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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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 $f:C\rightarrow \mathbb{R}$ ...
1
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1answer
19 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 ...
1
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0answers
30 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 ...
2
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1answer
35 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 ...
1
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0answers
49 views

Proof that equation is non-convex function

I have a objective function as following $$E(\phi)=\int_{\Omega}(I(x)-m_1)^2H(\phi(x))dx+\int_{\Omega}(I(x)-m_2)^2(1-H(\phi(x)))dx+\int_{\Omega}|\nabla H(\phi(x)|dx$$ where $I$ is an image; $I: \Omega ...
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0answers
16 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
33 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
57 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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0answers
17 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 ...
4
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2answers
117 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?
0
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1answer
21 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} ...
1
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0answers
23 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 ...
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0answers
20 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) = ...
0
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1answer
12 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
14 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?
2
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1answer
19 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) + ...
2
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0answers
23 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 ...
0
votes
1answer
18 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 ...
1
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1answer
26 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
31 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 ...
3
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0answers
18 views

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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0answers
15 views

Injectivity of a map on a non convex domain

Let $\Omega \subset \mathbb{R}^n$ open, bounded, and connected, a map $f \in C^1(\Omega)$ and $\alpha > 0$ such that $$ \langle \nabla f(x)\xi ; \xi \rangle \geq \alpha |\xi|^2,\quad \forall\, x ...
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0answers
8 views

Is this expression for the sub-dimensional volume of a convex polytope correct?

Let $\mathbf{S}$ be an $m\times n$ real matrix, with $m\le n$. Let $\vec{a}$, $\vec{b}$ be two real $n$-vectors, such that $a_i < b_i$ for all $i$. Consider the system of equations: ...
0
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1answer
16 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 ...
2
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2answers
25 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 $\dots$" ...
1
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1answer
26 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 ...
0
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1answer
21 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
34 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
39 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, ...
0
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1answer
29 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
26 views

K is convex.Is this true that ${K^ \circ } \ne \phi$?. [closed]

Let $ K$ is convex.Is this true that ${K^ \circ } \ne \phi$?
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1answer
76 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
22 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 ...
0
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1answer
22 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
31 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 ...
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0answers
16 views

min max duality

I have been introduced min max duality. When i have a problem : $f:X \to R$ $(Primal) inf (f(x)):x\in A$. Let say that $A=\{x \in X: h(x)=0\}$. I can express my problem : $inf_X sup_Y L(x,y):= ...
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1answer
36 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} ...
4
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0answers
22 views

Is a convex salient cone necessarily contained in an open half-space?

A cone $C$ in $\Bbb R^n$ is said to be salient if it does not contain any pair of opposite nonzero vectors; that is, if and only if $C \cap (-C) \subset \{0\}$. Obviously, a cone $C$ such that that ...
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2answers
57 views

Triangle Inequality Like Equation [closed]

If we are in $R^2$ and define $d(a,b)$ as the set of points between $a$ and $b$ we can create an equation like this: $$d(x,z) \subseteq d(x,y) \times d(y,z)$$ where the $\subseteq$ is the subset ...
2
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1answer
18 views

About the Affine hull and Span.

I'm learning Linear Algebra and Convex Optimization simultaneously, I notice that the affine hull is, to some extent, analogous to the span, but when I read the lines "We define the affine dimension ...
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0answers
46 views

Show that if A and B are strictly convex, then A + B is strictly convex or provide a counter example.

We have: If A is open: $\exists x,y \in A,$ $x \neq y$ such that $\lambda x+(1-\lambda y)\in \dot A $ (the interior) and $\exists u,v \in B,$ $x \neq y$ such that $\lambda u+(1-\lambda v)\in ...
2
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2answers
38 views

Why a convex cone cannot have more than one extreme point?

The way I define an extreme point is : A point which cannot be defined as a convex combination of two distinct points. I'm not able to extend this and show why a convex cone cannot have more than ...
0
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1answer
11 views

Discontinuous semiconcave functions

A function $u: \mathbb{R}^n \to \mathbb{R}$ is defined to be semiconcave if there is a positive constant $c$ such that for all $x,z$ $$ u(x-z) + u(x+z) - 2u(x) \leq c |z|^2. $$ Alternatively, one ...
0
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1answer
32 views

Check if ray intersects internals of $D$-facet

Given a ray $\overrightarrow{r_0} + \overrightarrow{v} \cdot t, t \in [0;+\infty)$ and a $(D - 1)$-simplex, defined by $D$-tuple of its vertices $p_i = (p_i^1, p_i^2, \dots, p_i^D), i \in \{1, 2, ...
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0answers
15 views

Discrete concavity of a log function

I want to prove that the function $f_i(P)=f_i(P_1,..P_i,..,P_K)=log(1+(\frac{a_iP_i}{\eta+\sum\limits_{i'\neq i}a_{i'}P_{i'}}))$ is discetely concave, which means that I should prove: $\forall \lambda ...
0
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0answers
11 views

Convergence of backtracking and gradient descent.

I am thinking a bit about the following exercise: Let $f(x) = x_1^2 + x_2^2$ with dom $f = \{ (x_1,x_2):x_1 > 0 \}$. The optimal value of this problem is $p^* =1$, but it is never attained since ...
1
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1answer
56 views

$\lambda f(x)+(1-\lambda)f(y)=f(\lambda x+(1-\lambda)y)$ implies $f $ linear?

Let $X$ be a Banach space. $f:X \to X$ a continuous function. If we assume that $f$ satisfies the following convexity condition: $$\lambda f(x)+(1-\lambda)f(y)=f(\lambda x+(1-\lambda)y),$$ for all ...
0
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0answers
50 views

Conditional expectation over a convex set

Let $\boldsymbol{X}$ be an $\mathbb{R}^d$-valued absolutely continuous and integrable random vector. Further, let the cdf $F$ of $\boldsymbol{X}$ be strictly increasing in each component on ...
0
votes
1answer
13 views

Are unions, sums (…) of quasiconvex functions again quasiconvex?

for a project I need to prove quasiconvexity of several general functions. Can I argue that the union (or sum, or difference...) of quasiconvex functions is again quasiconvex? I do know that the sum ...
0
votes
1answer
36 views

Prove that $|\int_C f(z)dz| \le M |z_2 - z_1|$ where $M \gt 0$ such that $|f(z)|\le M; \ \forall \ z \in \Omega$

Let $z_1$ and $z_2$ be any two points in $\Omega$ and let $C$ be any oriented contour in $\Omega$ from $z_1$ to $z_2$. Also, assume that $f:\Omega \to \Bbb{C}$ is analytic on an open convex set ...