All Questions
783 questions
1
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Analytic formula for minimizing the maximum inner product of a set of vectors
Given $x_j\in\mathbb{R}^n$, $j=1,\ldots,p$, find
$$
\widehat{w} \in \arg\min_{\Vert w\Vert=1}\max_{1\le j\le p} |\langle w,x_j\rangle|.
$$
I am also interested in the special case where we further ...
- 710
1
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0
answers
187
views
Strong Duality of Mixed Integer Linear Program
The problem at hand is to optimize a mixed-integer linear program closely related to the maximum flow problem. I would like to reformulate the problem with its dual and I'm concerned with the ...
- 11
1
vote
0
answers
173
views
Matrix exponential bounds
Let $A$ be a stochastic matrix, $q\in (0,1)$. How to bound $n$ such that $$q^n A^n e^A \leq e^A$$
Note that here $e^A$ is the matrix exponential, and $\leq$ is taken entrywise.
To be clear, what I ...
- 502
1
vote
0
answers
64
views
Maximize discrete harmonic function at given point
Let $n>0$, and let $S_n$ denote the discrete square
$S_n=[|-n,n|]^2$ (so $S_n$ has $(2n+1)^2$ elements). Let $K_n$ denote the set of four corner points $\lbrace (\pm n,\pm n)\rbrace$, and $C_n=S_n\...
- 621
1
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0
answers
55
views
Separation on discrete set
Consider the set $L = \prod_{i=1}^n\{1,0\}$, i.e. L consists of the element of n-tuples whose entries are 0 or 1. Also we can regard $L$ as a subset of $R^n$.
Define linear functions $f(x)= a_1x_1+ \...
- 61
1
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0
answers
364
views
The application of recursive SVD [closed]
Given an n*m matrix A, the SVD decomposition of A is ${\rm SVD}(A)$= $USV^t$.
The application of SVD to the product of U and S gives as a result the same matrices multiplied by the identity matrix, i....
- 11
1
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1
answer
219
views
approximate diameter of polytopes in high dimensions
I just came across the following problem:
Let us consider the unit corner of the n-cube
$$
\Delta^n = \left\{(t_1,\cdots,t_n)\in\mathbb{R}^n\mid\sum_{i = 1}^{n}{t_i} \leq 1 \mbox{ and } t_i \ge 0 \...
- 75
1
vote
0
answers
171
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\}...
- 175
1
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0
answers
120
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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 ...
- 111
1
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1
answer
279
views
Splines linearly independent
Let $N_1:=\chi_{[0,1]}$ be defined as this characteristic function and $N_n:=N_{n-1}*N_1$ then this leads to polynomials with support $[0,n]$. These splines are well-studied click for wikipedia My ...
- 231
1
vote
0
answers
85
views
Smallest sum of original column entries in 2d matrix [closed]
I have an interesting optimization problem I am trying to solve now and I thought I'd share it here in order to find the best answer. The problem itself is not complicated and it is stated like this:
...
- 111
1
vote
0
answers
101
views
How to fit a stochastic matrix to given data.?
Given a data sequence of noisy observations of a 3-state Markov chain $X$ -- $y_1$,$y_2$,...$y_n$, with two transition matrices $A_1$ and $A_2$ corresponding to different regions (**) in the (unit) ...
1
vote
1
answer
6k
views
Convert linear programming problem into its standard form [closed]
all,
I met a question that, the cost function of the linear programming problem is a function with absolute value. Here is the problem:
min 3x1+|6x2+3|
st.
|x1+4|+|2x2|<=3
How can I deal with it?...
- 13
1
vote
0
answers
140
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 ...
- 11
1
vote
2
answers
172
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 ...
- 11
1
vote
0
answers
75
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?
...
- 111
1
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0
answers
493
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:
$$\max(\...
- 13.2k
1
vote
0
answers
168
views
How to solve a divergent linear system using iterative methods?
I have a matrix A which is symmetric and non-diagonal dominant. I tried to use Jacobi/Gauss-Seidel/SOR to solve it but it diverges. Is there any mechanism to condition the matrix for convergence using ...
