All Questions
8 questions
8
votes
3
answers
526
views
Lower bound for spectral radius on $\operatorname{GL}(n,\mathbb{Z})$
Consider the group of matrices $G =\operatorname{GL}(n,\mathbb{Z})$ with integer entries and determinant $\pm 1$. For each matrix $D \in G$, the product of the eigenvalues of $D$ is equal to $\det D =\...
- 345
5
votes
0
answers
330
views
Best Approximation in Operator/non-Frobenius Norm
Since the Frobenius norm on matrices is generated by an inner product, solving the optimization/approximation problem of approximating an operator $X$ with a scalar multiple of another operator $Y$
$$\...
- 153
4
votes
1
answer
2k
views
lipschitz constant of a multivariate function
I have a function $f:\mathbb{R}^{50} \rightarrow \mathbb{R}$ and I need to compute the Lipschitz constant of $f$ to solve an optimization problem using a specific algorithm. Does any one have ...
- 41
3
votes
1
answer
775
views
Is there some quantitative version of interlacing of eigenvalues of a matrix under rank-1 update?
Given a real symmetric matrix $A$ and a vector $v$ of the same dimension we know that the eigenvalues of $A + vv^T$ are left interlaced by the eigenvalues of $A$.
But do we have any quantitative ...
- 1,893
3
votes
1
answer
154
views
Solving Matrix/Operator Equation $H P X + X P H + HQ = 0$
This problem arises when minimizing the operator equation $X P X^* + X Q + R$ over positive $X$ with respect to the positive cone on a Hilbert space $\mathcal{H}$.
The (reduced) task:
Given $P$ and $...
- 153
2
votes
2
answers
352
views
The set of matrices with same spectral radius
I am working on an optimization problem over the set of positive matrices (that is, matrices where all entries are positive numbers) that have the same spectral radius. My main problem is how to ...
- 51
2
votes
1
answer
226
views
Tighest Gap $\|x\|_1/\|x\|_2$ between $\ell^1$ and $\ell^2$ norms
I'm looking specifically at the optimization problem
$$
\begin{align*}
\text{maximize: }& n - \frac{\|\lambda\|_1^2}{\|\lambda\|_2^2}\\
\text{subj. to: }& \lambda \succeq \epsilon\mathbf{1}
\...
- 153
1
vote
1
answer
291
views
Norm of solution of quadratic program
In a quadratic program (QP), do linear equality constraints always reduce the norm of the minimizer? Specifically, let $P \succ 0$, $A \in \mathsf{M}_{m\times n}$ and $q\in\mathbb{R}^n$. Define
$$x^* ...
- 153