The linear-programming tag has no usage guidance.

**13**

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2k 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?

**6**

votes

**0**answers

98 views

### A specific case of the $p$-center problem

Given a fixed positive integer $m$, let $\cal{S}$ be the subset from $\mathbb{R}^m$ defined as $\cal{S} = \{u \in \mathbb{R}^m \mid \forall i \in \{1, \dots, m\}, u(i) > 0$ and $\sum_{i=1}^m{u(i) = ...

**5**

votes

**0**answers

61 views

### Finding the optimal mixture of two convex functions

I am trying to find an efficient way to solve the problem $$\min_{p,x_1,x_2} p\cdot f(x_1)+ (1-p) \cdot f(x_2)~~~~~ s.t.\\p\cdot g_1(x_1) + (1-p)\cdot g_2(x_2)\leq 1 \\ 0\leq p \leq 1$$ where ...

**5**

votes

**0**answers

170 views

### A linear optimization problem on a graph

Let $G=(V,E)$ be a finite graph and let $f$ be any positive function defined on the vertices. Put weights on the vertices $v_{i}$, way $w_{i}$ so that $\sum_{i=1}^{n}w_{i}\leq 1$. Assume that every ...

**5**

votes

**0**answers

183 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

**0**answers

141 views

### Variant of orthogonal Procrustes problem

The orthogonal Procrustes problem seeks a matrix $M$ that minimizes $||AM-B||_F$ subject to $M^TM=I$, where $M$ is $d\times d$ and both $A$ and $B$ are $n\times d$. Geometrically, $M$ rotates a set of ...

**3**

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883 views

### 0,1 solution to system of linear integer equations.

I have the following problem:
$A x = b$
where $A, b$ - $m \times n$-maxtrix and $m$-vector of nonnegative intgers (respectivelly).
$x \in \{0,1\}^n $ - vector of binary variables, which need to be ...

**3**

votes

**0**answers

186 views

### Could SVD be used to optimize the partial inner-products?

Suppose a set $N$ of $n$ distinct points in $m-$dimensional space is given in $X\in\mathbb{R}^{n\times m}$. Also, suppose a subset $L\subset N$, $|L|=l<m<n$, with
$m-$dimensional coordinates in ...

**3**

votes

**0**answers

545 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

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60 views

### Listing all Lattice Points in a Box

Let $B := [-1,1]^n$ be an $n$-dimensional box. Moreover, let $v_1,\ldots,v_n \in \mathbb{R}^n$ form a basis of $\mathbb{R}^n$, where the entries of the $v_i$ are explicitly irrational. We can assume ...

**2**

votes

**0**answers

45 views

### Derivation of gradient of SSE in Geodesic Regression

On page 79 (or page 5) of this this paper the gradient of the SSE of the Geodesic model is described explicitly. My question is how are these equitations derived in detail; where can I find the ...

**2**

votes

**0**answers

68 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

**0**answers

27 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

**0**answers

135 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

**0**answers

44 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

**0**answers

419 views

### 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. ...

**2**

votes

**0**answers

238 views

### 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 ...

**2**

votes

**0**answers

168 views

### 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 ...

**2**

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**0**answers

276 views

### 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 ...

**2**

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**0**answers

265 views

### 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 ...

**2**

votes

**0**answers

536 views

### Is it possible to use linear programming to solve this problem?

I am trying to write software to minimize pricing for cell phone subscription services, ie: choose the optimum plan for each customer in a large group.
Could someone comment on whether this is ...

**1**

vote

**0**answers

53 views

### Finding all feasible solutions

Let $u$ be a $n_{max} \times m$ matrix. Let $z$ be a $n_{max} \times s_{max} \times n_{max}$ cube. Let $w$ be a $n_{max} \times 1$ vector. All the three matrices can have values from the set $\{ 0, ...

**1**

vote

**0**answers

49 views

### Finding optimal linear transformation for intersection of convex polytopes

I previously posted this on MathSE and am now trying here.
I have the following situation, as shown in the following diagram:
$W=\{w_i\}_{i=1..|W|}$ is a set of vertices (show in diagram in ...

