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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3
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Permutation and Combinatorics Problem

A function $G$ is defined on a set $S$ with size $k$ : $G(a_1,a_2,a_3,\ldots,a_k)$. $G(a_1,a_2,a_3,\ldots,a_k) = 1$ if and only if a convex polygon can be created by taking these $k$ elements as the ...
0
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1answer
41 views

Is this combination of convex functional is still convex?

Let $u$, $v\in C_c^\infty$ and $\Omega\subset \mathbb R^N$ is open bounded, smooth boundary. We also assume that $0\leq v\leq 1$. Define $$ F(u,v):=\int_\Omega |\nabla u|^2v^2dx. $$ Do we have ...
2
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1answer
33 views

Convex function?

I have a positive function $f(x,y)$, where $x\in{\mathbb R}^n$ and $y\in{\mathbb R}$. I know that for $y$ fixed, $g(x)=f(x,y)$ is convex, and that for $x$ fixed, $h(y)=f(x,y)$ has positive second ...
1
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1answer
31 views

If $y\in C$ then $||x-y||>||y-z||$.

I'm trying to prove the Separating Hyperplane Theorem. Let $C\subset \mathbb{R}^n$ be a closed and convex set, and $x\not \in C$. Then there exists $d\in \mathbb{R}^n$ and $\delta\in \mathbb{R}$ ...
5
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2answers
59 views

Is $f(x) = x^T A x$ a convex function?

Is $f(x) = x^T A x$ a convex function, where $x \in \mathbb{R}^n$, and $A$ is a $ n\times n$ matrix? If not, my question can be reformed to: when is $f(x)$ convex, any restriction for $A$? For ...
0
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1answer
29 views

Supremum over ellipsoid set

In Boyd's Convex Optimization Textbook, page 157, it is stated: $ \mathrm{sup}\{a_i^T x\; |\; a_i\in\mathcal{E}_i \} = \bar a_i^T x + \mathrm{sup}\{u^T P_i^T x\; |\; \lVert u \rVert_2 \leq 1 \} = ...
2
votes
1answer
17 views

Concave functions

Given $(X,\mathcal{F},m)$ a measure space with $m(X)=1$ and $||f||_p<\infty$ for some $p>0$. Need to show that $\forall q \in (0,p)$ $$\int \log |f| dm\leq \log(||f||_q),$$ $$ \log ||f||_q ...
0
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1answer
25 views

What would be a description of this set in $\mathbb R^3$?

Suppose that $K=\{(x,y,z)\in \mathbb R^3|x\geq 0,y\geq 0, xy\geq z^2\}$. Let $K_0=\{x\in\mathbb R^3|\langle x,k\rangle\leq 0\forall k\in K\}$. What would be a description of the set $K_0$? I don't ...
0
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0answers
54 views

A maximal monotone operator (and not a subdifferential) with a non-convex domain

I am looking for a maximal monotone operator $T:X\to 2^{X^*}$ that has a non-convex domain $D(T)=\{x\in X\mid T(x)\neq\emptyset\}$ and $T$ is not cyclically monotone, that is, $T$ is not a convex ...
0
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1answer
20 views

A problem about $-\max$ and $\min$

Suppose $X$ and $Y$ are convex compact subsets in $\mathbb R^n$. Let $\langle.,.\rangle$ be the standard inner product. Does the following equality $$\max_{y\in Y} [\langle y, z\rangle- \max_{x\in ...
0
votes
5answers
75 views

Is $S = \{(x,y) : y = e^x\}$ is convex or not?

If $S = \{(x,y) : y = e^x\}$ was convex, the following relation holds \begin{align} &t y_1 + (1-t) y_2 = e^{t x_1 + (1-t) x_2} \tag{1}\\[2mm] \Longleftrightarrow \quad & t e^{x_1} + (1-t) ...
1
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1answer
36 views

Characterization of a projection operator in $\mathbb R^n$

Let $X$ be a closed convex set in $\mathbb R^n$ and $y\in \mathbb R^n$. Suppose a projection $T_X: \mathbb R^n \to \mathbb R^n$ satisfies $$T_X( \alpha y+ (1-\alpha) T_X(y))=T_X(y)$$ for all ...
1
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0answers
16 views

