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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Proof that the set of extreme points of a compact convex set is not empty.

The Krein–Milman theorem states that if $S$ is convex and compact in a locally convex space, then $S$ is the closed convex hull of its extreme points. In particular, such a set has extreme points. Is ...
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11 views

Where can I find the nowhere subdifferentiable example of rockafellar?

I'm told that Rockafellar gave an example of a real extended function defined on a locally convex space, whose subdifferential is empty at each point of its domain. The function is proper, lower ...
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24 views

Least expensive way to “walk” with a convex potential

Let $V(x)$ be a non-decreasing, convex function on $\mathbb{R}^+$. Let $A_{L,k}$ be the space whose elements are sequences of positive integers $r= (r_i)_{i=1}^{N}$ such that $\sum\limits^N_{i=1} r_i ...
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11 views

Positive hyperplane? Is there a name for these type of hyperplanes?

Consider an affine hyperplane $\{ x : \langle x,v\rangle=a \}$ where $a\ge 0$ and $v\in\mathbb{R}^n_{+}$. That is, both the level $a$ and the vector $v$ are non-negative. Is there any special name ...
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1answer
14 views

One definition of strong convexity (from textbook of Prof. Bertsekas in 2015)

In strong convexity, there are a few definitions, one of them is: $f$ is strongly convex over $\mathcal{C}$ with coefficient $\sigma$ if $\forall x,y \in \mathcal{C}$ and all $\alpha \in [0,1]$, we ...
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90 views

Write a random variable as a convex combination of other 2

I'm trying to prove that if $f:[0,1]\to\mathbb{R} $ is continuous and convex, then the Bernstein polynomials are too. The hint that I've got is this: "Let $p_1 < p_2 < p_3<1$ and consider ...
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57 views

Is there a name for this property among variables?

I have a convex function of four variables, $f(w,x,y,z)$, which when solving for the symbolic arg min of one variable assuming the other three are known I got something similar to the following. ...
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1answer
21 views

Is the closure of every bounded convex set , with non-empty interior , in $\mathbb R^n (n>1)$ homeomorphic to a closed ball?

Is the closure of every bounded convex set , with non-empty interior , in $\mathbb R^n (n>1)$ homeomorphic to a closed ball (by closed ball I mean $B[a,r]:=\{x \in \mathbb R^n : d(x,a)\le r\}$ , ...
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2answers
24 views

Convex function and second devirative

I would like to ask a question about the condition of a convex function. We know that a function $f(x)$ is convex if and only if $f''(x) \geq 0$. But what if a function has more than one variable? ...
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16 views

Is the closure of every bounded convex set in $\mathbb R^n (n>1)$ homeomorphic to a closed ball ? [on hold]

Is the closure of every bounded convex set in $\mathbb R^n (n>1)$ homeomorphic to a closed ball ?
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1answer
65 views

Convexity under diffeomorphisms

Let $K \subset \mathbb{R}^n$ be a compact convex subset with non-empty interior, and $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ a diffeomorphism. Then is it true that $f[K]$ is convex?
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1answer
86 views

Is the function $f(x) = |x|$ convex?

I am asking because some Internet pages contradict. In Wikipedia, the definition I found of a convex function is: "Let $X$ be a convex set in a real vector space and let $f: X \to R$ be a function. ...
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0answers
12 views

Does expectation operation preserve convexity? [on hold]

We have f(x) is a convex function. Is $E[f(x)] $ is convex? If yes, how can we prove it? Thanks, Tan-
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1answer
14 views

Minimum of biconjugate of a nonconvex $f(x)$ is the minimum of $f(x)$ also?

In Fazel (2002) Matrix rank minimization with applications, Ch. 5.1.4-5.1.5, the author finds an analytic expression for the convex biconjugate of their nonconvex function; however, they state that ...
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1answer
36 views

The maximum of several affine functions is a polyhedral function

A function $f: \mathbb{R}^n \mapsto (-\infty,\infty]$ is polyhedral if its epigraph is a polyhedral, i.e. $$\text{epi}f=\{(x,t)\in \mathbb{R}^{n+1} | \ \ C\left( \begin{matrix} x\\ t \end{matrix} ...
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1answer
35 views

Relative Interior of a Convex Hull

Given pts $y_0,...,y_k \in \mathbb{R}^n$, their convex hull is Co($y_0,...,y_k$):={$\sum_{i=0}^k a_i y_i$ : each $a_i \geq 0$, $\sum_{i=0}^k a_i =1$}. Their affine hull is ...
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1answer
34 views

Are the stationary points of a strongly convex function unique in each dimension?

Consider a strongly convex function $~f: \mathbb{R}^n \rightarrow \mathbb{R^+}~$ with a unique minimum at the point $x^* \in \mathbb{R}^n$. I am wondering: if I have another point $y \in ...
2
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1answer
37 views

How to solve the convex combination problem of matrix?

