# Tagged Questions

Convex analysis is the study of properties of convex sets and convex functions. For questions about optimization of convex functions over convex sets, please use the (convex-optimization) tag.

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### Is the ratio trace problem convex?

I have a ratio trace problem described as follows: $\arg\max_{w} \text{trace}((w^tAw)\cdot \text{inv}(w^tBw))$, where A and B are full rank matrices. This problem can be solved via generalized ...
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### Splitting the plane to fit convexes

I'm trying to show the following : Let $K,L$ two closed convexes of $\mathbb{R}^2,O=(0,0)$ If $O\notin K$ then there exists a straight line $D$ going through $O$ such that $K$ is in one of the half ...
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### Show that $y$ is convex in $x$ provided that $y=h(I)$ and $x=g(I)$

Suppose that $\phi:[0,\infty)\to[0,1]$ is strictly increasing, infinitely differentiable such that $I\mapsto(1-\phi(I))I$ is injective. Define $$y=\phi(I)I,\quad x=(1-\phi(I))I.$$ I would like ...
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### geometric interpretation of analytical hahn-banach theorem

I understand this interpretation. But how can I see this in example?
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### MVT and functions

Let $f$ be defined on an open interval $I := (a,b)$. (a) Let $x$ and $y$ be real numbers such that $x<y$. Show that if $z \in [x,y]$, then there is some $t \in [0,1]$ such that $z=tx+(1-t)y$. (b) ...
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### Will this optimisation give the global maximum?

My book optimises a type of functions using the lagrange method. From calculus I remember that we had to check the boundary when using lagrange, because it only gave local max, but it is not mentioned ...
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### Phase retrieval via SDP (semidefinite program) of 2D test image (Matrix completion)

In SDP based phase retrieval we have intensity measurement (abs(FFT2(x))^2) of the form A(xo) = |ak,x|^2 =b^2, phase retrieval is then find x that obeys A(xo)=b The quadratic measurements can be ...
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### Closedness of convex sets in a locally convex space

Let $C$ be a convex subset of a locally convex topological vector space. Consider the properties: a) $C$ is closed. b) $C$ is weakly closed. c) $C$ is weakly sequentially closed. d) $C$ is ...
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### Is there Lipschitz property for subdifferential?

I'm trying to bound the quantity $\langle \nabla \Psi(x),\bar{x}-x \rangle$ above, with the bound depending on $\|x-\bar{x}\|$ and perhaps also of $\|x-y\|$ for fixed (but not varying) points $y$. ...
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### Convex hull of a set of points

Let $a_1,a_2...a_r \in R^n$ be points in $R^n$. Prove:$$CH(\{a_1,...,a_r\})=\left\{\sum_{i=1}^r\alpha_ia_i|\sum_{i=1}^r\alpha_i=1,\alpha_i\ge0\right\}=:K$$i.e. the convex hull of the $a_i$ is the set ...
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### in normed space hyperplane is closed iff functional associated with it is continuous

E is a normed linear space . i have two questions Q1 why the complement of H is nonempty Q2 How then the functional is continuous?? Thanks
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### When does a continuous function defined on a closed and bounded convex set has a fixed point?

For a function $f$ defined from a domain $K$ to itself, we have a point $x$ in $K$ is said to be a fixed point of $f$ if $f$ maps $x$ to itself. When the domain K is a compact convex set with some ...
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### Show that scalar-valued function of a matrix is convex

Consider the mapping $$f(X) = g\left(\frac{b}{a^TXa}\right),$$ where $g$ is a convex function, $b$ is a strictly positive scalar, $a$ is a real vector, and $X$ is restricted to be symmetric and ...
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### Convergence rates of the radius of largest hypercube within a random convex hull

Let $X_1,\cdots,X_n$ be i.i.d drawn from uniform distribution on $[0,1]^d$, and let $$\eta_n:=\inf\{\eta\in (0,1):[\eta,1-\eta]^d \subset \mathrm{conv}\{X_1,\cdots,X_n\}\}.$$ Then it is easy to show ...
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### Is it true that $A \geq B$ implies $\|A\|_2 \geq \|B\|_2$ for $A,B \geq 0$?

All matrices are real and not necessarily symmetric. Denote by $A \geq B$ the condition that $(A-B)$ has eigenvalues with non-negative real parts. Denote by $\| \cdot \|_2$ the $L_2$ matrix norm. Is ...
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### Prove $\Xi (I - P)$ has eigenvalues in the non-negative real half-plane.

Let $P$ be a stochastic matrix (square, non-negative,rows sum to one). Let $\Xi$ be a diagonal matrix with any left principal eigenvector of $P$ on the diagonal and zeros elsewhere (stationary ...
Let $B(x):[0,1] \rightarrow [0,1]$ be piecewise linear increasing convex function with $B(0)=0$ and $B(1)=1$. (Think of the power of the neyman-persron test). Let $E(x)=-log(B(1-e^{-x}))$ a logaritmic ...