# Tagged Questions

Linear programming is the study of optimizing a linear function over a set of linear inequalities. The Simplex Method, Ellipsoid Method and Interior Point Method are popular algorithms to solve linear programs.

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### How to find a closest integer point to intersection of two lines?

Here's a question that originates from StackOverflow (and the SO crowd isn't really qualified to solve it). We're given two lines on the plane, specified by equations ($a x + b y = c$) whose ...
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### An Interesting Optimization Problem

You are given n non-negative integers $a_1, a_2 ,, a_n$. In a single operation, you take any two integers out of these integers and replace them with a new integer having value equal to difference ...
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### Why are optimization problems called “programming”?

Why are optimization problems often called programs? linear programming geometric programming convex programming Integer programming ...
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### Is the tensor product of polyhedra a polyhedron?

Conventions: A polytope in a finite-dimensional $\mathbb R$-vector space $V$ is defined to be a convex hull of finitely many points in $V$. A polyhedron in a finite-dimensional $\mathbb R$-vector ...
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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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### Deciding membership in a convex hull

Problem: Given points $u,v_1,\dots,v_n\in\mathbb{R}^m$, decide if $u$ is contained in the convex hull of $v_1,\dots,v_n.$ This can be done efficiently by linear programming (time polynomial in $n,m$)...
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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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### 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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### Does the Hirsch conjecture hold for $n < 2d$?

The Hirsch conjecture asserts that the graph (i.e. $1$-skeleton) of a $d$-dimensional convex polytope with $n$ facets has diameter at most $n - d$. After being open for decades, Francisco Santos has ...
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### A polytope with a bound on the sum of any $k$ variables

Let $2\le k\le n-1$ and define the polytope $$P_k(n) = \lbrace (x_1,\ldots,x_n) \in \mathbb{R}^n : -1\le x_{i_1}+\cdots +x_{i_k} \le 1 \text{ for all } 1\le i_1\lt\cdots\lt i_k\le n\rbrace.$$ There ...
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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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### Inverse of a totally unimodular matrix

A unimodular matrix $M$ is a square integer matrix having determinant $+1$ or $−1$. A totally unimodular matrix (TU matrix) is a matrix for which every square non-singular submatrix is unimodular. A ...
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### Has this generalization of a determinant (assigning multiplicities to the rows) been studied?

I'm working on some questions in tropical geometry, and my problem led me to create the following generalization of a determinant: Let $A$ be an $m \times n$ matrix with $m \le n$, and positive ...
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### Algorithm that solves every Mixed Integer Linear Program (to optimality)?

Given a Mixed Integer Linear Program with rational coefficients (both for the objective functions and all constraints), is it always possible to solve it algorithmically? I know that you usually ...
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### Speed up Linear programming

I have a linear programming problem like this: minimize $c^t X$ under the constraint that $AX \ge b$. I will need to solve this linear programming problem online many times. I need it to be as fast ...
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### Resource Constrained Routing with Refueling

What are good algorithms (resp. models) for calculating optimal or near optimal routes while taking into account fuel consumption, options for refueling and, limited tank capacity? Especially modeling ...
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### Algorithm to minimally connect line segments in Euclidean plane

Suppose you have finitely many line segments in the Euclidean plane. How do you "connect them to form one chain of line segments of minimal length?" More formally and generally, what I'm looking for ...
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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?
I have a linear program over a finite set of points $(x_1, x_2,\ldots, x_m)\in\mathbb{R}^n$: $$\max_j c' x_j$$ Suppose the solution of this LP is obtained at a point $x_{j_1}$, which is a vertex ...