Let $A \succeq B$ denote matrix $A-B$ is positive semidefinite, and here is the definition of redundant(all the matrix dimensions are $N\times N$ ): Given a set of matrix $\{B_i\}_{i=1}^{l}$, if ...
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0answers
14 views

Intersecting Simplices with Normballs

Let $e$ be the vector of all ones 's consider the standard simplex $$\Delta_m:=\{x\in\mathbb{R}^m_+: \langle x, e\rangle=1\}.$$ Then the truncated simplex $\Delta_m^d$ is given as ...
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2answers
124 views

Are convex functions enough to determine a measure?

Suppose we are talking about $\mathbb{R}^n$. We know that if $\mu$, $\nu$ are two finite Borel measures such that $$\int_{\mathbb{R}^n}f(x) \, d\mu(x)=\int_{\mathbb{R}^n}f(x) \, d\nu(x),$$ for all ...
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2answers
26 views

$A$ is a convex subset with non-empty interior and $D$ is dense in $\mathbb R^n$ ; then $\mathbb R^n$ , $U\cap D \cap A \ne \phi$? [closed]

Let $A$ be a convex subset , with non-empty interior , of $\mathbb R^n$ and $D$ be a dense subset of $\mathbb R^n$ ; then is it true that for every open subset $U$ of $\mathbb R^n$ , $U\cap D \cap A$ ...
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$A$ be a convex subset and $D$ be dense in a real NLS ; then for every non-empty open subset $U$ of the NLS , $U\cap D\cap V \ne \phi$?

Let $A$ be a convex subset of a real normed linear space $V$ and $D$ be a dense subset of $V$ ; then is it true that for every non-empty open subset $U$ of $V$ , $U\cap D\cap A $ is non-empty ?
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44 views

Is the projection function convex?

Define the following function $f:\mathbb{R}^n\rightarrow \mathbb{R}^n$ to be the projection function onto a convex and a closed set C $f(x)=\arg\min_{y\in C} ||x-y||_2^2 $ Denote $f_i(x)$ ...
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1answer
28 views

Is this function composition convex?

Say we have two functions $f:R^n\rightarrow R$ , $g:R^m\rightarrow R^n$. Given that $f$ is convex, under what conditions on $f$ and $g$ we will be able to say that the composition function ...
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1answer
10 views

Interior of polar cone and self-concordant function

Suppose $f$ is a self-concordant(see 9.6.2) barrier of a proper cone $K$ ( solid,convex, closed and pointed) in $\mathbb{R}^n$. It looks like the value of all $\nabla f$ is just the interior of the ...
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2answers
88 views

Convex Optimization Closed Form Solution

I'm currently studying for my exame in convex optimization. It covers problem formulation, first and second order conditions of optimality, unconstrained optimization, constrained optimization with ...
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1answer
25 views

Prove if C is midpoint convex and closed then its a convex set [duplicate]

Midpoint convexity. A set C is midpoint convex if whenever two points a,b are in C, the average or midpoint (a + b)/2 is in C.Prove that if C is closed and midpoint convex, then C is convex. ...
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2answers
51 views

Does this relation imply convexity?

I'm trying to figure out wheter the following condition inplies convexity or not. Let $\cal{X}$ be an inner product space with inner product $\langle \cdot, \cdot \rangle$ and a norm $\|\cdot\|$ (not ...
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2answers
46 views

Prove that the sum of convex functions is again convex.

I must to prove that the sum of convex functions is again convex. I know the definition of convex function: $f(tx_1+(1-t)x_2)\leq f(x_1)+(1-t)f(x_2)$ - this the first convex function, then I have the ...
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2answers
31 views

Convex and conic hull, geometric interpretation

$$\operatorname{conv}\,X=\left\{\sum_{i=1}^N \lambda_i x_i \,\Bigg\vert\, N\in\Bbb N,\, x_i\in X,\, \sum_{i=1}^N \lambda_i = 1,\lambda_i \geq 0\right\}$$ $$\operatorname{cone}\,X=\left\{\sum_{i=1}^N ...
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Unique solution of LP

Hi I am working on the following question: If $c \in int(N_P(x))$, then $x$ is a unique solution. I have proven that this is true if $x$ is a vertex. Well I am wondering if the following is a ...
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1answer
18 views

Can I write $\mathbb{S}_+^3$ as a norm cone?

Let $\mathbb{S}^3_+$ be the set of $3\times 3$ symmetric semi-definite positive matrix. I wonder whether I can write $\mathbb{S}^3_+$ as a norm cone, i.e., $$\exists A\in \mathbb{R}^{m\times 9}, C, ...
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1answer
22 views

Can I write $\mathbb{R}^n_+$ as a norm cone?