**1**

vote

**0**answers

34 views

### The column generation technique on a Train Unit Assignment Problem [Linear Programming]

I am doing an assignment where I need to implement a mathematical model that I can't wrap my head around. For the technique of column generation, one would need to my understanding, a master problem ...

**1**

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**0**answers

51 views

### Select n vectors from k vectors (in 3D) such that each component of the resultant vector >= each component of a given vector M

this was left unanswered for 1 week on MStackExchange, so I thought MOverflow would be more appropriate. Thanks :)
Let $R = (R_x, R_y, R_z)$ be the resultant vector of the n vectors and $M = (M_x, ...

**1**

vote

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127 views

### Reduce a Combinatorial problem

It is given n sets with k vectors. (k is element-wise positive or zero)
Choose one vector of each set so that the biggest element of the sum of the chosen vectors is minimal.
What i also know but is ...

**1**

vote

**0**answers

64 views

### integrality of a linear program — binary equality constaints

Consider the following linear program:
$\left\{
\begin{array}{l}
\underset{x}{max} \;\;c^Tx\\
[I, \;B]x = \mathbf{1}\\
x\geq 0
\end{array}
\right.$
where $c$ is a vector ...

**1**

vote

**0**answers

62 views

### Are there any known bounds on the value of solutions of linear integer programming?

Given a linear objective function and a system of linear constraints; are there any known bounds on the values of (positive) integral solutions in terms of the coefficient matrix of the constraints?
...

**1**

vote

**0**answers

138 views

### Complexity of Nested Linear Optimization

My question is motivated by the fact, that among other ways, it is possible to restrict a variable to two discrete values, e.g. the prototypical $0$ and $1$, via an optimization constraint:
...

**1**

vote

**0**answers

109 views

### Interior point optimisation using big M for L1 norm on linear system using Dikin's Affine method

I am a 4th year undergrad surveying student studying computations, specifically $L_{1}$ norm minimisation of residuals in large data sets. To start with (and probably to finish with) I'm using a set ...

**1**

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169 views

### Equal maximum and minimum in a large-scale linear programming

For a linear optimization of an integral (with integral constraints), I perform a linear programming for the equivalent series. Maximum and minimum of the LP problem tend to be equal as I increase the ...

**1**

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**0**answers

76 views

### Arrangements of graphs of linear functions

Fix $n>0$ and $X\subseteq\mathbb{R}^n$. Assume $X$ is unbounded.
A function $f:X\longrightarrow\mathbb{R}$ is linear if it is of the form
$$
f(\bar{x})=a_1x_1+\ldots+a_nx_n+b
$$
for some ...

**1**

vote

**0**answers

103 views

### Matrix Minimax problem

I have the equation $\Sigma_k(M_k{p_k})V=EV$, where the $M_k$ are n*n real Hermitian matrices, $V$ is a n*n eigenvector matrix, $E$ a dim-n energy eigenvector and the $p_k$ scalar parameters. The ...

**1**

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**0**answers

164 views

### Consistency of a system of linear equations

I have a system of linear equations in form of $AX=b$ where $A_{m\times n}$, $X_{n\times 1}$ and $b_{m\times 1}$. Coefficient matrix $A$ is quite sparse. However, using a practical LP solver like ...

**1**

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**0**answers

253 views

### Totally Uni-modular Matrices

A matrix is totally uni-modular if the determinant of any (square) sub-matrix is {+1, 0, -1}. My question is, "Is there a way to transform(linear or non) a general matrix into a totally uni-modular ...

**1**

vote

**0**answers

738 views

### How to solve simple bilinear equations under extra linear constraints

Hello,
This is the full version of a question I asked earlier. I am trying to understand whether finding a solution to the following bilinear system is computationally hard or easy:
$\lambda_i^T ...

