The linear-programming tag has no usage guidance.

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### a closed form lower bound solution for linear programming

Given a linear objective function and a system of linear constraints, is there any known closed form lower bounds for it?
to clearly express the problem assume that
$$
z(\mathbf{a,B,c})=\mathop {\inf} ...

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**1**answer

328 views

### Name search for special Linear Integer Program

I am looking for a name for the following question in literature!
(and if you know it, then it would be great)
I couldn't find it and due to wide audience here, presumably you know more. Thank you
...

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**1**answer

151 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 ...

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**1**answer

48 views

### Large scale least squares of non symmetric and non square problems

Given a system like $b=Ax$ with an non symmetric and non square $A$ I would like to solve it having many elements in $x$ (lets say $10^7$).
There is a large amount of algorithms for symmetric ...

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**1**answer

57 views

### Heuristic for choosing n-vectors from n-sets

my given problem is:
choose n-vectors from n-sets (one vector from each set) so that the biggest element in the sum of the chosen vectors is minimal. Unfortunately the problem is NP-hard. So I'm ...

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**1**answer

166 views

### Calculate the discrete set of points B which are in the convex hull of the set of points A

This problem is likely best described with the following picture:
Given the discrete set of points $A$ (shown in blue), I wish to calculate the discrete set of points that are contained within the ...

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**2**answers

121 views

### sensitivity analysis in conic optimization

I have a conic optimization of the form:
$\min_x \langle c, x \rangle$, s.t. $Ax = b$, $x \in K$.
Where $x \in \mathbb{R}^{n}$, $A$ is an $m \times n$ matrix, $b \in \mathbb{R}^m$, $K$ is a self ...

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**0**answers

69 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 ...

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**2**answers

58 views

### Linear Programm with matrix [closed]

Is there a name for problems like this
min norm(Cx)
Ax = b
where C is a matrix and norm is the maximum norm.
This is kind of like a linear Programm. Could this be rewritten as linear programm? Or Any ...

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110 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.
$$
...

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**1**answer

171 views

### submatrix of a given size with maximum frobenius norm

Let $I\subset \{1,2,\ldots,n\}$, and let $|I|$ denote its cardinality. Now given a Hermitian matrix $\mathbf{A}\in\mathbf{C}^{n\times n}$. I am interested in finding the subset $I$ that maximizes the ...

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66 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 ...

**2**

votes

**1**answer

300 views

### optimization of inverse matrix with constraint on matrix elements

everyone! I have this optimization problem with constraint.
$D$ and $T$ are symmetric matrices, where T is known and D is the unknown parameter.
$x$ and $v$ are two known p-dimensional vectors.
The ...

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**2**answers

928 views

### 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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**2**answers

142 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 ...

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**1**answer

68 views

### Integer point in a non-empty polytope

I have a high-dimensional, non-empty polytope $Ax\geq b$ sitting inside the cube ($0\leq x_i \leq 1$). Is there any general theory or technique to show that this polytope contains an integer point, ...

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**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?
...

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**1**answer

1k views

### Maximizing linear objective function with absolute values

This has be asked on other forums, though couldn't
find authoritative answer.
I have a linear program over the reals and don't
want to introduce integer or binary variables.
The objective function ...

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**2**answers

244 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 ...

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144 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:
...

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**2**answers

235 views

### Minimax theorem on a non convex domain

A minimax theorem is a theorem which states that under certain conditions on $\mathcal{X}$, $\mathcal{Y}$ and $f$:
$$ \inf_{x \in \mathcal{X}}{\sup_{y \in \mathcal{Y}}{f(x,y)}} = \sup_{y \in ...

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**1**answer

58 views

### generalization from linear programming solution [closed]

I have a series of similar linear programs that depend on an input vector $a\in A$ and whose solution is an output vector $b\in B$. I can solve them individually, but this is wasteful. I suspect that ...

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**1**answer

302 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 ...

