# Tagged Questions

41 views

240 views

### Convexity of the squared Frobenius norm of a matrix

I was reading this paper where the define an optimization problem as where K and L are kernel matrices and $\pi$ is the permutation matrix. They have explained that the function is convex because ...
39 views

### Looking for a “Neat” Transform to Yield a Convex Set

Optimizing on a unit sphere $\mathbb{S}^n$ is almost a convex problem (if the function is convex in the new set) if we make our "new" set $\mathbb{R}^n$, via the stereographic projection. Clearly ...
280 views

### convex hull of orthogonal matrices

Where can I find the proof of the fact that the convex hull of the set of orthogonal matrices is the set of matrices with norm non greater than one? It is easy to show that a convex combination of ...
128 views

### optimizing a logdet function with respect to a scalar and the Hessian matrix

Given a logdet function $\mathcal{L}(\gamma)$, $$\mathcal{L}(\gamma) = \log\vert \mathbf{I} + \gamma\mathbf{S} \vert - \mathbf{q}^T(\gamma^{-1}\mathbf{I} + \mathbf{S})^{-1} \mathbf{q},$$ where ...
181 views

212 views

### Proof of Non-Convexity

Am looking for a proof of non-convexity of the quotient of two matrix trace functions as given by $\frac{\operatorname{Tr}X^TAX}{\operatorname{Tr}X^TBX}$, when $TrX^TBX>0$ for two different ...
392 views

### Proof of Convexity?

Given a positive semidefinite matrix $A$, is $\operatorname{Tr}X^TAX$ a convex function in $X$? Am looking for a proof of convexity or non-convexity, whichever is true.
56 views

### How to construct a matrix satisfying two semidefinite constraints

You are given matrices $A$, $B$ and $C$. $C$ is symmetric and positive semidefinite. How would you go about constructing a matrix $X \succeq 0$ such that $X \succeq AXA^T$ and $C \succeq BXB^T$? ...
112 views

### Inverse mapping for $\bar{y} = \frac{A\bar{x} +\bar{b}}{\bar{c}^{T}\bar{x} + d}$?

Let $\bar{y} = \frac{A\bar{x} +\bar{b}}{\bar{c}^{T}\bar{x} + d}$, where $A$ is $n \times m$ matrix with $n, m \in \mathbb R_{+}$. Let $f(x) := \bar{y}$ so $f : \mathbb R^{m} \mapsto \mathbb R^{n}$. ...
### Cones of positive semidefinite matrices generated by matrices of rank $1$
Let $S_n$ be the space of real $n \times n$ symmetric matrices and let $S_n^+$ be the convex cone of positive semidefinite matrices in $S_n$. The extremal rays of this cone correspond to the positive ...