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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Convex and concave functions

This maybe a silly question... So mercy me. Let $m,v:[0,S]\to \mathbb{R}$ be two Lebesgue integrable, monotone functions, say $m$ decreasing and $v$ increasing and set: $$M(s):=\int_0^s m(\sigma)\ ...
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Convex hull area of projected points are convex respect to rotations?

Let $A$ be a finite list of points in $\mathbb{R}^3$ and $c$ the centroid of $A$. Let $P$ be an orthographic projection onto a plane in $\mathbb{R}^2$ and $h$ be the convex hull of $P(A)$. Let further ...
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Sets whose intersection with line segments have finite components

Certain subsets $A$ of $\mathbb{R}^n$ satisfy the property that given any line segment $L$, $A \cap L$ has a finite number of connected components. For instance, $A$ can be any finitary union of ...
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Distance between a point to a $2d$ ellipse in $3d$ ambient space

Suppose we are working in the 3D Euclidean space. We are given an arbitrary point $p$ and a 2d ellipse: $$E=\{x:x^TQx\leq1,x^Tq=0\},$$ where $Q$ is a positive definite matrix and $q$ is an ...
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145 views

Is this matrix function convex or non-convex?

Given, $g(Z)=Tr(Z^Tf(Z)Z)$ , where $f(Z)=h(Z)-ZZ^T$ is a p.s.d matrix formed using entries in $Z$, where again $h(Z)$ is a diagonal matrix with its $i$'th diagonal entry being ...
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249 views

Relationship between a convex function and a convex set

Here is an assertion I have read from these lecture notes: Let $f(x)$ be a convex function, then the set $I_\beta= \{f(x)\leq \beta\}$ is convex for every $\beta$ This is not hard to prove. we ...
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233 views

Projection operator property

Let $\pi_M(a)$(projection operator) be the closest point of $M$ from the point $a$ . How one can prove if $M$ is convex set of $\mathbb R^n$ then projection operator has this property? ...
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280 views

$(d-1)$-rectifiability of a boundary of compact convex set

Let us have a compact convex set $A\in \mathbb{R}^d$. Then $\delta A$ should be a $(d-1)$-dimensional rectifiable set. I don't seem to be able to show that it can be covered by a countable union of ...
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0
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80 views

Quasi-Convexity

Can I get the conclusion that the function of matrix $P$ and $Q$ \begin{equation} \mathrm{tr}\left( PQ\right) \end{equation} is a quasi-concave function for $P>0$, and $Q>0$? It is true for ...
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When finding root, does Newton's method fail if the function is non-differentiable?

According to wikipedia's description, the Newton's method finding a root presumes a differentiable function. Then, will it fail when encountering non-differentiable function? For example, can it find ...
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340 views

Convex function from Hessian

Am I correct to say that the following function is convex? $$\begin{align} & f(x,y)=-\sqrt{xy} \\ & x>0,y>0 \\ \end{align}$$ After computing the Hessian: $$ Hf =\left[ \begin ...
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264 views

Fenchel conjugate of non smooth function

Is it valid to derive Fenchel conjugate for a non-smooth function? Checking its definition $f^*(y) = sup_{x \in \mathsf{dom}f} (y^Tx - f(x))$, I think this would be OK, but I'm not sure about that. ...
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Is it problematic when using Newton Descent with discontinuous Hessian?

Is there any side effect when applying Newton Descent to a convex function whose Hessian is discontinuous?
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154 views

Proving that $|x|^p,p \geq 1$ is convex

I want to show that $|x|^p,p \geq 1$ is convex, for this i have to prove the inequality $|(1-\lambda )x+\lambda y)|^p \leq (1-\lambda)|x|^p+\lambda |y|^p $ for $\lambda \in (0,1)$ Can anyone prove ...
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1answer
115 views

Some convex optimization questions

Is minimizing number of $\{{i : x_i \ne 0}\}$ subject to $Ax=b$ a convex problem? Why is it computationally hard? What is polar cone of $\{x \in \mathbb{R}^2:0\le x_1 \le x_2\}$? Are ...
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1answer
81 views

Projection: two closed convex sets

I am really struggling with this problem: $C$ and $D$ are closed, convex subsets of ${R}^n$ with non-empty intersection, i.e. $C \cap D \neq \emptyset $ . Is it true that projection $p_{C\cap ...
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Definition of direct products of two cones or of two convex subsets?

When reading a comment after this reply, I was wondering what the definitions of direct product of two cones? More generally, what is the direct product of two convex subsets? This case is what I ...
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812 views

Log-concave functions whose sums are still log-concave: possible to find a subset?