Let $\mathbb{R}^n_+=\{x=(x_1,\dots,x_n):x_i\geq 0,\forall i \},n\geq 2$. I wonder whether I can write $\mathbb{R}^n_+$ as a norm cone, i.e., $$\exists A, c, \|\cdot\|, s.t. x \in \mathbb{R}^n_+ \iff ...
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22 views

What is the right isomorphism for convex set in $\mathbb{R}^n$

Like we have linear transformation for vector space, I wonder what kind of 'transformation' or 'homomorphism' or 'isomorphism'( when the map is bijective) to look at for convex set in $\mathbb{R}^n$. ...
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2answers
35 views

Question about the proof of Caratheodory's theorem

In the proof available here, I do not understand why $\alpha>0$. How can we know for sure that $\lambda_i>0$?
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1answer
44 views

What is a proximity operator? why do we need it?

I am going to deal with convex optimization problems and I am not a math student so I may have some problems in understanding some topics. As you know, many of the optimization problems lead to a cost ...
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1answer
41 views

About the convexity of $\sin x$ for $\pi\leq x\leq 2\pi$ [closed]

To prove the convexity of $\sin x$ over $[\pi,2\pi]$ through the second derivative is easy, but I would be interested in a (possibly) simple proof of convexity that avoids derivatives. Can you provide ...
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4answers
89 views

Direct proof for convexity of $e^x$ [closed]

Is there any direct proof without using second derivative for convexity of $e^x$?
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27 views

Strong convexity of quadratic function

Assume that $Q$ is a positive definite matrix, is it true to say that the function $f(v)=v^TQv$ is strongly convex with respect to the norm $||u||=\sqrt{u^TQu}$? Thanks
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1answer
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Convexity increases the “cost” of long steps

Let $V(n)$ be a non-decreasing, convex function on $\mathbb{N}$ such that $V(0)=0$, $V(1)=1$. Let $(r_i)_{i=1}^{N}$ and $(r^{\prime}_i)_{i=1}^{N^{\prime}}$, $N^{\prime} > N$, be two sequences of ...
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Proof Attempt of Brouwer (via Separating Hyperplane Theorem)

In part motivated by the discussion here, I have been playing with trying to prove Brouwer's theorem appealing as minimally as possible to topology. In the 1-dimensional case I believe one can ...
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1answer
21 views

(Just for clarification) - Is a convex, piecewise continuous function f on an closed interval continuous?

Lets say f is defined on an Interval $I = [a,b] $. Since f is convex, one immediately knows that f is continuous on $I^°$ , however left are the points $a$ and $b$ The piecewise continuity of f ...
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Notion of outer normal cone and supporting cone if $x \in$ relint($C$)

In my lecture we defined the outer normal cone $ N_c(x^*)= \{ c\ \in \mathbb{R^n} : \max\limits_{x \in C} \ \ c^Tx = c^Tx^* \}$ and the supporting cone $S_C(x^*)= \bigcap\limits_{c \in N_c(x^*)} ...
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minimal distance betwen a point and and the halfspace containing a convex set

Let $L^2(I)$ be the usual $L_2$ space with $L_2$ norm and $S$ a convex and compact subset of $L^2(I)$. Suppose $g^*\notin S$ and $$\min_{f\in S} \|f-g^*\|$$ has the unique solution $f^*\in S$. ...
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1answer
41 views

Convexity of a certain set

Would someone please help me? I know that the set $$\{(x,y)\mid \cos(x+y)\geq \frac{\sqrt 2}{2}\}$$ is convex, but I am seeking for a simple proof?
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1answer
38 views

Function defined by integrals convex?

Let $g$ be a positive integrable function in $[0,\infty)$, and $G$ its integral, that is $G(t) = \int_0^t g(u) \, du$. Is the function f, defined as $$ f(t) = \int_0^\infty g(u) e^{-(G(u+t) - G(u))} ...
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30 views

Convex analysis of a vectorial function

In the context of equilibrium equations on structural concrete study, I deal with an system (3 non linear equations) that relates an entry of variables $(\epsilon_0,\kappa_x,\kappa_y)$ (which ...
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23 views

How to sample from a convex hull?

Let us consider a couple of points $x^{(i)}\in \mathbb{R}^m$ where $i=1,\dots,n$. Convex hull is defined as $$ C = \left\{\sum_{i=1}^{m} \alpha_i x^{(i)} \mathrel{\Bigg|} (\forall i: \alpha_i\ge ...
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20 views

Indicator function to zero-set of a function

Given the indicator function $I_{C}: \mathbb{R} \rightarrow \mathbb{R}$ to a convex set $C \subset \mathbb{R}$ and a function $g(x): \mathbb{R}^n \rightarrow \mathbb{R}$ $$ I_{C}(g(x)) = ...
0
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1answer
38 views

On convexity of $\frac{1}{x}$

I would like to prove convexity of $\frac{1}{x}$. It can be proved by using second derivative but I want without using second derivative. Can someone help me?