Tangential surface of an extreme point of a convex subset of a simplex

Suppose that there is a convex set (polyhedra) $H$ which is a convex subset of a simplex $G = \{x\in R^d ~|~ \sum_{i=1}^d x_i=1, x_i \ge 0, i=1,2, ..., d \}$. Clearly, $H$ has extremal points $x^*$ ...
0
votes
1answer
126 views

Given the sides of a polygon, determine if it is convex or concave

We are given the lengths of all sides of a polygon. We need to determine if the given polygon is convex or concave. How can this be done? What is the propery applied to determine this?
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1answer
29 views

Convergence of second derivatives of uniformly convergent convex functions

Set-up: Let $\{f_n \}_{n=1}^\infty$ be a sequence of smooth convex functions on $(0,1) \subset \mathbb R$ that converge uniformly to the continuous (not necessarily differentiable) convex function ...
0
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1answer
30 views

Implicit function for a convex gradient.

Let $Q \colon \mathbb{R}^n \to \mathbb{R}$ denote a convex function with $g(x) = \nabla Q(x)$ well-defined. I am interested in defining the following variables \begin{eqnarray} c & = & x - ...
-2
votes
2answers
28 views

differentiable functions, concave or convex?

Suppose U and g are two twice differentiable functions of x, both of them increasing and concave, with U’ ≥ 0, U” ≤ 0, g’ ≥ 0, and g” ≤ 0. Prove that the composite function f(x) = g(U(x)) is also ...
0
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2answers
15 views

Hessian at a non-stationary point

I have a function $G(Q) : \mathbb{R}^n \rightarrow \mathbb{R}$ that is known to be convex. I also know that $Q^*$ is a minimum of $G(D)$. If I apply Taylor's theorem to $G(Q)$ at $Q^*$, I get: $$ ...
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0answers
7 views

conditions on a pdf such that a ratio is unimodal

Consider a pdf $f(\cdot)$ of a positive continous random variable with support $[0,1]$. And let $G(\theta) = \frac{\int_0^{\theta} x f(x)dx}{\theta}$. What conditions can be found for $f(\cdot)$ ...
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1answer
30 views

The sum of two strongly convex functions is strongly convex

Assume that $f$ is $a$-strongly convex and $g$ is $b$-strongly convex. Is the sum $f+g$ strongly convex, and with what constant? Definition: $f$ is $a$-strongly convex if $$ f(x)-f(y) \le ...
0
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2answers
56 views

What is the mathematical notation for Convex Hull?

I've been scanning through scientific papers, this site and just googling for it, but I can't find a commonly accepted notation for the convex hull. So my question is; if there is, what is the ...
0
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2answers
77 views

Prove $\frac{1}{2} \|y-x\|^2$ is a strongly convex function

Get stuck in proving $f(y)=\frac{1}{2} {||y-x||}^2$ is strongly convex function (Assume $x$ is fixed). My Proof: $y_1$, $y_2$ are two variables. \begin{align} & f(\lambda y_1+(1-\lambda) ...
2
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1answer
46 views

How to show that $f(x) = \frac{\sqrt{\cos x}}{1-x^2}$ is convex and has a minimum value

I want to show that $f(x) = \frac{\sqrt{\cos x}}{1-x^2}$ is convex on the interval $]-1,1[$. How do I have to proceed? I did take the derivative of the function which is $f'(x) = \frac{2x\sqrt{\cos ...
0
votes
1answer
54 views

Convex function and local extrema problem

Let $f$ be twice differentiable and strictly convex on $[a,b]$. Assume also that at a point $x_0 \in (a,b)$ the derivative $f'(x_0) = 0$. Show that $x_0$ must be a strict local minimum. I find that ...
0
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0answers
19 views

normal cone inclusion and non-symmetric matrix and optimization problem

I have the following normal cone inclusion $$-(A x + b) \in \mathcal{N}_\mathcal{C}(x) \qquad (1)$$ where $\mathcal{N}_\mathcal{C}$ denotes the normal cone to the convex set $\mathcal{C}$ at the ...
2
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1answer
87 views

