Tagged Questions
3
votes
2answers
66 views
Maximize the determinant
Over the class $S$ of symmetric $n$ by $n$ matrices such that the diagonal entries are +1 and off diagonals are between $-1$ and $+1$ (inclusive/exclusive), is
$$\max_{A \in S} \det A = \det(I_n)$$
...
0
votes
1answer
17 views
Extremum of a multidimensional quadratic function
I have the following function:
$$
g(h) = h'\Sigma\Sigma'h-h'm-r,
$$
where $h$ is a vector in $\mathbb{R}^M$, $\Sigma$ is a $M\times K$ matrix such that $\Sigma\Sigma'$ is positive definite and has ...
0
votes
0answers
17 views
Lower triangular constraint
How can I solve this optimization problem:
$\min_L (X^T L L^T X)$ s.t. $L$ is lower triangular
I want to use gradient descent but do not know how to add the constraint.
0
votes
1answer
33 views
Monotonically Increasing Mapping?
$\mathbf{h}_1, \mathbf{h}_2\in\mathbb{C}^{n}$ are given column vectors and $a>0$ is a given constant. Consider the matrix ...
1
vote
1answer
60 views
Estimate Euler angles between rotated coordinate system via Newton-method based on position vectors
I've got $N$ position vectors $\mathbf{a}_i = \begin{pmatrix} a_{i,x} \\ a_{i,y} \\ a_{i,z} \end{pmatrix}$ in one coordinate system and $N$ corresponding position vectors $\mathbf{b}_i = ...
0
votes
1answer
45 views
Minimisation of Concave Function
$f\left(\mathbf{x}\right):\mathbb{R}_+^n\rightarrow\mathbb{R}_+$ is a concave monotonically increasing function to be minimised over the feasible region
$\sum_{i=1}^n x_i=1$ and $x_i\geq ...
1
vote
1answer
64 views
Optimization problem to find an optimal matrix
I need to find a $n\times m$ matrix $N$ with binary values $(0,1)$ which will maximize an objective function. N(i,j)=0 or 1 indicates whether jth offer is made to ith customer
$m$ represents number ...
6
votes
2answers
114 views
What is the maximum possible value of determinant of a matrix whose entries either 0 or 1?
My question is simply the title:
What is the maximum possible value of determinant of a matrix whose entries either 0 or 1 ?
3
votes
1answer
64 views
Finding the smallest subset of a set of vectors which contains another vector in the span
Consider a set $S=\{ \underline{v_1},\dots , \underline{v_n} \} $ of vectors of dimension $d<n$. Suppose for some vector $\underline{b}$ that the solution space for the matrix equation
$\left[ ...
2
votes
2answers
62 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.
$$
...
0
votes
0answers
70 views
using fmincon in matlab
Say i have a function f(X) which i want to minimize with constraints such that some other functions- A(X) = 0 and ...
0
votes
0answers
35 views
Least square distance between matrices under linear constraints
I have got a set of $n$ elements and a square matrix
I = $
\begin{bmatrix}
I_{11} & I_{12} & \dots & I_{1n} \\
I_{21} & I_{22} & \dots & I_{2n} \\
\vdots & \vdots & ...
0
votes
1answer
38 views
Notation minimum of a column vector
I'd like to know the notation to express the minimum of a column vector.
Is this notation correct?
\begin{equation}
\min
\left[\matrix{
\left|b_{n}-b_{n+1}\right| \cr
...
2
votes
1answer
44 views
Ask a question about an example in a course note on optimization problem with equality constraint
I have two difficulties on understanding the solution to an example in a course I took this semester on optimization. This example is given to illustrate the usage of Lagrange multiplier method ...
0
votes
1answer
46 views
Efficient (approximate) projection onto the special orthogonal group
I need to carry out an optimization on the special orthogonal group $SO(n)$. For the line search I use a simple back-projection method
$$\mbox{minimize}_\tau f(\pi(X+\tau Z))$$
where $X\in SO(n)$ ...
0
votes
2answers
106 views
Transpose of matrix inverse: $(AA^T)^{-1}A^Tb \stackrel{?}{=} (A^TA)^{-1}A^Tb$
Given the matrix equation:
$$ x^TA^TA = b^TA $$
I'm trying to find the least squares solution (i.e.; trying to minimize $r=||Ax-b||$). The matrix $A$ is not necessarily symmetric.
