0
votes
1answer
41 views

On the decomposition of Stochastic matrices

Let "stochastic" matrix be the matrix whose rows sum to one and deterministic matrix be a stochastic matrix whose all rows consist of a one and zero. For example $\left [ \begin{array}{ccc} 1 & ...
0
votes
0answers
28 views

Extreme points by intersection of extreme rays and hyperplane

I just met one question and have no idea about the proof, hope someone can give me some ideas on how to attack this question. Given a graph $G=(V,E)$ with $|E|=n$. Define a set $S$: ...
0
votes
1answer
95 views

Matlab - Generate square convex function with positive definite Hessian Matrix

So, I have to generate a square convex function in Matlab and it's Hessian Matrix must be positive definite but I can't find any function that can help me do that. Is there anything I should search ...
0
votes
1answer
78 views

Largest eigenvalue of symmetric matrix

I am trying to understand why the $\lambda_{\max}$ function is convex given an $n\,x\,n$ symmetric matrix, let's call it $A$. I know from elementary property of eigenvalues that all the eigenvalues of ...
1
vote
0answers
57 views

Properties of matrix functions

Can I say that a certain matrix function is absolutely continuous or monotonically increasing in $\lambda$(assuming that the matrix is a function of $\lambda$)? In other words, are these ...
0
votes
1answer
35 views

convexity of a Hessian matrix.

Suppose I have $f(x_{1},x_{2}) = x_{1}^2 + x_{2}^2, S = \mathbb{R}^2$. How do I determine whether the function is concave or convex based off of the Hessian of what is above? I know the Hessian is ...
0
votes
1answer
69 views

Confusion solving constant function

Find $f:\mathbb{R}\to\mathbb{R}$ which is not a constant function which is neither star-concave, nor star-convex, but both concave and convex. Please help me how to solve for this function?
2
votes
0answers
64 views

Maximizing the smallest eigenvalue of a linear combination of matrices

I have an engineering back ground. Due to work, I came across this problem \begin{align} &\max_{\lambda,y_i\in \mathbb{R}}~\lambda \\\ ...
0
votes
1answer
62 views

Proximal operator

What is a proximal operator and how would one derive it in general for a function? In particular, if I had a function: $ f(x) = x^TQx + b^Tx + c $ How would I get the proximal operator for this if Q ...
1
vote
0answers
41 views

convexity of a function involving inverse matrix

The function has the following form: $f(a_1,a_2,b_1,b_2) = bA^{-1}c$, where \begin{align*} ...
0
votes
1answer
54 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
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 ...
1
vote
0answers
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 ...
2
votes
1answer
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 ...
3
votes
1answer
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 ...
2
votes
2answers
181 views

Optimize a log det function with respect to a matrix, and the saddle point analysis

Suppose I want to to find the local minima of a logdet function $\mathcal{L}$ with respect to a Matrix $\mathbf{A}$, $$ \mathcal{L} = \log\vert \mathbf{I} + \mathbf{A}\mathbf{S} \vert - ...
1
vote
1answer
74 views

Is $f(X)=-tr(AXBX^T)$ convex?

Given $A,B \in \mathbb{R}^{N \times N}$ and they are non-negative matrix. Is $f(X)=-tr(AXBX^T)$ convex when $X$ is also non-negative? If yes, how can I show that?
1
vote
1answer
103 views

Trace Constraints and rank-one positive semi-definite matrices.

Let $C_1$,$C_2$...$C_N$ be $M\times M$ hermitian matrices and $c$ be a given positive constant. Let $W$ be a positive (possibly semi) definite matrix such that \begin{align} \text{trace}\{WC_1\}\geq ...
8
votes
1answer
226 views

Proof that the set of doubly-stochastic matrices forms a convex polytope?

Does the set of all doubly-stochastic matrices form a convex polytope? In general, I wonder how the proofs of convexity and geometry can be established for sets of matrices of this kind? Anything to ...
3
votes
2answers
82 views

About the convexity of Ky Fan's norm

As we know, the Ky Fan norm is convex, and so is the Ky Fan k-norm. My question is, does this imply that the difference between them is a non-convex function, since it results from "difference between ...
2
votes
1answer
54 views

How to test the convexity of mutual information using leading principal minors?

I read from textbooks that the mutual information function $I(X;Y)$ is a concave function of $p(x)$ for fixed $p(y|x)$ and a convex function of $p(y|x)$ for fixed $p(x)$. I tried to test the ...
1
vote
0answers
30 views

Matrices produced by bivariate convex functions

Suppose we have an $n \times n$ real-valued matrix $A = (A_{ij})$. When is it the case that there exists a bivariate "convex" function $f: \mathbb{R}^2 \to \mathbb{R}$, and a permutation $\pi$ on ...
2
votes
2answers
82 views

A Quadratic Problem (which looks very simple)

This arises as a part of my work. \begin{align} \min_{x^{H}x=1}~&x^{H}A_1x \\ subject~to~&x^{H}A_2x=0 \end{align} $A_1$ and $A_2$ are $N\times N$ hermitian matrices and $x$ is a unit norm ...
1
vote
1answer
115 views

Convex Combination of Hermitian Matrices

Assume all the matrices I discuss about are $N \times N$. Consider any two hermitian matrices $A_1$ and $A_2$ which are indefinite. The question is, In general, for any $A_1$ and $A_2$, does there ...
1
vote
0answers
82 views

Duality between extremal points and extremal maps

Suppose I have a convex set $C\subset\mathbb{R}^n$ such that $0\in C$ ad every Cauchy sequence in $C$ converges in $C$, but $C$ need not be bounded. (Actually I want unbounded $C$). Consider the set ...
4
votes
1answer
97 views

Is this function convex when the input vector is positive?

I am wondering if $f(\mathbf{x})$ is convex on the input of a vector of $n$ positive reals $\mathbf{x}$: $$f(\mathbf{x})=\operatorname{Tr}[(\mathbf{A}+\operatorname{diag}(\mathbf{x}))^{-1}]$$ where ...
2
votes
1answer
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 ...
1
vote
2answers
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.
1
vote
0answers
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 $C \succeq BXB^T$? ...
0
votes
2answers
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 $f : \mathbb R^{m} \mapsto \mathbb R^{n}$. ...
6
votes
1answer
289 views

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 ...
1
vote
3answers
192 views

Generating an N-Dimensional Convex Quadratic Function

I am currently working on a research project that is trying to show that some numerical integration techniques that do well on separable convex functions will not necessarily do well on non-separable ...