the hiring cost is only 1 instead of 1.2.
Example: if 30 workers are hired in ...

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### Maximum average value within a rectangular bounding box

The goal is to expedite detection using the sliding window approach. In other words, an object classifier is known and I need to find where the possible locations of this object are in an image. This ...

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### Inequality-constrained linear-regression, what is the covariance of the estimator?

If you do a linear regression: $||Ax - e ||^2$, where e is iid Gaussian, mean 0 and variance 1, then your answer is $x_{hat} = (A' A)^{-1} (A' * e)$ and the covariance of $x_{hat}$ is $(A' A)^{-1}$
...

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### How to find which subset of bitfields xor to another bitfield?

I have a somewhat coding-oriented problem. I have a bunch of bitfields and would like to calculate what subset of them to xor together to achieve a certain other bitfield, or if there isn't a way to ...

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### what is the difference between the revised simplex method andthe full tableu?

No to sound naive but they look like they include the same steps to me, one's just the algorithmical representation of the other. Thanks in advance.

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### Recovering a piecewise affine function

Lets say I have an piecewise affine convex function $f(x_1,x_2)$, on which the following operations are possible:
Computing $f(x_1,x_2)$.
Computing a subgradient to $f$ at $(x_1,x_2)$
Computing all ...

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### Sorting a binary matrix diagonal in polynomial time while preserving rows

Is there a polynomial time solution to sort an arbitrary binary square matrix in polynomial time by rows so that the diagonal contains a 1 if any row contains a 1 in that column?
For example given ...

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### For Ax = b, x and b unknown vectors, how do I solve the x that maximizes min(b_i)?

Given a matrix $A$, each element $A_{i,j} \geq 0$, find the vector $\vec x$ that maximizes the minimum element in $\vec b$ ($\vec b = A \vec x$). Note that this is not a linear equation system as I ...

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### Continuous Linear Programming: Estimating a Solution

I have a "continuous" linear programming problem that involves maximizing a linear function over a curved convex space. In typical LP problems, the convex space is a polytope, but in this case the ...

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### Symmetry of the integer gap

Are there results that bound the asymmetry of the duality gap of an integer program? That is to say, if the difference between the LP solution and the IP (primal) solution is $a$, is there a function ...

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### Applications of minmax theorem(s)

Intro We suppose $X$ and $Y$ are nonempty sets and f: $X\times Y \rightarrow \mathbb{R}$. A minimax theorem is a theorem that asserts that, under certain conditions,
$$ \inf_Y \sup_X f = \sup_X ...

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### How to solve Linear Programming problem with tighter Integer Programming constraints

I want to learn a bit about Linear Programming.
After some research, I decided to solve the Cutting Stock problem as an example to learn. After doing some more research, I feel like I finally ...

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### Proving that a binary matrix is totally unimodular

I'm working on a set of problems for which I can formulate binary integer programs. When I solve the linear relaxations of these problems, I always get integer solutions. I would like to prove that ...

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### Solving a system of linear inequalities — what is the dimension of the solution set?

It is well known how to solve a system of linear equations $A{\bf x} = {\bf b}$, but how do we solve a system of linear inequalities $A{\bf x} \leq {\bf b}$?
For the applications I have in mind the ...

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### Existence/Uniqueness of Nonnegative Solutions of Linear Systems of Equations

Suppose we have an $m$x$n$ matrix $A$, with $m\lt n$, and an $m$x$1$ vector $b$. Are there existence and uniqueness conditions characterizing nonnegative solutions of the system of linear equations ...

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### When is a triangular matrix totally unimodular?

I have an {0,1}, invertible, triangular matrix, that I would like to show to be totally unimodular. Are there any known results on the total unimodularity of classes of triangular matrices?

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### Feasibility of linear programs

It's known that finding the intersection of n halfplanes in 2-d takes $\Omega(n\log n)$ time. Does the lower bound apply if we change the question to deciding whether the intersection is non-empty?

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### A system of linear equations with linear constraints

Mathematical problem.
Suppose we have $2n$ indeterminates $x_1,\dots,x_n$ and $y_1,\dots,y_n$ (which are denoted by $q$ with indices and called abundances below) and $m$ subsets $P_1,\dots,P_m$ of ...