- 121
1
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0
answers
377
views
The rationale of QR algorithm for computing eigenvectors
For a symmetric matrix $A$, the rationale for the success of applying QR to compute the spectral decomposition of $A = UDU^T$ is, for large $k$, the QR factorization of $A^k = Q_kR_k$ obeys,
\begin{...
- 55
1
vote
0
answers
196
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 ...
- 111
1
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0
answers
256
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 ...
- 111
1
vote
0
answers
1k
views
Robust optimization in matlab using fmincon [closed]
I am trying to implement the following optimization (from this paper) in Matlab using fmincon:
$\min_\omega\omega'\Sigma\omega$ subject to $\min_Ur_p \geq r_0$
where $\Sigma$ is a positive definite ...
1
vote
0
answers
126
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 $M_k$...
- 4,829
1
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0
answers
628
views
Totally unimodular 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 ...
- 11
1
vote
1
answer
241
views
Covering max flow arcs by arc disjoint paths
Let $(N,A,s,t,u)$ be a network with node set $N$, arc set $A$, source $s\in N$, sink $t\in N$ and capacity vector $u\in\{1,2,\ldots,T\}^A$, and let $x=(x_a)_{a\in A}$ be a maximum $(s,t)$-flow. Is it ...
- 2,966
1
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0
answers
1k
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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 u_{...
- 53
1
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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 ...
- 111
0
votes
1
answer
1k
views
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 ...
- 135
0
votes
1
answer
769
views
Proving that the eigenvalues of a certain matrix product are positive
Let $A$ be an $m \times n$ matrix, and define:
\begin{align*}
U &= {\rm diag} \{ \frac{1}{\beta_j} \}, \beta_j = \sum_{k=1}^m |a_{kj}|, j = 1 \dots n \\
V &= {\rm diag} \{ \frac{1}{\alpha_i} \}...
- 3
0
votes
1
answer
320
views
Correct way to conduct equilibrium scaling of linear/integer/MIP program
I would like to scale my linear/integer program and also mixed-integer program using the equilibrium scaling method. I have worked on two research papers and one research book. However, they did the ...
- 21
0
votes
2
answers
97
views
Optimization algorithms for Kronecker approximation of high-dimensional covariance matrices
I'm working with a high-dimensional covariance matrix and exploring Kronecker product approximations to make it computationally manageable.
Here's the setup:
I have a graph $G$ represented by a $D\...
- 1
0
votes
1
answer
539
views
Method for (binary) optimization under constraints
I would like to know if there is a method to solve the Problem.
Problem:
Maximize the following function: $$f(p_{1,i},p_{2,i},\dotsc,p_{m,i})=\sum_{i=1}^{n}\begin{bmatrix}p_{1,i} & p_{2,i} & \...
- 3
0
votes
1
answer
226
views
Fractional values in linear programming
Consider the linear programming problem
\begin{align}
f^* = \max_{x}&~p^Tx~\\~&A^Tx\leq b~,~0\leq x_i\leq 1
\end{align}where $p$ is a $n\times 1 $ vector, $A$ is a $n\times c$ matrix and $b$ ...
- 1,421
0
votes
1
answer
144
views
Maximize function on rotation matrices [closed]
Let $A$ be a fixed 3-by-3 matrix and $Q$ be a rotation matrix whose yaw, pitch, and roll angles are $\phi\in[0,\pi]$, $\theta\in[0,\pi]$, and $\psi\in[0,\pi/2]$, respectively:
\begin{equation}
Q=
\...
0
votes
1
answer
540
views
Computing spectrum of convex combination of SPD matrices given individual spectral decompositions
Given the spectral decompositions of a non-commuting collection of symmetric positive definite $N\times N$ matrices $$\left\{ K_{i}\right\} _{i=1}^{M}, U_{i}D_{i}U_{i}^{T}=K_{i},\quad i=1,\dots,M,$$ ...
- 133
0
votes
1
answer
110
views
How to show inconsistency of a linear system of equations? [closed]
Consider the (real) linear system of equations $A\mathbf{x}=\mathbf{c}$ of size $N$ as
$$
\begin{bmatrix} a_{N}-a_{2}& a_{2}& 0 &\dots& 0 & -a_{N} \\-a_{1} & a_{1}-a_{3}&a_{...
- 105
0
votes
1
answer
338
views
Is spectral properties a general term for condition number?