Proof involving Ramsey numbers

$S$ is a set of R(m,m;3) points in the plane in which no 3 points are collinear. I am trying to prove that $S$ contains $m$ points that form a convex $m$-gon. I have tried using similar logic to the ...
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2answers
37 views

Prove a linear combination of a convex set is convex

Suppose $S$ is a convex subset of $\mathbb R^n$ , and suppose $T: \mathbb R^n \rightarrow \mathbb R^m $ is any linear transformation. Prove that the set $\,\{T(x)\,|\,x \in S \}$ is also convex. ...
0
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1answer
18 views

Positive distance between sets of a compact convex family

So I'm struggling with this exercise from Lay's Convex Sets: Let $\mathcal{F}$ be a family of compact convex sets in $\mathbb R^n$ containing at least $n+1$ members. Suppose that for each ...
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0answers
24 views

Differentiable iff unique subgradient

Let $f:\mathbb{R}^n\to\mathbb{R}\cup \{ \infty \}$ be convex, $D:=Dom(f)$, $x\in D^o$. Then f is differentiable at x iff $\partial f(x)=\{ d \}$ (a singleton), in which case $\nabla f(x)=d$. ...
0
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0answers
30 views

Show that $f$(sum of log terms) is convex with Jensen's inequality..

We have the equation \begin{equation} f(\mathbf{x})=\mathbf{c}^T\mathbf{x}-\sum_{i=1}^m\log{(b_i-\mathbf{a}_i^T\mathbf{x})} ,\;\;\; \mathbf{x} \in \mathbb{R}^n \text{ and } m>n \end{equation} ...
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0answers
11 views

Determining Euler-Lagrange equations for a nonsmooth functional

Is there any good resource for understanding how to derive the EL equations for non-smooth problems? In particular, I would like to get them for: $\mathcal{J}(u,v)=\iint_{\Omega} \sqrt{u_x^2 + u_y^2 ...
0
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0answers
21 views

Why does $f$ being convex and twice continuously differentiable imply that the domain of $f$ is open?

On the slide here: It says: $f: \mathbb{R}^n \to \mathbb{R}$ is convex and twice continuously differentiable imply that the domain of $f$ is open What is the reason behind the implication? Is it ...
0
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1answer
32 views

Hypograph of a quasiconcave function

If $f$ is a quasiconcave function, is its hypograph necessarily a convex set? My intuition says yes, but somehow I feel that this is not true, does anybody has a conterexample if the statement is not ...
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2answers
58 views

Prove that if $f(x)$ is convex and nonnegative, then $g(x)=(f(x))^2$ is convex too

Let $$f: \mathbb{R}^d \longrightarrow \mathbb{R}\\ g: \mathbb{R}^d \longrightarrow \mathbb{R}$$ such that $$f(x)\geq 0 \quad \forall x \in \mathbb{R}^d.$$ Prove that if $f(x)$ is convex, then ...
0
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1answer
23 views

For any two disjoint convex open sets there is a hyperplane that strictly separates them

How to prove the affirmation?: If $K_1$ and $K_2$ are nonempty, nonintersecting, convex and open sets, there exists a closed hyperplane $M$ such that $K_1$ and $K_2$ are strictly on opposite sides ...
0
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0answers
42 views

Recall that in the context of Fenchel–Rockafellar duality, the primal problem is defined by

Let X and Y be two Hilbert spaces. Let $f : X \rightarrow ]-\infty, +\infty]$ be convex and lower semicontinuous, let $A : X \rightarrow Y$ be linear, and let $g: Y \rightarrow ]-\infty,+\infty]$ be ...
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1answer
34 views

Let $a_1,\ldots,a_m$ be elements of $\mathbb{R}^n.$ Then the convex cone $K_{\Omega}$

I am having a problem with one aspect of the following proof I came across in "An Easy Path to Convex Analysis and Applications" by Mordukhovich and Nam. It is Proposition 3.9 and it is the line ...
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0answers
32 views

Find the distance between two convex sets

Let's say that ||.|| is an Euclidian norm in $R^3$ and we have two sets in $R^3$ defined by inequalities: $Y = \{y| f(y)<a\}, Z = \{ z| g(z)<b \} $ Let's say that $f$ and $g$ are convex and ...
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61 views

Explain why the function $f(x)=\frac{1}{2}(x-d)^2+\alpha|x|$ is strictly convex.

