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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Support function of an ellipse

I defined the support function $h_A:R^n→R$ of a non-empty closed convex set $A\subseteq \mathbb{R}^n$ as $$h_A(x)= \sup\left\{x⋅a |\ a \in A\right\}$$ Everything I know about this topic I found it. I ...
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Minimal Ellipsoid in $R^{2}$; why is it the Ellipsoid 2 in the figure?

It is stated in the book Convex Optimization, Boyd in page 47 that the ellipsoid 2 is the minimal because no other ellipsoid (centered at the origin) contains the point (top point) and is contained in ...
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1answer
20 views

Is it possible to move vertices of a regular polygon to shape a given convex polygon?

can vertices of a regular polygon (n-gon) in the plane be moved (slide) one at a time to form a given convex polygon so that the polygons in between remain convex?
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Dual cone and sum of closed cones

Picture below is from the 35 page of Schneider R.-Convex Bodies_ The Brunn-Minkowski Theory-Cambridge University Press (2013) , I think $C^o$ is always closed no matter $C$ is closed or not. Because ...
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27 views

convex conjugate

$X$ is a Banach space and $X^{*}$ denotes its dual. Let $f:X\rightarrow\mathbb{R}$ be an arbitrary convex function. The Fenchel conjugate of $f$ is the function $f^{*}:X^{*}\rightarrow\mathbb{R}$, ...
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$f:\mathbf{R}^n \to \mathbf{R}$'s derivative in each argument has the same sign everywhere. What is $f$'s shape?

We have a differentiable $f:\mathbf{R}^n \to \mathbf{R}$ with the property that each partial derivative has the same sign everywhere in its domain. Does this mean that the sublevel sets of $f$ (sets ...
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8 views

A question of the proof of the duality mapping for convex bodies.

In picture below ,why the set $A$ is convex ? Below picture is from the A characterization of the duality mapping for convex bodies ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~...
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A question of the duality mapping for convex bodies

In picture below ,why the define of $\varphi$ is independent of the choice of sequence $(K_i)_{i\in N}$ ? Below picture is from the A characterization of the duality mapping for convex bodies ~~~...
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discrete convexity arising in a simple discrete optimization problem

Let $S$ be a fixed integer satisfying $S \ge 1$, let $a$ range over the integers between $1$ and $S$ inclusive, and for $i = 1, \dotsc, a$, let each $x_i$ range over the nonnegative integers, such ...
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1answer
17 views

How to Calculate the normal cone of a covex set at a point?

Let $C$ be a convex set of $\mathbb{R}^d$ and $\overline{x}\in C$ we define the normal cone of $C$ at $\overline{x}$ by \begin{equation} N_C(\overline{x}) = \{ y \in \mathbb{R}^d \ <y ,c-\...
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36 views

Proving an inequality involving a strictly convex function

Given, $f$ is a strictly convex function. Based on what assumptions on '$x$' and '$y$', can I say that the following inequality stands true : $$f(x) \; + f(y) \; > \; f(x + y) \; \; ?$$
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Set of marginals is convex [closed]

Let $[n] = \{1,2,\cdot,n\}$ and $[m] = \{1,2,\cdot,m\}$. Let $Z_{1,2}$ denote the set of all probability distribution on the Cartesian product $[m]\times [n]$. Let $S_{1,2}$ denote a convex closed ...
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1answer
58 views

A term for a function $f$ such that $x f'(x)$ is decreasing

Consider a differentiable, monotonically increasing function $f$. If $f'(x)$ is increasing, then $f$ is convex. If $f'(x)$ is decreasing, then $f$ is concave. Is there a term that describes $f$ when ...
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1answer
28 views

The standard n-simplex is compact set

$ " Let \ K\subseteq \mathbb R^n \ be \ a \ compact \ set,\ then\ the\ convex \ hull \ of\ K\ is\ also\ compact \ set\ " $ In order to prove this we use that the standard n-simpex as defined ...
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1answer
14 views

Shape of polar set

Let $K$ be a subset of $R^n$, which contains the origin $\theta$ , maybe , it is needed that it is not very strange . The polar set of $K$ is $$ K^0=\{x\in R^n : \langle x,y \rangle \le 1 ~~~\forall ...
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1answer
30 views

Convexity of a set generated by $A^TA$

Problem: Let $X_k = \lbrace A^T A \mid \operatorname{rank} A \ge k, A \textrm{ is an $n \times n$ matrix} \rbrace $. Show that $X_k$ is convex. I know from the classes that $\operatorname{rank} ...
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28 views

Proving the concavity of a function

I want to prove that the function $x \mapsto \Phi(\Phi^{-1}(x) + \lambda)$ defined for $x \in [0,1]$ is concave for any $\lambda \geq 0$. $\Phi$ is the cumulative distribution function of a standard ...
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36 views

