# 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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### Proof of Convexity Using Quasiconvexity

Suppose $f_{1}, \ldots, f_{n}$ are continuously differentiable functions defined on $\mathbb{R}$ such that for each $i\in \{1,\ldots,n\}$, we have: (i) $f_{i}(v_{i}^{0})=0$ and ...
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### Log-concave function changes when scalar is added

A function $f$ is log-concave if $\log(f)$ is concave. Intuitively, one might guess that adding a scalar to a function would not affect properties like concavity, log-concavity, quasi-concavity etc., ...
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### Inequality involving the Hessian matrix of a convex function

Let $f \in C^2(\mathbb{R}^d)$ be a convex function with Hessian $H$. Is it true that $$(x^T H(x) - y^T H(y)) (x-y) \ge 0 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(1)$$ for all $x,y \in \mathbb{R}^d$? ...
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### Dual of a maximization problem

We have a positive, smooth, increasing concave function $f:\mathbf{R}^n\to \mathbf{R}^+$ and $k$ smooth, increasing constraint functions $f_i:\mathbf{R}^n\to\mathbf{R}$. I've recently encountered two ...
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### The subset of $\mathbb R$ with $x\geq 0$ is closed and convex

Let $C=\{x\in \mathbb R|x\geq 0\}$. Prove that $C$ is a closed convex subset of $\mathbb R$ and show that for $x_0 \in \mathbb R$, the closest element in $C$ to $x_0$ is $max\{x_0,0\}$. I have looked ...
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### How to Compute or Find an Upper Bound for the Diameter of a Convex Set?

Let $\mathcal{R}$ be an $n$-dimensional bounded hyperrectangle, and consider a $n\times n$ matrix $A$ with real entries. Given set $\mathcal{R}$ and matrix $A$, I want to compute or find an ...
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### How can I prove the concavity of $f(p_1,p_2,\ldots,p_n) = \sum_{i = 1}^n p_i(1-p_i)$?

Assume $p_n$ is the probability of being in class $n$ which mean that $f(0) = 0$ , $f(1) =0$ , and $p_1+p_2 = 1$ I need to come up with a concave function that show the relation between $p_1$ and ...
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### Can a figure inside a circle be seen at right angle from any point on the circle?

A convex, closed figure lies inside a given circle. The figure is seen from every point of the circumference at a right angle (that is, the two rays drawn from the point and supporting the convex ...
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### Is the set of probability density functions convex?

Given is the set of probability density functions defined as $P:=\left \{ p(x)\mid p(x)\, is\ a\ probability \ density \ function \right \}$ Is $P$ a convex set? I am not sure that here i have to ...
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### An inequality involving expectation and a concave function

I have a real-valued random variable $X$ and a function defined on the real line $x \mapsto U(x)$ which is concave and strictly increasing. What I am wondering is whether the following is true. For ...
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### How to get explicit solution of the fractional minimization problem?

I need to minimize $f(x) = \frac{x^tQx}{a^tx-b}$, $Q$ is positive definite, $a,b$ are constant vector. But when I take gradient and set it equal to $0$, it's hard to get an explicit expression for ...
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### Proving function is convex argmin

How can I show that the following function is convex in which Z is a random variable? $$\rho(Z)=\frac{2}{3}argmin_{t}\{t+10\mathbf{E}[Z-t]_{+}\}+\frac{1}{3}argmin_{t}\{t+5\mathbf{E}[Z-t]_{+}\}$$ I ...
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### Closed-form solution for a simple system of concave equations

I am trying to solve what looks like a simple system of equations: $$x_j = A_j\left(\sum_{i=1}^n B_{ij} x_i\right)^\alpha$$ for all $j\in\{1,\dots,n\}$, where $n$ is a positive integer, ...
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### Non-subdifferentiable convex function on a normed space

Is there any convex function $f$ on a normed space $X$ such that $\partial f(x)=\emptyset$ at every $x\in X$?
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### Under which conditions is $f(x)=\frac{1}{2}x^TPx+q^Tx+r$ convex?

I am given the function $$f(x)=\frac{1}{2}x^TPx+q^Tx+r$$ and am asked to establish under which conditions $f(x)$ is a convex function. I have to use the definition of a convex function where we look ...
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### integrality of generators for the dual cone semigroup

$\newcommand{\R}{\mathbb{R}}$ $\newcommand{\Z}{\mathbb{Z}}$ Let $N$ be a lattice in $\Z^d$, $N \otimes_\Z \R:=N_\R$, the dual lattice be $M=hom(N,\Z)$, and $M_\R:=N_\R^*=N^* \otimes_\Z \R^*$. Let ...
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### Affine transformation on compact non-convex sets?

I understand that the affine transformation on compact convex set is convex. However, is affine transformation on compact non-convex set always non-convex?
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### Do corner points optimise a linear function over a bounded convex region?

This proof says if $Z_P \ne Z_Q$... ...then $Z$ is maximised (or minimised, I guess) at one of the $\color{red}{\text{endpoints}}$ -- of what exactly? $\overline{PQ}$? So the maximum value of ...
$f:R^n\rightarrow R$ , if $\forall x,y \in R^n \text{ and } \lambda \in[0,1]$ $$f(\lambda x+(1-\lambda) y )= \lambda f(x)+(1-\lambda )f(y)$$ How to show $g(x)=f(x)-f(0)$ is linear ? I try to prove ...
### bound $\delta_{s+1}$ from $\delta_s - \frac{1}{2\beta \| x_1 - x^\star \|^2} \delta_{s+1}^2$
The origin of the problem is on page 271, Convex optimization: Algorithm and complexity Given a function $f$ convex and $\beta$-smooth. Define $\delta_s = f(x_s) - f(x^\star)$, where $x_s$ is the ...