All Questions
15 questions
7
votes
1
answer
373
views
Is the solution of this optimization problem always positive semidefinite?
We are given a set of unit vectors $U \subset \mathbb{C}^n$ which spans the space $\mathbb{C}^n$. Given another unit vector $x$, consider then the following optimization problem:
$$ \sup_H \left\{ x^*...
- 73
7
votes
0
answers
217
views
Characterizing matrices with rank constraint
Given matrix $M\in\Bbb\{0,1\}^{n\times n}$, is there a nice method to characterize $$\mathscr{D}[M]=\{Q\in\Bbb\{0,1\}^{n\times n}:\mathsf{rank}(M-Q)= \mathsf{rank}(Q),\quad M-Q\in\Bbb\{0,1\}^{n\times ...
- 13.9k
5
votes
2
answers
480
views
Decomposition of a semi-definite matrix into sums of sparse semi-definite matrices
I'll first provide the background.
Let $x\in\mathbb{R}^N$ be decomposed into $n$ non-overlapping blocks of variables
$x^{(1)},\ldots,x^{(n)}$.
We say that $f:\mathbb{R}^N\rightarrow\mathbb{R}$ is ...
- 51
5
votes
2
answers
429
views
Simultaneous maximization of two Generalized Rayleigh Ritz Ratios
Consider hermitian positive semi-definite matrices $A_1$ and $A_2$. Consider also positive definite matrices $B_1$ and $B_2$. I want to maximize the minimum of the two Generalized Rayleigh Ritz ratios ...
- 1,421
4
votes
3
answers
1k
views
Solving a quadratic matrix equation with fat matrix
I am trying to find an $n \times m$ fat (i.e., $m > n$) matrix $T$ that solves
$$T^T T = X$$
where $X$ is a given $m \times m$ symmetric, positive semidefinite matrix.
I saw this post, but ...
- 143
3
votes
1
answer
416
views
What's the best orthonormal matrix to align two matrices in the operator norm sense?
Let $A,B \in R^{n\times r}$ with $A^\top B $ invertible. It is known that
\begin{equation}
UV^\top :=\arg\min_{R \in \mathcal{O}^{r\times r}}\|AR-B\|_\mathrm{F},
\end{equation}
where $USV^\top$ is ...
- 515
3
votes
0
answers
83
views
How many local maxima can $(x_1,\dots,x_r)\mapsto\|x_1A_1+\dots+x_rA_r\|_\infty/\|(x_1,\dots,x_r)\|_2$ have for Hermitian $A_1,\dots,A_r$?
Let $K\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$. Suppose that $A_1,\dots,A_r\in M_n(K)$ are all Hermitian.
Define a function $f_{A_1,\dots,A_r}:\mathbb{RP}^{n-1}\rightarrow[0,\infty)$ by setting
$$f_{...
2
votes
2
answers
765
views
Why eigenvectors optimize this orthogonally constrained nonlinear minimization problem?
Given a $p \times p$ positive definite matrix $\Sigma$, why eigenvectors of $\Sigma$, stacked as columns of a matrix $R \equiv [r_1 \, r_2 \, \ldots \, r_p]$, optimize the following orthogonally ...
- 362
2
votes
2
answers
338
views
Of all probability matrix $P$ having stationary distribution $\pi$, find the one having smallest diagonal
I am requesting your help today trying to solve a somewhat odd problem. Is there a way to find through some numerical algorithm such as Newton's method the stochastic matrix $\boldsymbol{P}$ having ...
- 23
1
vote
0
answers
150
views
Minimax optimization of diagonal entries of function of matrix
Let $\mathbf{A}$ and $\mathbf{U}$ be arbitrary complex $M\times N$ and $N\times M$ matrices, respectively. Let denote superscript $(\cdot)^{\dagger}$ and $(\cdot)^{\mathrm{H}}$ as pseudo-inverse and ...
- 287
1
vote
0
answers
138
views
Matrix completion in $2\times2$ case by nuclear norm minimization to guarantee rank $1$?
Does fixing diagonal entries and minimizing nuclear norm under weighted sum of entries conditions produce a rank $1$ matrix? I think the answer for this is no.
At least could it be true in $2\times2$ ...
- 13.9k
0
votes
1
answer
272
views
A certain type of quadratic problem.
I am interested in solving the following equality constrained quadratic (?) problem.
\begin{align}
\min_{u^{H}u=1}~(u^{H}A_1u) \\\
s.t.~ u^{H}A_2u=0
\end{align}
$A_1$ and $A_2$ are $N\times N$ ...
- 1,421
0
votes
0
answers
72
views
Minimizing the Spectral Norm of the Hadamard Product of a Quadratic Form Using CVX
I am trying to use CVX to minimize the spectral norm of the Hadamard product of two matrices, one of which is in quadratic form. Specifically, I am trying to minimize $\|{\bf A} \odot {\bf XX}^H\|_2$, ...
- 43
0
votes
0
answers
46
views
Lipschitz solutions to linear complementarity problems (LCP)
Let $M\in\mathbb{R}^{n\times n}$.
For $q\in\mathbb{R}^n$, define the set:
$$S_M(q)=\{y\in\mathbb{R}^n|y\ge 0,q+My\ge 0, y^\top (q+My)=0\}.$$
This is the set of solutions to the LCP $(q,M)$.
We say $...
- 183
-2
votes
1
answer
183
views
Property of positive semi-definite
Let $A$ is a positive semi-definite matrix like this:
$$ A = \begin{bmatrix}
1 & \alpha_{1,2} & \alpha_{1,3} & \alpha_{1,4}\\
\alpha_{1,2} & 1 & \alpha_{2,3} & \alpha_{2,4}\\
\...
- 25