**1**

vote

**0**answers

3k views

### 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 ...

**1**

vote

**0**answers

1k views

### Covariance matrix formula interpretation - what am I missing?

I'm reading a paper that outlines the calculation of a covariance matrix like the following:
$C=\displaystyle\sum^{N_b}_{i=1}\vec{x}_i\vec{x}_i^T$
What is the order of this matrix? My interpretation ...

**0**

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**0**answers

56 views

### Is there a space in which the $\vec a$ in $\sin(a_1\cdot x)+\sin(a_2\cdot x)$ is linear?

Suppose one has equations of the form $\sin(a_1\cdot x_i)+\sin(a_2\cdot x_i)=y_i$ for $i=1...n$ (there are also amplitudes and phase shifts, but let's ignore those for now). I want to solve for $\vec ...

**0**

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**0**answers

44 views

### Finding a movement taking out of a convex set

There is a convex set $S$ as the hull of M points in an D-dimensional Euclidean space and a point $\vec P$ in the set. Then, there is a set of vectors $\vec W$ taking the form $\vec W=\sum_{i=1}^N ...

**0**

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**0**answers

15 views

### How to solve a special linear system with noise?

Sorry the title may be confusing. I'm not so sure how to categorize this problem.
Anyway, We have a real-valued vector $\overrightarrow a=(a_0,a_1,...,a_{2^n-1})$. Each $a_i$ comes from the inner ...

**0**

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**0**answers

27 views

### Find relationships between events

I have a set of Events $(E_i)_i$ which have a probability $(P_i)_i$.
I am able to write each event as a sum of distinct events that form a partition of the space.
My goal is to find all the ...

**0**

votes

**0**answers

107 views

### Inverse problem with a rank-1 update

I hope you can help me out with this. I have to find the solution x to an inverse system
$$
x=A^{-1}b
$$
This inverse problem is basically a least square problem with a rank-1 update.
$$
...

**0**

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**0**answers

74 views

### Big eigenvalues of a special stochastic matrix

Given a matrix $M$ of size $n\times n,$ we write its different eigenvalues by $x_1,x_2,\ldots,x_m$ with $m\leq n$ such that $|x_1|>|x_2|>|x_3|>\cdots|x_m|,$ and call $x_2\doteq ...

**0**

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93 views

### Gauss-Newton for quotient functions

I'm optimizing a function of the form
$$
\sum \frac{ \|\mathbf{f_i}(x)\|^2 }{ g_i(x)^2 + h_i(x)^2 }
$$
where $x$ is a real vector, $\mathbf{f}(x)$ is a real vector, and $g(x)$ is a scalar. My first ...

**0**

votes

**0**answers

180 views

### A linear program related question

Dear all, recently, I encountered the following problem. It is closely related to the order of growth for UMD constants of all $n$-dimensional Banach lattice.
Let $\alpha^k \in (\alpha_1^k, ...

**0**

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71 views

### Computing maximum point for minimal function of a family of linear functions

Let $x \in S^n $ where $S^n = ${$ [x_1,x_2,...,x_{n+1}]\in \mathbb{R}^{n+1} \mid x \ge 0 , \sum x_i = 1 $} and let $f_i : I^n \to \mathbb{R}$ be a finite $m$-sized family of LINEAR functions such ...

**0**

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204 views

### LP relaxation for ILP\IP (integer linear programming)

I am familiar with LP relaxation for ILP (or IP). Assume we concern with integer minimization problem, which we formalize using ILP; we then relax the ILP into LP and we say that the LP provides a ...

**0**

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111 views

### sparsest cut always has solution

Hi!
How to prove that sparsest cut always has an optimal solution which is the cut for some vertex-subset.
Looks like it's should be a kind of fundamental theorem for sparsest cut. But I didn't ...

**-1**

votes

**0**answers

51 views

### About perturbation of spectral radius of a matrix because of diagonal perturbations

Say I have an off-diagonal symmetric $0,1,-1$ matrix $B$ and a set of $2k$ diagonal matrices, $D_{11}, D_{12}, D_{21}, D_{22},..,D_{k1},D_{k2}$. (you can assume the diagonal matrices to be such that ...