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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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 $$\text{...
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63 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 $(x,...
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$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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1answer
17 views

$x=argmin_{x\in A}||y-x||_2$ 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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1answer
130 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
66 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
48 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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1answer
162 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 ...
5
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414 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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1answer
29 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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1answer
31 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 ...
3
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1answer
66 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 0<b_2<1,\...
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1answer
62 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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0answers
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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1answer
80 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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1answer
74 views

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 ...
2
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1answer
59 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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1answer
85 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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2answers
1k views

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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1answer
145 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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0answers
54 views

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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1answer
49 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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0answers
26 views

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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3answers
206 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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2answers
64 views

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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31 views

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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408 views

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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1answer
148 views

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} J(x)=\{j(...
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48 views

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 P\},...
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1answer
175 views

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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1answer
19 views

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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1answer
76 views

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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1answer
44 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) + A^\...
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34 views

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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1answer
30 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
425 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 ...
3
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1answer
45 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 ...
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1answer
23 views

Example of convex subset (unbounded) with $\text {rec} (C) = {0}$

Example of convex subset (unbounded) with $\text {rec} (C) = {0}$ I've proved that for a bounded convex subset $C$ it always holds that $\text {rec} (C) = {0}$. However, now I'm looking for an ...
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2answers
48 views

Number of local minimums and nonconvexity

I came across the following in my reading, and I like to know why this is true. "$\dots$ but, the fuction $F:\mathbb{R}^n \to \mathbb {R}$ is nonconvex since it has several local minima $\dots$" ...
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1answer
64 views

Lower bound on Hessian, mean-value theorem

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 $\nabla^2f(x) + A^\top A \succ 0$. Is it ...
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1answer
37 views

Find the edges of a polyhedron P.

Given the polyhedron $P = \{v \in \mathbb R^2 \mid Av \le b\}$ with $A = \begin{bmatrix} -1 & -1 \\ 2 & -1 \\ -1 & 2 \\ 1 & 2 \end{bmatrix}$ and $b = \begin{bmatrix} 0 \\ 1 \\ 1 \\ 2 ...
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2answers
43 views

What is condition for a convex polyhedron to be uniform?

A uniform polyhedron has all its vertices exactly lying on a spherical surface with a certain radius. Condition: A convex polyhedron will be uniform (i.e. all the vertices will exactly lie on a ...
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1answer
133 views

Curvature and convexity of a plane curve

Let $\mathbf{r}:[a,b]\to\mathbb{R}^2$ be a $C^2([a,b])$ regular curve. Is it true that $\mathbf{r}$ is convex if and only if its curvature $\kappa(t)\leq 0, \forall t\in [a,b]$ or $\kappa(t)\geq 0, \...
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1answer
113 views

Definition of a fan of a polytope

In Fulton's book Introduction to Toric varieties (page 25), he says that: A rational convex polytope $K$ in $N_{\mathbb{R}}$ determines a fan $\Delta$ whose cones are the cones over proper faces ...
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1answer
284 views

Finding the vertices of a convex set of matrices

I'm a little new here so wasn't sure if this was the right area. I've been trying to figure out how to generate a set of random $K \times N$ (for $K<N$) matrices that are subject to a several ...
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1answer
155 views

Proximal Operator of $\ell_{\infty,1}$ norm of a matrix

How can I calculate the proximal operator of mixed norm $\ell_{\infty,1}$ for any general matrix, $X\in R^{m\times n}$ i.e., $X^* = \arg \min_X ||X||_{\infty,1} + \frac{1}{2\tau} ||X-Y||_F^2$ where $...
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1answer
44 views

Proof of convexity of a quadratic function

I have the next problem: If $f(x)$ is a quadratic function with n variables: $f(x) = 0.5$$\mathbf{x}^T$$A$$\mathbf{x}$$+$$\mathbf{b}$$^T$$\mathbf{x}$$+$$\mathbf{c}$ were $A$ is a symmetric matrix ...
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0answers
56 views

Gluing two strongly convex function

Definition: We call $f:\mathbb{R}^n\rightarrow \mathbb{R}$ a $\lambda$-strongly convex function iff for every $x,y\in \mathbb{R}^n$ and $t\in[0,1]$ it follows $$f(tx+(1−t)y)\leq tf(x)+(1−t)f(y)−λt(1−t)...
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2answers
81 views

Weighted least squares with nuclear norm minimizaiton, how to optimize?

Nuclear norm minimization is very popularization and formulation is least squares term with nuclear norm term as following, $$\min\limits_{X} \frac{1}{2}\|X_{3\times3}-Y_{3\times3}\|_F^2+\lambda\|X_{...