How to prove coercivity

I have a problem in understanding how to prove if a function is positive or negative coercive. I understood the definition of coercivity, which is: $$\lim_{||x|| \to +\infty}f(x) = +\infty$$ However, ...
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1answer
21 views

Convex set in a vector space gives a norm

Given an $\mathbb{R}$ or $\mathbb{C}$ vector space $X$ and a function $p:X\rightarrow[0,\infty)$ with $p(x)=0$ iff $x=0$ and $p(\alpha x)=|\alpha|p(x)$ for all $x,\alpha$, I want to show that $p$ is a ...
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1answer
25 views

Show that $f$ is convex if and only if $f\left( \sum_{i=1}^m\lambda_ix_i \right) \leq \sum_{i=1}^m\lambda_if(x_i)$

I need to prove the following statement Let $S \subset \mathbb{R}^n$ a nonempty convex set and $f: S \to \mathbb{R}$. Then $f$ is convex in $S$ if and only if $f\left( \sum_{i=1}^m\lambda_ix_i \...
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1answer
35 views

Can Extragradient method be expressed with proximal steps?

As we know, for solving saddle point problems, forward-backward algorithm is generally not guaranteed to converge. But extragradient method converge Structured Prediction via the Extragradient Method ...
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Proof of Convexity Using Quasiconvexity

Suppose $f_{1}, \ldots, f_{n},~n\geq 3$, are continuously differentiable functions defined on $\mathbb{R}$ such that for each $i\in \{1,\ldots,n\}$, we have: (i) $f_{i}(v_{i}^{0})=0$ and $f_{i}^{'}(...
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1answer
20 views

Inequality involving the Hessian matrix of a convex function

Let $f \in C^2(\mathbb{R}^d)$ be a convex function with Hessian $H$. Is it true that $$ (x^T H(x) - y^T H(y)) (x-y) \ge 0 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(1) $$ for all $x,y \in \mathbb{R}^d$? ...
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40 views

Solution of a sublinear elliptic problem.

In a lecture notes, the author showed the problem $\tag{$P$}$ $\begin{cases} -\Delta u = |u|^{q-2}u \textrm{ in } \Omega, \\ u(x)= 0 \textrm{ in } \partial\Omega, \end{cases}$ where $\Omega \...
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1answer
19 views

Closure of intersection of convex sets

Let $C_i$ be a convex set in $R^n$ for $i\in I$, suppose sets $ri \, C_i$ have at least one point in common, then how to prove this: $cl\bigcap\{C_i\mid i\in I\} = \bigcap\{cl \, C_i \mid i\in I\}$
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Conditions for all positive $x$ solved by $\min \bf{x}^T\bf{x}$ $s.t. Ax=b$

I want to find a condition for having all nonnegative $x_i$ in $\min \bf{x}^T\bf{x}$ $s.t. \bf{Ax}=\bf{b}$ where $\bf{x}\in \mathbb{R}^{n\times 1}$, $\bf{A}\in \mathbb{R}^{m\times n}$, $\bf{b}\in \...
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Monotony and convexity of $U(t) = w(t) - t. w'(t)$

Let $\mathcal{D} = (\mathbb{R}^{+*})^2$. $c \in ]0,1[$. Moreover we have $\theta < 1$ and $\theta \ne 0$. We consider the following function (which is called CES or constant elasticity of ...
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Existence of direction for polyhedral set

I refer to Lemma 4.42 of this lecture notes on linear programming, about the relationship between the boundedness of a polyhedral set and its directions. Let $P$ be a non-empty polyhedral set. ...
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30 views

Concavity condition for function of more than one variable

We know for single variable function $f(t)$, the necessary and sufficient condition for concavity is $$ f((1-\lambda)x+\lambda y) \ge (1-\lambda)f(x)-\lambda f(y) $$ for every $x$ and $y$ and $0 \...
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How to Compute or Find an Upper Bound for the Diameter of a Convex Set?

Let $\mathcal{R}$ be an $n$-dimensional bounded hyperrectangle, and consider a $n\times n$ matrix $A$ with real entries. Given set $\mathcal{R}$ and matrix $A$, I want to compute or find an ...
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An inequality involving expectation and a concave function

I have a real-valued random variable $X$ and a function defined on the real line $x \mapsto U(x)$ which is concave and strictly increasing. What I am wondering is whether the following is true. For ...
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When is a mapping the proximity operator of some convex function?

Sorry for cross-posting from MO. It's been a few days and the question hasn't received any attention there. So, is there a characterization of mappings $p : \mathbb R^n \rightarrow \mathbb R^n$ which ...
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1answer
23 views

Concave up theorem for $f:A \to\mathbb R, A \subseteq \mathbb R $ - True or false?

