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 linear function convex or concave?

I was wondering if linear function is convex or concave? For example f(x)=x, is function whose second derivate is 0 so we cant tell anything using this criteria. Can someone help?
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Existence of points in closed and bounded convex sets that cannot be expressed as convex combination of other elements of the set.

I have an intuition about convex, closed, bounded sets but I can't really find a way to prove whether it's right or wrong. Let $\Sigma$ be a convex set, that means, that given any $A,B \in \Sigma$, ...
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75 views

Is $g(x^n)$ a convex function of $x$, if $g$ is a convex function of $x$ and $n>2$; given $x$ is nonnegative?

Is $g(x^n)$ a convex function of $x$, if $g$ is a convex function of $x$ and $n>2$; given $x$ is nonnegative? like $g(x^2)$ or $g(x^3)$
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1answer
24 views

Conxex combinations of max and min

Is the following true? $$α \left( \max_{p\in P}\int g\mathrm dp\right)+\left (1-\alpha \right ) \left( \max_{q\in Q}\int g\mathrm dq \right )=\max_{z\in\left (\alpha P+\left(1-\alpha \right )Q ...
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1answer
52 views

What is the name of this object?

Suppose I have a convex set $K\subset X$, where $X$ is say a real Hilbert space (for simplicity). Then, given some $a\in \Bbb{R}$, let $$ \hat{K}=\{x:\langle x,y\rangle \leq a \;\forall y\in K\} $$ ...
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2answers
127 views

What is the motivation behind strong convexity

Definition : A function is said to be $\beta$-strongly convex if, $f(\theta w + (1-\theta) w') \le \theta f(w) + (1-\theta) f(w') - \frac{\beta}{2}\theta(1-\theta)(w-w')^2$ What is the motivation ...
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67 views

Prove $x^y>y^x$ by using convexity

For $y>x>e$, show that $x^y>y^x$. It is not hard to prove this inequality by using the monotonicity of $\frac{\ln t}{t}$. I am curious if this inequality can be proved by using convexity of ...
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1answer
86 views

Does $\phi\circ g$ convex imply $\phi\circ f\circ g$ convex?

Assume that $\phi$ is an increasing function and $g$ is a decreasing function with $\phi\circ g$ convex. If $f$ is an increasing convex function does this imply $\phi\circ f\circ g$ is a convex ...
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1answer
58 views

Normed space problem

I am currently dealing with the following problem: Imagine you have two points $x,y$ in a normed space $(X,||.||)$ and in a convex set $K \subset X$. Now you know that $B_{\varepsilon}(x) \subset ...
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1answer
87 views

Prove that this set is convex

Prove that $$ \left\{x=(x_1,x_2) \in \mathbb R^2 \mid \cos(x_1 + x_2) \ge \frac{\sqrt{2}}{2}, x_1^2 + x_2^2 \le \frac{\pi^2}{4}\right\}$$ is convex. How should I do this? Hessian is made out of ...
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4answers
211 views

Is the convex hull operator continuous?

Is the convex hull operator continuous? I am trying to prove that the CONVEX HULL OF a finite union of non-empty convex compact sets is compact. It is easy to prove that the union of compact sets is ...
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0answers
67 views

Dominated Convergence on risk measures

This is a quite specific question and I am not able to provide the whole background (e.g. what a risk measure is). If someone knows that would be great. I am having difficulties understanding a ...
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1answer
46 views

Proving $x<y<z \implies \frac{f(y)-f(x)}{y-x} \le \frac{f(z)-f(y)}{z-y}$

Suppose $f$ is convex on $I$ and $(x,y,z)\in I^3$: How to prove that: $$x<y<z \implies \frac{f(y)-f(x)}{y-x} \le \frac{f(z)-f(y)}{z-y}$$
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55 views

Prove that the set is convex

$$x\in \Bbb R^2$$ $$4x_1^2 + 4x_2^2 \le 2x_1x_2 - x_1 + 2$$ I don't know how to prove that this set is convex, I can't find anything understandable either. The only thing I found is: $f(\theta x + ...
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167 views

Application of Fenchel Young- Inequality

i'm stuck on the weak duality ineqiality. For $X,Y$ euclidean spaces: $f: X\rightarrow (-\infty,\infty]$, $g: Y\rightarrow (-\infty,\infty]$ and $A:X\rightarrow Y$ linear bounded mapping. I want to ...
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1answer
17 views

$ d $ is ascent direction iff $ (\nabla\theta(w))^t d > 0 $

I have a doubt in this exercise. Let be $ \theta: R^n \rightarrow R $ a concave function differentiable at $w$. $ d $ is a ascent direction of $ \theta $ i.e. exists $ \delta > 0 $ such that ...
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1answer
25 views