Let $d\in\mathbb{R}$ and $\alpha>0$ be given. (i) Explain why the function $f(x)=\dfrac{1}{2}(x-d)^2+\alpha|x|$ is strictly convex. (ii) verify that \begin{equation} ...
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0answers
12 views

bounding the hausdorff distance between a convex set and a template polytope.

How can we find an upper bound on the hausdorff distance between a convex set and its enclosing template polytope whose facets directions are given in advance?? Note that the bound should tend to zero ...
0
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0answers
48 views

Show that $f + g$ is still strictly convex.

Let $d \in X$ and set $$f: X \to \mathbb{R}: x \mapsto \left(\frac{1}{2}\right) \parallel x-d \parallel^2.$$ Use (*) to show that $f$ is strictly convex. Now let $g$ be any convex function. Show that ...
2
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1answer
46 views

In the Krein-Milman theorem, can the weak closure of the convex hull be replaced by norm-closure?

I have a question on the following formulation of the Krein-Milman theorem: Consider a vector space $X$ equipped with the weak topology induced by a separating space $X^*$ of functionals on $X$. ...
3
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52 views

How to prove the matrix fractional function is convex by definition

It is well known that the matrix fractional function $f(\mathbf{w},\boldsymbol{\Omega})=\mathbf{w}^T\boldsymbol{\Omega}^{-1}\mathbf{w}$ is jointly convex with respect to $\mathbf{w}$ and ...
0
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1answer
38 views

Let $f:\mathbb{R}^n\rightarrow ]-\infty,+\infty]$ be proper and strictly convex. [duplicate]

Let $f:\mathbb{R}^n\rightarrow ]-\infty,+\infty]$ be proper and strictly convex. Show that $f$ has at most one minimizer. I am hoping someone can give me some feedback on the proof for this. I feel ...
0
votes
1answer
14 views

Upper bound for slope and the finiteness of Legendre transform

Let $f$ be a convex function and denote with $f^{*}$ the legendre transform. Is it true that if $f^*(p)=+\infty$, then $p$ is an upper bound for the slope of $f$? Do you have that if the slope of $f$ ...
0
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2answers
39 views

Given $\min\limits_x \|x\|_2^2 \quad \text{s.t.} \quad Ax = b$ show $x^* = A^T(AA^T)^{-1}b$

Given $$\min\limits_x \|x\|_2^2$$ $$\text{s.t.} Ax = b$$ show $x^* = A^T(AA^T)^{-1}b$ where $A \in \mathbb{R}^{m \times n}, m < n$ This is projection $x$ onto the hyperplane $Ax - b = 0$ ...
1
vote
1answer
63 views

Examples of $f$ strictly convex, either with one minimizer or with no minimizer.

Let $f\colon X \to [ -\infty, +\infty]$ be proper and strictly convex. Show that $f$ has at most one minimizer. Give examples where $f$ is strictly convex, and either (i) $f$ has one minimizer; or ...
0
votes
0answers
13 views

Proof that a set is a closed convex cone given feasible directions

I have some problems trying to show the following problem. Could you guys please lend me a hand? Let $d_1$, $d_2$ $\in$ $\bf{R}$$^2$ two linearly independent vectors. Consider the rays that begin ...
1
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1answer
47 views

Gradient of the form $(\textbf{x}-\textbf{x}_k)^TA(\textbf{x}-\textbf{x}_k)$

In the context of a convex optimization problem I came across with the following function: $$f_1(\textbf{x})=(\textbf{x}-\textbf{x}_k)^T\textbf{A}(\textbf{x}-\textbf{x}_k) - t^2$$ EDIT $f_1$ is a ...
0
votes
1answer
54 views

Is this function strongly convex? or could I find a value space to make this function strongly convex?

I want to judge if this function $f(x_1,x_2,...,x_n)=(\frac{x_1}{\sum_{i=1}^{n}{x_i}})^2+(\frac{x_2}{\sum_{i=1}^{n}{x_i}})^2+...+(\frac{x_n}{\sum_{i=1}^{n}{x_i}})^2$ strongly convex for each ...