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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Find the intersection between two convex hulls, in this specific case

We work over $\mathbb{R}^K$. Let $V$ the set of vectors whose coordinates take values $0$ or $1$, or equivalently the corners of the unit cube $[0,1]^K$, . Let $d:\{0, \ldots, K\} \to \mathbb{R}_+$ ...
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Proving $x\ln(\frac{x}{a})+y\ln(\frac{y}{b})\geq (x+y)\ln(\frac{x+y}{a+b})$

Let a,b,x,y be positive reals. Prove $x\ln(\frac{x}{a})+y\ln(\frac{y}{b})\geq (x+y)\ln(\frac{x+y}{a+b})$ I don't have any olympic background, so I may be missing some standard trick. The ...
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20 views

Interior of difference of two convex sets

Let $A, B$ be two nonempty convex set in normed space $X$. We always have $$ \text{int}(A)\bigcap B\ne\emptyset\;\Longrightarrow\; 0\in\text{int}(A-B). $$ Indeed, suppose that $\text{int}(A)\bigcap ...
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Is it convex function?

I have a function and I don't know it is whether convex or non-convex: $$J(c,\alpha)=\int_\Omega ( \alpha c-I(x))^2u \, dx+ \|\alpha\|^2$$ where $0 \le u \le 1$, $I(x): \Omega \to R$, $c$ is constant ...
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If $A\subset B$, then $\text{ri}\, A\subset\text{ri}\,B$?

Is it true that If $A\subset B$, then $\text{ri}\, A\subset\text{ri}\,B$? Let $u\in\text{ri}\,A$, then there is $\epsilon>0$ such that $$\mathbb B(u;\epsilon)\cap\text{aff}\,A\subset A\subset B$$ ...
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1answer
53 views

Lower semi-continuity of particular function

Let $F : \mathbb{R}^n \rightrightarrows \mathbb{R}^n$ be a set-valued map, locally bounded, upper semi-continuous, and taking nonempty, convex and compact values. Let $f : \mathbb{R}^n \to \mathbb{R}$ ...
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proving that $\text{aff}C-\text{aff}C\subset\text{aff}\,(C-C)$

In proof of Theorem 6.4.1 of Auslender's book about asymptotic cones, the author assumes that $\text{rge}\,A\subset\text{aff}\,C$ and for $\epsilon>0$ claims that ...
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21 views

Curvature of strictly concave function: scaling of derivative

I am looking at a strictly concave function $f: R_+ \to {R}$ with $f' >0$ and $f'' < 0$. I require the following property which I don't fully understand: $$ k f'(kx) \geq f'(x) \quad \forall ...
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Supporting hyperplane of a convex set

Let $\Omega$ be a bounded convex set in $\mathbb{R}^n$, and let $\partial \Omega$ denote its boundary. Fix a point $p$ in $\Omega$, and let $c$ denote the point on $\partial \Omega$ that is closest ...
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1answer
36 views

any good way to approximate this non-convex function with convex function?

There is a non-convex constraint in my optimization problem, which is given by $\displaystyle -xy\log\left(1+\frac{z}{xy}\right)$. Obviously, it is neither convex or concave. Is there any good convex ...
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372 views

Are convex hulls of closed sets also closed?

If $X$ is a compact set in $\mathbb R^n$, how can we show that $\operatorname{conv} X$ is compact as well? Can we say something similar without assuming boundedness, i.e., are convex hulls of closed ...
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230 views

Proving that $\text{ri rge}\,A=\text{ri conv rge}\,A$

"If $A:\mathbb R^n\rightrightarrows\mathbb R^n$ is maximal monotone,then $\text{ri rge}\,A$ is convex". This is a proposition in auslender's book about the asymptotic cones. We can prove that ...
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50 views

Existence of a continuously differentiable function from the projection of a convex set

Let $V$ be an open convex set of $\mathbb{R}^{n+1}$ and let $U$ be the projection of $V$ onto $\mathbb{R}^n$ - i.e. the set of $x \in \mathbb{R}^n$ such that there is some $y \in \mathbb{R}$ with ...
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42 views

$f$ convex: exists linear map $g$ s.t. $f\geq g$?

Let $f:\mathbb R\to\mathbb R$ be convex. Does there exist a linear map $g(y)=ay+b$ such that $f(y)\geq g(y)$ and $f(x)=g(x)$? Clearly, if f is differentiable we can argue with the tangent line, but ...
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$x=argmin_{x\in A}||y-x||_2$ iff $\langle y-x,z-x\rangle \leq0$ for all $z\in A$

Consider $x\in A\subset\mathbb{R^n}$ with A closed and convex. How can you see that $$x=argmin_{x\in A}||y-x||_2$$ iff $$\langle y-x,z-x\rangle \leq0$$ for all $z\in A$. I tried using ...
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54 views

Interior of closed ball

I'm an absolute beginner in Convex analysis. I'm wondering how the following statement is true. I just got this from a lecture notes and unfortunately no proof is provided. "The interior of the ...
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3answers
54 views

Prove $||\lambda x_1 + (1-\lambda) x_2 - y|| \leq ||x_1 - y||$

Assume we have have $3$ points $x_1, x_2$ and $y$ and $||x_1-y||=||x_2-y||$. How do we prove that the distance between $y$ and the convex combination of $x_1$ and $x_2$ is smaller than that between ...
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1answer
28 views

Interpreting a condition about CDF

Let F(X) be a strictly increasing CDF which admits a positive density f(x). Is the condition x/F(x) being non-increasing (aka, weakly decreasing) equivalent to saying that F(x) is convex? If not, what ...
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86 views

Is there any way to make the following function convex?