When I solve it ...
2
votes
0answers
49 views
Use of low rank approximation of a matrix
I am trying to figure out why do we need a low rank approximation of a matrix. Why is it used and where? Any insights?
0
votes
1answer
93 views
Solve: This System of equations for $X$ (does a real solution, exist?)
How can I solve $AX + diag(X)[I-c]=0$ for $X$?
All matrices have real entries, $diag(X)$ is a diagonal matrix with the diagonal entries being the diagonal entries of $X$, and $c$ is a constant, real ...
0
votes
2answers
56 views
Is this a linear programming problem?
So I'm trying to solve this problem.
We are given an image ( a two dimensional matrix ).
The image is all white except for some red dots. We are given the list of the red dots ( (X,Y) pairs ). The ...
3
votes
2answers
243 views
Solution of a Sylvester equation?
I'd like to solve $AX -BX + XC = D$, for the matrix $X$, where all matrices have real entries and $X$ is a rectangular matrix, while $B$ and $C$ are symmetric matrices and $A$ is formed by an outer ...
1
vote
3answers
146 views
A constrained linear least Frobenius norm problem:$\min_{X} \|A-XB\|_F$ subject to $Xv=0$?
Assume we are given two matrices $A, B \in \mathbb R^{n \times m}$ and a vector $v \in \mathbb R^n$. $\|\cdot\|_F$ is the Frobenius norm of a matrix. How can we solve
$$\min_{X \in \mathbb R^{n ...
0
votes
1answer
34 views
inequalities for optimization over psd matrices with constraints
Consider two p.s.d. matrices $A$ and $B$ both in $\mathbb{R}^{d \times d}$. Define $$a = argmax_{x \in \mathbb{R}^d} x^\top A x $$ and $$b = argmax_{x \in \mathbb{R}^d} x^\top B x $$ both subjected to ...
2
votes
0answers
34 views
maximal m-elements of the matrix inversion
Suppose the $n\times n$ matrix $A$ is invertible, and all its elements are between 0 and 1. The existing matrix inversion operation of $A^{-1}$ will take $O(n^3)$ time. Now I just want to find the ...
2
votes
0answers
148 views
Optimization problem about large matrices
I'd like to solve the following optimization problem:
Find non-negative scalar $a$, $b$, $c$ to minimize
$\| (D-(aA+bB+cC+D^{-1})^{-1})y\|^2+2\operatorname{trace}((aA+bB+cC+D^{-1})^{-1})$
where ...
0
votes
0answers
45 views
Matrix Partioning with number of rows being optimized.
Given a ternary matrix G which is an N*M matrix with values containing all entries 0,1,2.
Is there a methodology/algorithm wherein we could write another binary matrix H of dimension 2N*M such that ...
2
votes
5answers
205 views
How to minimize $\| y- Ax\|$ subject to $\|x\|=1$ and $x \geq 0$?
Given $y \in \mathbb R^n$ and $A \in \mathbb R^{n \times n}$, whis is some way for
$$\min_x \| y- Ax\|$$ subject to $\|x\|=1$, and $x \geq 0$ (which means every components of $x$ is nonnegative)?
...
1
vote
2answers
187 views
Linear optimization problem: Minimizing a linear function over an affine set.
The problem is as follows:
Give an explicit solution of the linear optimization problem below.
$$
\text{minimize}\ c^Tx \\
\text{subject to}\ Ax\ =\ b
$$
No other information is given.
My ...
2
votes
1answer
602 views
Armijo's rule line search
I have read a paper (http://www.seas.upenn.edu/~taskar/pubs/aistats09.pdf) which describes a way to solve an optimization problem involving Armijo's rule, cf. p363 eq 13.
The variable is $\beta$ ...
2
votes
0answers
74 views
Binary optimization
Let me first make my background clear. I am a PhD student with not much knowledge in optimization but I need to do some optimization as a part of my research work. My problem is as follows:
There are ...
2
votes
1answer
83 views
Proof: Ratio of matrix traces and difference of traces
$\newcommand{\Tr}{\operatorname{Tr}}$
Am looking for a proof that shows that the minimization of $\frac{\Tr X^TAX}{\Tr X^TBX}$ is equivalent to the minimization of $\Tr X^TAX-\lambda \Tr X^TBX$ for ...