I am reading an article about solving large sparse linear systems, in this paper it’s said that most of the iterative methods to solve $Ax = b$ are very much influenced by the spectral properties of ...
- 253
0
votes
3
answers
7k
views
Find an $N$-dimensional vector orthogonal to a given vector
I'm writing an eigensolver and I'm trying to generate a guess for the next iteration in the solve that is orthogonal to to all known eigenvectors calculated thus far. This means that if I have only ...
0
votes
1
answer
36
views
Benefit of adding a trivial constraint to ILPs
let ILP be an integer linear program with constraints-matrix $\boldsymbol{\mathrm{M}}\in\mathbb{Z}^{m\times n}$ and cost vector $\boldsymbol{\mathrm{c}}\in\mathbb{Z}^n$,
${\boldsymbol{\mathrm{x}}^*}\...
- 13.2k
0
votes
1
answer
93
views
How quickly can this IQP or its MILP relaxation be solved
Let $A\in\{0,1\}^{(n,n)}$ be a $n$ by $n$ boolean matrix (in particular think of an adjacency matrix of a graph), and consider the following optimization problem:
$$\begin{align*}&&\max_{P\in\{...
- 71
0
votes
1
answer
131
views
How hard is a linear programming with a bounded constraint?
Background: I am reading Greg Kuperberg's answer to the question Deciding membership in a convex hull. I am thinking about the complexity of ''Deciding membership in a convex hull''.
Restate the ...
0
votes
1
answer
214
views
How do you call a linear programming problem when the solution should be "constrained" to a norm?
(apologies for the n00b question)
Let's say we have a vector of length $n$, with to-be-determined values: $a_1, a_2, ...,a_n$.
And we have information that partial sums of these elements are equal to ...
- 143
0
votes
1
answer
112
views
Is there a specific name for this optimization problem?
Let $A$ be an $n\times n$ symmetric positive definite matrix with eigenvalues and eigenvectors $\lambda_1\ge\lambda_2\ge\cdots\ge\lambda_n>0$ and $v_1,v_2,\cdots,v_n$ respectively.
We know that the ...
- 205
0
votes
1
answer
535
views
Conditions to solve linear system with matrix blocks
How to verify if a linear system of symmetrical matrix blocks has solution?
I have the matrix:
$\left[M\right]_{p \times p}$, symmetrical
$\left[G\right]_{p \times q}$
and then, I would like to ...
- 253
0
votes
1
answer
320
views
Sub optimal algorithm for linear programming
Consider the linear programming problem
\begin{align}
f^* = \max_{x}&~p^Tx~\\~&A^Tx\leq b~,~0\leq x_i\leq 1
\end{align}where $c$ is a $n\times 1 $ vector, $A$ is a $n\times c$ matrix and $b$ ...
- 1,421
0
votes
1
answer
39
views
Gluing simplices through a common point/ realisation of a convex simplicial polytope
Given $m≥d+1$
a positive integer, is it always possible to find m d-dimensional simplices $\Delta_i=\mathrm{Conv}(M,V_{i,1},…,V_{i,d})$ such that
1) they all share the common vertex M
2) the ...
- 1,435
0
votes
1
answer
212
views
How to find out if a polytope contains a sphere?
Given a polytope described by linear inequalities $Ax \le b, x \in \mathbb R^n$, how do you find out if there exist a (non degenerate) sphere of dimension $n-1$ contained in the polytope?
Thanks!
- 225
0
votes
1
answer
130
views
Cascading minimization problems
Hi all. Suppose I have a linear programming problem on the vector variable $x$ that has many solutions and let $U$ be the set of these solutions. Suppose I have a second LP problem on $y \in U$. ...
- 57
0
votes
2
answers
891
views
Find both maximum and minimum values in linear programming problem
Hi all. I have a linear programming problem where I need to find both maximum and minimum values of the objective function. The optimal points are not relevant.
Is there an efficient way to do so?
- 57
0
votes
2
answers
4k
views
Linear programming piecewise linear objective
I am fairly new at linear programming/optimization and am currently working on implementing a linear program that is stated like this:
max $\sum_{i=1}^{k}{p(\vec \alpha \cdot \vec c_i)}$
$s.t. $
$|\...
- 3