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**1**answer

664 views

### Closed form solution for Least Squares Problem [closed]

I'm looking for closed form solution for the following equation:
$argmin_{X,y}(\sum_i{\parallel{a_i-Xb_i-y}\parallel^2})$, where $X \in\mathbb R_{m\times n}$ is a matrix and $y\in\mathbb R_{m\times ...

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**1**answer

157 views

### Equivalent method for maximum likelihood estimation of covariance parameters

My goal is to estimate the parameters of a covariance matrix $\Omega$, by maximizing the following log-likelihood function:
$$\log L(\vec\tau, \rho, \sigma \mid W, X) = -m\ln(\left | \Omega \right |) ...

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**1**answer

458 views

### Minimizing sum of absolute deviations

Suppose we want to find coefficients $b$ in $\underset{b}{\operatorname{argmin}} \displaystyle\sum\limits_{i=1}^n | y_{i}-b_{1}x_{i}-b_{0}\mid$.
If we rewrite this problem in terms of linear ...

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**0**answers

143 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 ...

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**1**answer

150 views

### Eigenvalue problem with quadratic constraints

$\circ$ Consider the following eigenvalue problem : $$Ax=\lambda x \hspace{0.5cm} (1)$$
where matrice $A \in \mathbb{R}_{n \times n}$ is a positive semi-definite with eigenvectors $x = ...

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**2**answers

194 views

### Survey on Compared Running Time: Ellipsoid Method vs. Simplex Method

If you look through papers on the Ellipsoid Method, there is a large agreement, that the Ellipsoid Method, although theoretically polynomial, is in practice way slower than the Simplex Method. ...

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1k views

### 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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**2**answers

487 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 ...

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**1**answer

68 views

### About the suboptimality of linear estimators

Let $X$ be a random variable and $N$ a Gaussian noise independent from $X$. We observe $Y=X+N$ and want to estimate $X$ based on $Y$ to minimize the mean square error $mmse(X|Y):=E(\hat X(Y)-X)^2$.
...

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**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 ...

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**1**answer

193 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 ...

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113 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 ...

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**2**answers

236 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 ...

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**1**answer

686 views

### eigen-decomposition solution? is it unique?

Assume an N*N covariance matrix (Q) which is a positive definite matrix. The decoder X is assumed to be N*s, where s<=N. X is calculated to be s eigenvectors corresponding to s minimum eigenvalues. ...

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votes

**1**answer

477 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 ...

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**0**answers

172 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 ...

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**1**answer

167 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 ...

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**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 ...

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**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 ...

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**1**answer

293 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 ...

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**1**answer

378 views

### Nonconvex optimization problem

I have a nonconvex optimization problem. It is actually optimizing a linear objective function over a set of linear constraints and a set of nonlinear, non convex constraints.
Is this problem ...

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135 views

### Rewrite optimization objective

Hi,
I wanted to ask, under which conditions can one rewrite the optimization objective
$\min_x f(x)\;\;\;s.t.\;\;\;g(x) \leq s$
as
$\min_x g(x)\;\;\;s.t.\;\;\;f(x) \leq t$
I have particular ...

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**2**answers

174 views

### Name of operations on two vectors

Suppose we have two vectors $x\in \mathbb{R}^n$ and $y\in \mathbb{R}^m$.
I could define the mapping
$$
T: \mathbb{R}^n\times \mathbb{R}^m \rightarrow \mathbb{R}^{n\times m}
$$
as follows
$$
T(x,y) = ( ...

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**2**answers

182 views

### positive semidefiniteness: a psd matrix substracted by another rank 1 psd matrix

Given that $A$ is a positive semidefinite matrix, $x$ is a vector, $\lambda_0 \in [0, +\infty) $ is a real non-negative number. I want to know the answer to the following optimization problem.
$$
...

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**2**answers

99 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 ...

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**1**answer

119 views

### Deducing Linear Inequalities

Let $X_1,X_2,\ldots,X_n $ be indeterminates. Denote by $S$ the set of all linear inequalities of the form
$X_{i_1}+X_{i_2}+\ldots+X_{i_k} \geq k,$
with $k \in \{ 1,2,\ldots,n \}$ and $1 \leq i_1< ...

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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 ...