I need to find optimal lagrangian multiplier vectors for a quadratic programming problem subject to three quadratic equality constraints and several other linear inequality constraints. I would like ...
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91 views

Proving $x^4$ is strictly convex

I'm not sure how to prove $f(x) = x^4$ is strictly convex using just the definition of strict convexity: $$f((1-t)x+ty) < (1-t)f(x)+tf(y)$$ for $0<t<1$. Is this just an algebra slog? If so, I ...
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22 views

Is $f(x) = \frac{(x+\alpha)^3}{(x+\beta)(x+\gamma)}$ quasiconvex?

Is the function \begin{equation} f(x) = \frac{(x+\alpha)^3}{(x+\beta)(x+\gamma)} \end{equation} where $0 \leq \beta \leq \alpha$ and $0 \leq \gamma\leq \alpha$ quasiconvex? $x$ can be assumed to ...
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26 views

Affine, surjective map between convex sets

The setup for my question is the following: I have a compact and convex subset $K$ of some locally convex topological vector space. Within $K$ there is a $T\subset K$ which is compact and convex and ...
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1answer
62 views

Determining the sign of a term

I have a problem in proving the sign of a term. It is as follows: $$x=\dfrac{1-a}{b_1b_2-a}+1,\qquad y=\dfrac{1-a}{b_1-a}+\dfrac{1-a}{b_2-a},\qquad z=x-y$$ with $0<b_1<1,\quad ...
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1answer
29 views

Proof of relationship of subgradients of function to convexity of function

I am trying to follow the proof of the first claim of Proposition 7 on this page: https://blogs.princeton.edu/imabandit/2013/02/05/orf523-advanced-optimization-introduction/ Basically, we are given: ...
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45 views

How to prove $f(x_1,\ldots,x_n) = \sum x_i\ln x_i - (\sum x_i )\ln(\sum x_i)$ is convex on R++ [duplicate]

How can i prove $f(x_1,\ldots,x_n) = \sum x_i\ln x_i - (\sum x_i )\ln(\sum x_i)$ is convex on R++
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41 views

convexity proof of a function including ln and sums

$$f(x_1,\dots,x_n)=\sum\limits_{i=1}^nx_i\ln x_i-\left(\sum\limits_{i=1}^nx_i\right)\ln\left(\sum\limits_{i=1}^nx_i\right)\rightarrow R_{++}^n$$ How can I prove this is convex on $R_{++}^n$? I tried ...
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Image of a bounded sequence by a convex continuous function in a Banach space

Let $(X, \Vert \cdot \Vert)$ be a Banach space, and $f : X \longrightarrow \mathbb{R}$ a convex function, continuous for the norm topology. Suppose that $x_n$ is a sequence which weakly converges to ...
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33 views

contractivity to prove that an entire function is of finite exponential type?

Suppose that $G(z)$ is an entire function. If I can show that G is contractive within some compact and convex subset of $\mathbb C$, is it easier from that point to establish that $G$ is of finite ...
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40 views

Theorem 3.1 in Erwin Kreyszig's “Introductory Functional Analysis With Applications”: Is the notion of convex set valid in complex vector spaces?

Is the notion of convex sets valid and meaningful for complex vector spaces? Or, do we need to restrict ourselves to real vector spaces and normed spaces when we talk about convex sets? The ...
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48 views

Approximate model of a convex/concave surface

I have a set of measurements in 3d that yields a concave surface of a function $f(x,y)$ that I don't know its expression. I am thinking to approximate the function to a model where any point from the ...
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Prove local minimum of a convex function is a global minumum (using only convexity)

I'm studying for a calculus exam, and have come across this question in the textbook which I have problem solving; Let $C\subseteq \mathbb{R}^d$ a convex set, and let $f:C\rightarrow \mathbb{R}$ ...
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35 views

Strong convexity and strong smoothness duality

A function $f$ is said to be strongly convex with respect to a norm $\|\cdot\|$ at a point $y$ if $f(x) \geq f(y) + \nabla f(y)^T(x-y) + \frac{1}{2}\|x-y\|^2.$ It is said to be strongly smooth with ...
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tangent cone to the set

I'm supposed to solve this problem: Let us consider the set $M=\{(x, \sin{x}):x\in\mathbb{R}\}\cup\{\big(\cos(x)-1,x\big):x\in\mathbb{R}\}$ The question is to find the tangent cone to the set $M$ in ...
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38 views