I am a 1st year-2nd semester student of the department of Applied Mathematics @ NTUA. Some days ago, I had a really interesting conversation with one of my old school mathematics teacher about a ...
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Minima of two convex functions that are “close” to each other

Consider two convex functions $f_1 : \mathbb{R}^n \to \mathbb{R}$ and $f_2 : \mathbb{R}^n \to \mathbb{R}$ such that $\hspace{2cm} |f_1(x) - f_2(x)| \leq \epsilon \hspace{2cm} \forall x \in \mathbb{R}^...
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Is $h(x,y) = g(x)f(y)$ convex, if $g(x)$ and $f(y)$ are convex in $x$ and $y$, respectively?

I want to check if $h(x,y) = g(x).f(y)$ is convex, given that g(x) is convex and decreasing in $x$ on its and $f(y)$ is increasing and convex in $y$. Is there any way to check/prove the convexity of $...
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Good text book recomendation

I would like to do some reading about a technique called sequential convex programming. There is a lot of material about sequential quadratic programming out there, including books (Nocedal & ...
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25 views

Alaoglu & Krein-Milman to show bounded convex weak-$*$ closed subset has extreme point

Let $X$ be a normed linear space and $K$ a bounded convex weak-$*$ closed subset of $X^{*}$. I need to show that $K$ possesses an extreme point; however, I am not entirely sure how to do this. I ...
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1answer
26 views

Are odd functions that are concave and increasing everywhere necessarily linear?

The title gives away pretty much the entire question except for the fact that the class of functions I am interested in is a subset of twice continuously differentiable functions. I think that if an ...
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What value of 'a' will be the function is convex, concave or not either?

$$f(x,y) = -6x^2 + (2a+4)xy - y^2 + 4ay$$ The solution has to be : $$-2-\text{gyök}(6) \leq a \leq -2 + \text{gyök}(6)$$ I tried to define the derivation of the function accordance with $x$ , and ...
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Caratheodory's theorem for a point in boundary

I am wondering whether the following holds: if $x$ in $\mathbb{R}^d$ lies in the boundary of the convex hull of a set $P$, then $x$ can be expressed as a convex combination of $d$ points in $P$. We ...
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How to prove this inequality about the arc-lenght of convex functions?

Let be $f,F:[0,1] \to \mathbb{R}$ with $f,F \in C^2([0,1])$ two convex functions such that $f \le F$ in all points and $f(0)=F(0)$, $f(1)=F(1)$. Considering the plane-curves $\gamma(t)=(t,f(t))$ and $...
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maximum of a concave function in a minkowski sum

Let: $f(x,y)$ - a strictly-concave function, monotonically increasing in $x$ and $y$; $A,B$ - two compact and convex sets in the positive quadrant; $C$ - their Minkowski sum, $A+B$; $(x_A,y_A)$ - ...
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convex function on open interval

I have a quick question. If a continuous function $f$ is convex on $(a,b)$, then the following is true? Could you explain why or why not it is true? $for \,\,x\in(a,b)$, $t=\frac{x}{b}<1$ Thus, $...
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How to get inequality representation of cone give by generators

To be as specific as possible, I have a subspace $I \subset R^{14}$ of dimension $3$ and an $11\times14$ matrix $A$ with linearly independent rows spanning the orthogonal complement of $I$. I have an ...
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23 views

Property of Conical Hull

Let $H$ be a real Hilbert space and $C$ be a nonempty convex subset of $H$. The conical hull of $C$ is defined by $$ \operatorname{cone}{C} := \bigcup_{\lambda >0}{\lambda C}. $$ (it is a cone in ...
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Dual of a point which is in the convex cone of a set, contains the dual cone of that set

Let $\Lambda\subseteq R^n$ contains $m$ elements, where $\lambda_i$ is the $ith$ element, and $co(\Lambda)$ is the smallest convex cone contains $\Lambda$. Also, consider any point $u\in R^n$. Now, I ...
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Correctness of use induction in the proof

"Let $S$ be a subset of vector space $V$. Let $P_1, ... , P_n$ be elements of vector space $V$. Let $S$ be the set of all linear combinations $t_1 P_1 + ... t_n P_n$, with $0 \le t_i$ and $t_1 + ... ...
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39 views

Proving function is convex argmin

How can I show that the following function is convex in which Z is a random variable? $$\rho(Z)=\frac{2}{3}argmin_{t}\{t+10\mathbf{E}[Z-t]_{+}\}+\frac{1}{3}argmin_{t}\{t+5\mathbf{E}[Z-t]_{+}\}$$ I ...
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6 views

Closed-form solution for a simple system of concave equations

I am trying to solve what looks like a simple system of equations: $$x_j = A_j\left(\sum_{i=1}^n B_{ij} x_i\right)^\alpha $$ for all $j\in\{1,\dots,n\}$, where $n$ is a positive integer, $0<\...
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Under which conditions is $f(x)=\frac{1}{2}x^TPx+q^Tx+r$ convex?

I am given the function $$f(x)=\frac{1}{2}x^TPx+q^Tx+r$$ and am asked to establish under which conditions $f(x)$ is a convex function. I have to use the definition of a convex function where we look ...