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22
votes
5answers
4k views

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

Hello. 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, each of them has at least two integer points ...
20
votes
4answers
1k views

Why are optimization problems called “programming”?

Why are optimization problems often called programs? linear programming geometric programming convex programming Integer programming ...
13
votes
0answers
1k views

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?
10
votes
1answer
514 views

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 ...
10
votes
2answers
720 views

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 ...
10
votes
1answer
1k views

Sum of difference moduli vs. sum of modulus differences

This is a failed attempt of mine at creating a contest problem; the failure is in the fact that I wasn't able to solve it myself. Let $x_1$, $x_2$, ..., $x_n$ be $n$ reals. For any integer $k$, ...
7
votes
1answer
265 views

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 ...
6
votes
1answer
233 views

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 ...
6
votes
2answers
3k views

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 ...
5
votes
1answer
238 views

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?
5
votes
2answers
248 views

relation between solution of a linear program and its perturbation

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 ...
5
votes
0answers
182 views

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 ...
4
votes
2answers
764 views

How do you tell if a system of linear inequalities has a solution?

A naive solution would be to optimize a dummy variable via linear programming and see if a result is returned. I imagine there must be a more direct way.
4
votes
2answers
194 views

point in polytope

Suppose I have the convex hull $P$ of a finite collection of points in $\mathbb{R}^d,$ and I want to see whether a point $p$ is contained in $P.$ This is a standard (some would say the standard linear ...
4
votes
1answer
206 views

Checking if one polytope is contained in another

Hi, I have two sets of inequalities, say, $Ax \leq 0$ and $Bx \leq 0$. I would like to know if they both define the same polytope. Or, even, whether one is contained in the other. At the moment I am ...
4
votes
2answers
356 views

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
4
votes
2answers
120 views

combinatorial and linear duality

Let $S$ be a finite set, and let $W$ be a nonempty set of subsets of $S$; we will identify every subset of $S$ with its characteristic function, a 0-1 vector in $\mathbb R^S$. The combinatorial dual ...
4
votes
2answers
479 views

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 ...
4
votes
1answer
2k views

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 ...
4
votes
2answers
354 views

Simplified knapsack problem

There is a problem that I can not solve. Given a set of items (each item has some integer weight) we have to fill bag with some number of copies of these items, with the only restriction that the ...
4
votes
1answer
224 views

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 ...
4
votes
1answer
740 views

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 ...
4
votes
2answers
1k views

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 ...
3
votes
1answer
2k views

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 ...
3
votes
1answer
326 views

Solving for Hamiltonian path with constraints on allowable routes through vertices

Suppose you have a complete graph with N vertexes, with a distinguished vertex $n=1$ ("start"), and you wish to find a route traveling exactly once through each vertex so that the distance along the ...
3
votes
2answers
2k views

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 ...
3
votes
2answers
2k views

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 ...
3
votes
1answer
149 views

Explicit formula for an LMI solution

Suppose we have a linear matrix inequality (aka LMI aka spectahedron aka linear matrix pencil): $$A_{0}+x_{1}A_{1}+x_{2}A_{2}+\ldots+x_{m}A_{m} \succeq 0.$$ (The notation $X \succeq Y$ means that ...
3
votes
1answer
258 views

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 ...
3
votes
1answer
316 views

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 ...
3
votes
0answers
475 views

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 ...
2
votes
2answers
230 views

convex polytope integer points

is there a simple proof for the following lemma: An unbounded convex polytope (defined by linear constraints) has either zero integer points or infinite many integer points.
2
votes
4answers
349 views

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 ...
2
votes
2answers
728 views

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 ...
2
votes
1answer
140 views

For interior point methods of linear programming, what is the “L” in the computational complexity $\mathcal{O}(n^3 L)$?

My question is about interior point methods of linear programming. Suppose the constraint matrix $A$ has $m$ rows and $n$ columns, and $m<n$. The state-of-the-art methods, like primal dual interior ...
2
votes
1answer
269 views

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 ...
2
votes
2answers
123 views

Finding the maximum of a multivariate polynomial of degree one

I need to find the global maximum of the function \begin{align} f\left(x\right) & = p_1 \max\left(\sum a_{1i} x_{1i}, \sum b_{1i} x_{1i}\right) - \sum c_{1i} x_{1i} \\ &+\ldots \\ &+ p_n ...
2
votes
1answer
323 views

Algorithm for satisfiability of inequalities.

I am looking for an algorithm for checking the satisfiability (with natural values) of a set of inequalities made of variables and natural numbers, for example: $u < v, u \leq z, 3 \leq v$. In ...
2
votes
2answers
511 views

how to model a linear program with step-like cost function in the objective

I have a large linear program with the following details. d1 to di are the variables, where di is an integer. The constraints are a series of inequalities of the form d1 < d3 < d7 < d23 ...
2
votes
2answers
1k views

Solving a system of equations/inequalities that have trigonometric functions on the left-hand side

Is there any known (symbolic) method that solves a system of equations/inequalities that have trigonometric functions on the left-hand side of the system? Ex) Find $x,y,\theta \in \mathbb{R}$ that ...
2
votes
1answer
135 views

Optimization problem whose cardinality never exceeds 7 for some reason

I am working on a problem in which I have a collection of $n$ points, $x_1,\dots,x_n$, in the plane, as well as a positive definite matrix $\Sigma$ and another point $\mu$ in the plane. I am trying ...
2
votes
1answer
195 views

Why does the LP Formulation of the MST Problem need Topology Constraints?

I am looking for an example that demonstrates the necessity of either subtour-elimination or of connectivity constraints in the LP formulation of the MST In the internet I only could find the LP ...
2
votes
1answer
747 views

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 ...
2
votes
1answer
268 views

Finding integer points inside of a parallelogram

Suppose $P = \{p_1,\ldots,p_4\} \in \mathbb{R}^2$ defines a quadrilateral (here, specifically, a parallelogram). In the particular case I'm dealing with, I know that there exists at least one point ...
2
votes
2answers
94 views

LP constraint enconding

I have an objective function to be maximized $obj(x) = \sum_i \gamma_i x_i$ with $x_i \in \mathbb{R}$ With multiple constraints of the form: $\min_{y \in 0,1} (\sum_{i \in A} \alpha_i x_i + \sum_{i ...
2
votes
0answers
51 views

Algorithms to find the solutions of a homogenous matrix equations for non-commutative rings

In one paper from 1980 I found a note that there are no known algorithms for solving homogenous matrix equations $x \cdot M = 0$ for matrices which elements belong to a non-commutative ring. (The ...
2
votes
0answers
25 views

In what paper was the shrinkage parameter introduced to the nelder-mead simplex direct search algorithm?

I have read lots of papers referencing a 4th shrinkage parameter when talking about the Nelder Mead Simplex method. However, I cannot see any shrinkage parameter in the flow chart of the original ...
2
votes
0answers
120 views

existence of lattice point in polytope

This question was probably asked before but here goes. I have a convex polytope given by $Ax\leq b$ for a specific integer matrix $A$ and integer vector $b$. I need a simple method/result on how to ...
2
votes
0answers
38 views

Put positive polynomial in finite intersection of half-spaces

This is a cross-posting of a MSE question (which did not attract any attention there so far). Denote by $V={\mathcal P}_{n,d}$ the space of polynomials in $n$ variables with degree at most $d$, ...
2
votes
0answers
126 views

How to implement linear constraints that include several absolute values

Dear all, I am trying to implement a linear constraint that includes several absolute values in the form: Abs(A) + Abs(B) + Abs(C) + Abs(D) + ... = 1 Since the minimization problem includes quite a ...