Prove $f(\sum^k_{i=1} \alpha_i x_i) \leq \sum^k_{i=1} \alpha_i f(x_i) $ for a convex function f

I'm learning for an exam in calculus and have come across this question which I can't seem to prove: Let $C \subseteq V$ be a convex set. Let there be a function >$f:C\rightarrow \mathbb{R}$ a ...
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Difference of linear transformation of convex function

I'm trying to show that for constants $a,b > 0$, and a convex, continuously differentiable function $f$ with $f(0) = 0$ that $x_1 > x_2 > 0$ implies $f(-a-b x_1) - f(-b x_1) \geq f(-a-b x_2) ...
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44 views

Proving a half-plane is convex

Given a half plane in $\mathbb{R}^2$ described by the equation $2x-3y \leq6$, how would one go about proving this vector space "$S$" is convex? Clearly it is when graphed, but I'm a bit puzzled by how ...
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What is the relation between $\nabla f(\mathbf x), \mathbf y - \mathbf x, \text{ and } f(\mathbf y) - f(\mathbf x)$?

The Problem: Let $f(\mathbf x)$ be a convex function on $\mathbb R^n$. Given two points $\mathbf x$ and $\mathbf y$, what is the relation between $\nabla f(\mathbf x), \mathbf y - \mathbf x, \text{ ...
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Understanding extremal Lipschitz functions

I am new to concept of extremal Lipschitz functions and I have several basic question I'm still unsure about. To fix notation let $(X,d)$ be a metric space, $Lip(X)$ Banach space of Lipschitz ...
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Increasing concave function

Let $f:[0,1]\rightarrow\mathbb{R}$ be a concave function with $f(1)=\sup_{t\in[0,1]} f(t)$. Then $f$ is non-decreasing in $[0,1]$. Does someone know how to prove this?
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Is the normalized duality mapping symmetric?

In the area of functional analysis, nonlinear operator theory, the normalized duality mapping on a Banach space $X$ is a set valued map from $J:X\rightarrow 2^{X^*}$ given by \begin{align} ...
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Linear programming in Hilbert spaces

Let $H$ be a real Hilbert space. Let $b,c\in H$, $P\subset H$ be a convex cone and a continuous linear mapping $A:H\rightarrow H$. Consider the following sets: $$ B:=\{(Ax, \langle c,x\rangle:x\in ...
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Product of linear and convex function

More specific, how many maxima are there for product of these two functions: $ f(x) = ax + b $, and $ a > 0 $ $ g(x) $ is (strongly) decreasing convex function, $ \lim_{x\rightarrow\infty} g(x) = ...
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Checking the convexity of a parametric set

Let $r\in\mathbb{R}$ and $|v|\leq \frac{1}{2}$. Prove that $$ \{x\in[0,1]:\sqrt{x}+vx\leq r\} $$ is convex. Thank you for all kind help.
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Concavity for multivariate functions

What ways are there to prove that a function with more than 2 variable is concave?ٍ I know we can check that the associated Hessian matrix is negative (semi)-definite, but are there other ways?
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24 views

Upper and lower bound on Hessian

Let $x \mapsto f(x) \in \mathcal{C}^2$ be convex, i.e. $\forall x \in \mathbb{R}^n$, $\nabla^2f(x) \succeq 0$. Let $A \in \mathbb{R}^{m \times n}$ and suppose we have $M I_n \succeq \nabla^2f(x) + ...
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Show that F can have at most two fixed points

Consider a function $F:\mathbb{R}^n\to\mathbb{R}^n$, where $F=(F_1,...,F_n)$. Suppose that $F$ is strictly quasi-concave, and that for all $i=1,...,n$ the function $F_i:\mathbb{R}^n\to\mathbb{R}$ is ...
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21 views

Differentiability of the composition of a Lipschitz, convex function and a power function

$f:\mathbb{R}^n\rightarrow \mathbb{R}$ is a positive, convex and Lipschitz function. Is the fuction $|f|^{2+\alpha}$, $\alpha>0$, twice continuously differentiable? How to prove it, or there is ...
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1answer
47 views

Square of a convex non-negative function is still convex

Let $f: \mathbb R \rightarrow \mathbb [0, \infty)$ be a convex function. If $f$ is twice-differentiable, then $$ (f^2)'' = (2ff')' = 2(f')^2 + 2f f'', $$ which is $\geq 0 $ since $f, f'' \geq 0.$ But ...
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32 views

Defining functions for connected sets

Let $\Omega \subset \mathbb{R}^n$ an open, bounded and connected set with a $C^2$ boundary and a function $\rho \in C^2(\mathbb{R}^n)$ such that $$ \Omega = \{ x \in \mathbb{R}^n : \rho(x) < 0 ...
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Are not all neighborhoods of $0$ in a locally convex space absorbent?

A locally convex space (LCS) can be defined as a topological vector space (i.e. scalar product and sum are continuous) whose topology is generated by translation of a family of balanced and absorbent ...