Converting a Probability constraint to a Norm constraint

Let $\mathbf{z}$ be a $N\times 1$ complex vector. Let $\mathbf{u}$ be a $N\times 1$ random Gaussian vector whose entries are i.i.d with zero mean and $\sigma^2$ variance. Consider the following ...
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1answer
439 views

convexity of log of moment generating function

Why is log of a moment generating function of random variable Z is convex? that is $\log \mathbb{E}[\exp(\lambda.Z)]$ My logic says since expectation is linear so it is in particular convex and ...
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1answer
78 views

Convex polygon is connected by broken lines with each chain parallel to one of two vectors

Given is convex polygon $P$ on the plane. Settle if there are two vectors $\vec{a}$, $\vec{b}$ such that any two points belonging to the polygon $P$ you can connect a broken line contained in the ...
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1answer
88 views

Trying to make sense of convex combinations of more than 2 points

I know that a line segment from point $a$ to point $b$ can be defined by $\{x:x=\alpha a + \beta b\}$ such that $\alpha + \beta = 1$. That is, the convex combination of $\{a, b\}$. We can generalize ...
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1answer
37 views

Graph of polytope and hyperplane

Suppose that $P$ is a compact and convex polytope in $R^d$ and let $G$ be the graph of $P$ ($V(G)$ are the vertices of $P$ and $E(G)$ are the $1$-dimensional faces - for example polyedral graphs are ...
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1answer
107 views

Property for the subdifferential and duality mapping in context of the Moreau-Yosida regularization

I have a question arising from the Moreau-Yosida regularization in Banach spaces. The real Banach space $X$ and its dual $X^*$ are both reflexive strictly convex, $f:X \rightarrow \mathbb{R} \cup ...
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1answer
59 views

Convexity of a second order cone

Does the following set define a second order cone? Anyway, is it a convex set? $(x,t)$ so that $\lVert(Ax+b)\rVert^{2} \le t(c^{t}x+d)$ $x \in R^{n}$ (A being a matrix, b,c vectors of the ...
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1answer
27 views

Question about finite functinos in Rockafellar

In the famous book "Convex Analysis" by R.T. Rockafellar, we have the following Corollary (10.1.1): A convex function $f$ finite on all of $\mathbb{R}^n$ is necessarily continuous. What is the ...
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1answer
90 views

Concavity / concavity in one direction and the cross partials

I just have a question with regards to convexity and concavity (in one direction) in relation to its cross partial derivatives. Suppose we have a smooth function $f(x,y)$ on well defined domains. And ...
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104 views

Is there an easy characterization of functions satisfying the condition $f(x) - x f'(x) \leq 0$?

Is there any easy way to characterize non-negative functions satisfying conditions of the form $$f(x) - f'(x)x \leq 0$$ or the integrated version $$2\int_a^b f(x) dx \leq f(b)b-f(a)a$$ ? From the ...
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If $f$ is continuous, why is $f$ with the property $f\left(\frac{x+y}{2}\right)\le \frac{1}{2}f(x)+\frac{1}{2}f(y)$ is convex?

If $f$ is continuous, why is $f$ with the property $$f\left(\frac{x+y}{2}\right)\le \frac{1}{2}f(x)+\frac{1}{2}f(y)$$ ,where $0\le x,y\le 1$ is convex?
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Introduction to Analysis: Convexity

A friend and I were trying to figure out this problem from our assignment. Prove that on an open $I$, a geometrically convex function $f(x)$ is continuous. To better assist the audience, it is ...
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1answer
51 views

Convex hull of infinite points

Does there exist such a convex hull of infinite points? For example, consider infinite number of points of which form a circle in $\mathbb{R}^2$. Is this considered as a convex hull?
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883 views

Prove Convex Hull of Minkowski sum

I want to prove that the following holds, where the $+$ means Minkowski sum: $$ conv(A+B)=conv(A)+conv(B) $$ Let the convex hull of $A+B$ be $$ conv(A+B)=\sum_{j,k}\lambda_j\mu_k(a_j+b_k) $$ I ...
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What is first edge position in the Minkowski sum of two convex polygons in the plane?

I am trying to understand the informal algorithm of the Minkowski sum of two convex polygons in the plane as described here: ...
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1answer
73 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?
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Are convex function from a convex, bounded and closed set in $\mathbb{R}^n$ continuous?

