Suppose we have some functions f 1 </msub> ( x ) , f 2

Some have proposed that for a natural centrality measure, the most central a node can get is the center node in the star network. I've heard this called "star maximization." That is, for a measure $M(\cdot )$, and a star network $g^{\star }$ with center $c^{\star }$,
$\{(c^{\star },g^{\star })\}\in {\arg max}_{(i,g)\in N\times \mathcal{G}(N)}M(i,g)$
where $N$ is the set of nodes and $\mathcal{G}$ considers all unweighted network structures.

I'd like to learn about some centrality measures that don't satisfy this property, but "star maximization" isn't a heavily used term, so I am having trouble in searching for many such measures. What are some such measures of centrality?

I need help with the following optimization problem

$max\alpha \ln (x(1-y^2))+(1-\alpha )\ln (z)$
where the maximization is with respect to $x,y,z$, subject to
$\begin{array}{rl}\alpha x+(1-\alpha )z& =C_1\\ \alpha y\sqrt{x(x+\gamma )}-\alpha x& =C_2\end{array}$
where $0\le \alpha \le 1$, $\gamma >0$, and $x,z\ge 0$, and $|y|\le 1$.

Generally, one can substitute the constraints in the objective function and maximize with respect to one parameter. The problem is that in this way things become algebraically complicated, and I believe that there is a simple solution.

Let $X\in \{0,1{\}}^d$ be a Boolean vector and $Y,Z\in \{0,1\}$ are Boolean variables. Assume that there is a joint distribution $\mathcal{D}$ over $Y,Z$ and we'd like to find a joint distribution ${\mathcal{D}}^{\prime }$ over $X,Y,Z$ such that:

1. The marginal of ${\mathcal{D}}^{\prime }$ on $Y$,$Z$ equals $\mathcal{D}$.
2. $X$ are independent of $Z$ under ${\mathcal{D}}^{\prime }$, i.e., $I(X;Z)=0$.
3. $I(X;Y)$ is maximized,

where $I(\cdot ;\cdot )$ denotes the mutual information. For now I don't even know what is a nontrivial upper bound of $I(X;Y)$ given that $I(X;Z)=0$? Furthermore, is it possible we can know the optimal distribution ${\mathcal{D}}^{\prime }$ that achieves the upper bound?

My conjecture is that the upper bound of $I(X;Y)$ should have something to do with the correlation (coupling?) between $Y$ and $Z$, so ideally it should contain something related to that.

I want to solve the following optimization problem
$$\begin{gathered} \text{maximize }f(X)=-\log \mathrm{d}\mathrm{e}\mathrm{t}(X+Y)-a^T(X+Y{)}^{-1}a \\ \text{subject to }X⪰W, \end{gathered}$$
where the design variable $X$ is symmetric positive semidefinite, $W,Y$ are fixed symmetric positive semidefinite matrices, and $a$ is a given vector. The inequality $X⪰W$ means that $X-W$ is symmetric positive semidefinite.

I was wondering if there's hope to find an analytical solution to either the constrained or unconstrained problem. And if there is none, could I use convex optimization techniques to solve this numerically?

"A common theme in linear compression and feature extraction is to map a high dimensional vector $x$ to a lower dimensional vector $y=Wx$ such that the information in the vector $x$ is maximally preserved in $y$. Opten PCA is applied for this purpose. However, the optimal setting for $W$ is in generall not given by the widely used PCA. Actually, PCA is sub-optimal special case of mutual information maximisation."

Can anyone elaborate why PCA is a sub-optimal special case of mutual information maximisation ?

The problem is like
$ma{x}_{\mathbf{x}}u(x_1,x_2,...,x_L)=-\sum _{i=1}^L\mid x_i-a_i\mid$,
$s.t.\sum _i^L{x}_i\le C$
for each $i$, $a_i>0$ is a scalar;
$C$ is a constant that is strictly greater than 0;
$\mathbf{x}=(x_1,x_2,...,x_L)\in {\mathbf{R}}_{+}^L$. Characterize the optimal $\mathbf{x}$ as a function of $C$ or $a_i$.
Hint: to solve the problem we should discuss the cases when $C\le \sum _i{a}_i$ and $C\ge \sum _i{a}_i$.
Thank you!

This is a utility maximzation problem

maximize $x^a+y^b$ subject to $p_{1}x+p_{2}y=w$ (utility maximization problem)

Anyone has any idea, there are no restrictions on $a$ and $b$, as far as i can see it. many thanks!!!