If I have a convex function $f:A\to \mathbb{R}$, where $A$ is a convex, bounded and closed set in $\mathbb{R}^n$, for example $A:=\{x\in\mathbb{R}^n:\|x\|\le 1\}$ the unit ball. Does this imply that ...
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1answer
67 views

Prove range of f',$\{f'(x),x\in X\}$ dense in $X^*$

Let $X$ be a Banach Space and let $f: X\rightarrow \Bbb R$ be a Fre'chet differentiable function. Suppose that $f$ is bounded from below on any bounded set and satisfies $lim_{||x||\rightarrow ...
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1answer
60 views

How to prove the compactness of the set of Hermitian positive semidefinite matrices

I am dealing with convex optimization problems. There are some useful theories for optimization problems where real-valued vector parameter, e.g., $x \in \mathbb{R}^n$, is considered. I manage to ...
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Convexity of $\int f(\frac{dP}{dQ})dQ$ for some convex $f$

I want to prove the convexity of $\int f(\frac{dP}{dQ})dQ$ for some convex $f$ and here is what I've done so far: Since $f$ is convex, $f(\frac{dP}{dQ})$ is also convex w.r.t. $dP$ because $dQ$ ...
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Proof of convexity of $f(x)=x^2$

I know that a function is convex if the following inequality is true: $$\lambda f(x_1) + (1-\lambda)f(x_2) \ge f(\lambda x_1 + (1-\lambda)x_2)$$ for $\lambda \in [0, 1]$ and $f(\cdot)$ is defined ...
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1answer
36 views

Characterizations of convexity relying only on gradient

Suppose $f:\mathbb{R}^n\rightarrow \mathbb{R}$ is once (and only once) continuously differentiable. Are there any characterizations of convexity that rely only on the gradient $\nabla f$? In the ...
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1answer
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Converting to canonical Polyhedral Sets

In many places, results about Polyhedral sets (for example the Characterization Theorem of Polyhedral sets) are proved for the canonical polyhedral set $\{x \in \mathbb R^n: Ax = b\}$ with $b\in ...
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Is this set convex?

I have been trying to show that the following set is convex, with no luck. I am not even entirely convinced that it is in fact convex. A small hint would be greatly appreciated. For $M>0:$ $$ ...
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Minimum of two convex functions

I'm having trouble showing the below statement is true. $\hat{\alpha}=\arg\min_\alpha \frac{1}{n} \sum_{i=1}^{n} f(u_i - h(v_i, \alpha))$ where $h(v_i, \alpha)$ is linear in $\alpha$. ...
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1answer
308 views

Unlike Jensen's Inequality, can we upper bound $\log \sum_{i}{u_i \exp(x_i)}$?

According to Jensen's Inequality, it is not hard to derive the lower bound for $\log \sum_{i}{u_i \exp(x_i)}$ due to the convexity of $\log(\cdot)$: $\log \sum_{i}{u_i \exp(x_i)} \geq \sum_{i}{u_i ...
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Prove that $\{(x, y): x\in ri (dom f), y >f(x)\}\subset ri (epi f)$

$f: R^n \to (-\infty, \infty]$ is a convex function. Prove that $ri (epi f)=\{(x, y): x\in ri (dom f), y >f(x)\}$ I have used definition of convex functions to prove that the right-side is ...
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2answers
63 views

When does it help to write a function as $f(x) = \sup_\alpha \phi_{\alpha}(x)$ (an upper envelope)?

In this question I was shown a very elegant solution based on writing a function as the upper envelope of a family of linear functions: $$f(x) = \sup_{y\in C} f(y) + \langle \nabla f(y), x-y ...
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Expected values of some properties of the convex hull of a random set of points

$N$ points are selected in a uniformly distributed random way in a disk of the unit radius. Let $P(N)$ and $A(N)$ denote the expected perimeter and the expected area of their convex hull. For what ...
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1answer
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How to determine convexity of the set

Let $S=\left\{(x_1,x_2)\in \mathbb{R}^2: \sqrt[4]{2x_1^4+2x_1^2x_2+x_2^2}\leq 5 \right\}\cap\left\{(x_1,x_2)\in \mathbb{R}^2: \cos(x_1)+3x_1^2+x_2\leq 5 \right\}$ I want to determine, whether S is ...
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1answer
42 views

How can the function's composite be convex function?

$h$ is a continuous function which is convex and strictly decreasing; $t$ is a continuous function which is strictly increasing; $\operatorname{ran}(t)$ is included in $\operatorname{dom}(h)$. Under ...
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66 views

proving that a function is convex

Suppose that $f(x)=\frac{1}{k}|x|^k$ where $k>1$ and $k<\infty$. $x$ here is in $\mathbb{R^n}. $ Is $f$ convex? I am trying to use the definition of convexity but it seems like I would need some ...
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1answer
42 views

PSD matrices properties

If I have a matrix $X \in R^{n \times n} $ and an index set $ I \subseteq \{1,\dots,n\} $, Is $X_I$ also positive-semidefinite $\forall \ \ I $? Why ? $X_I $ is the submatrix that is formed by ...
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1answer
54 views

Projection on a convex set

If I have a convex set $ S$ and if I project an $ x$ onto $S$. Is it true that $x $ would project onto a unique element of $S$. Why? What would be considered different if the set $S$ was non-convex?