Rationale: I am puzzled by a problem of log-concavity, which arises in population dynamics where the curvature of the logarithm of sums is a quantity of interest. It is well-known that sums of ...
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502 views

cosh x inequality

While reading an article on Hoeffding's Inequality, I came across a curious inequality. Namely $$\cosh x \leq e^{x^2/2} \quad \forall x \in \mathbb{R}$$ I tried many ways to prove it and finally, ...
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39 views

How can I reformulate my problem to make it convex?

I would like to find the symmetric positive definite matrix $S\in \mathcal{M}_{m,m}$ that minimizes the function $f(S)=\mathrm{trace}(S)+m^2\mathrm{trace}(S^{-2})$ which has been proven to be convex ...
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1answer
868 views

A question regarding the convex envelope of a function

I know that by definition, the convex envelope of a function $f$ ($f$ not necessarily convex), denoted $\operatorname{conv}f$, is the largest convex function majorized by $f$. That is, it is a convex ...
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371 views

lsc function on compact set it attains its maximum minimum?

Is this true if so how to show it? if not true can you give a counter example: A lower semicontinuous function f on a compact set K attaings its minimum on K. A lower semicontinuous function f on a ...
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$\int_0^1 u(t)\phi''(t)dt \geq 0,\ \forall \phi\in C_0^1((0,1)), \ \phi\geq 0$. Is $u$ convex?

Suppose that $u\in C([0,1])\cap C^1((0,1))$ satisfies for all $\phi\in C_0^2((0,1))$, $\phi\geq 0$ $$\int_0^1 u(t)\phi''(t)dt \geq 0$$ Can we conclude that $u$ is convex? Note: $C_0^2((0,1))$ is ...
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Continous map assuming positive value in the closure of a convex set

Let $\phi:\mathbb{R}^d\rightarrow\mathbb{R}$ be a continuous map w.r.t. Euclidean norm. Let $C\subseteq\mathbb{R}^d$ be a convex subset. Assume that there exists $y\in\overline{C}$ such that ...
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Convexity of $\mathrm{trace}(S) + m^2\mathrm{trace}(S^{-2})$

I have the following function $f(S)=\mathrm{trace}(S)+m^2\mathrm{trace}(S^{-2})$ where $S\in \mathcal{M}_{m,m}$ symmetric positive definite matrix. I'm trying to prove the convexity of this function ...
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98 views

An eigen problem

$K$ is a symmetric positive semidefefinit matrix. $K1 = 0$ (i.e. The sum of elements in each row is $0$. Or in other words matrix $K$ is centered. From this we conclude the smallest eigenvalue of $K$ ...
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Cyclically monotone sets on four points

A subset of $\mathcal{X} \times \mathcal{Y} \subset \mathbb{R}^d \times \mathbb{R}^d$ is cyclically monotone if $\sum_{i=1}^n \langle x_i,y_i\rangle \ge \sum_{i=1}^n \langle x_i,y_{i+1}\rangle$, where ...
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Prove that there are no convex functions on compact manifolds

This one seems intuitively obvious to me but I don't know how to prove it. Suppose you have a compact manifold $M$ with a function $f$ defined on it. Given two points $x$ and $y$ on the manifold, ...
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Show by example that the Minkowski sum of two sets $X+Y$ may be convex even if neither $X$ nor $Y$ are convex

There were two parts to this question. I proved that the Minkowski sum of two sets $X+Y$ is convex whenever $X$ and $Y$ are convex, but how do I prove this second part? "Show by example that the ...
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Mappings preserving convex polyhedra

It is known that linear mappings between euclidean spaces map convex polyhedra to convex polyhedra. Can you give a characterization of the class of mappings that preserve convex polyhedra?
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Why is this composition of concave and convex functions concave?

Please forgive my ignorance. I have a quick silly question about a statement given without proof in Convex Optimization by Boyd and Vandenberghe (page 87). Suppose $\mathbb{R}_+^n$ is the set of ...
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Is the set of all concave functions a convex set?

How can I prove this? I saw a similar question here: (But this was only for when g(x) is ≥0) Prove that a set defined by concave functions on $R^n$ is convex
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Are these sets not convex?

Definition of convex set says: an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object. From: ...
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how to solve the following expectation? closed-form expression or approximation

Suppose there is a binomial random variable $X\sim B(n-1,p)$,how to solve the following expectation $$E[(1- b^{X})^{m}]$$ where $b\in (0,1]$ and $m\in \mathbb{N} $ are all constants.I have tried my ...
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Need advice: what should be my next step?($2$) (does Cauchy-Schwarz help here?)

This question is based on the question that I asked here Need advice: what should be my next step? I did a little more progress and wanted to share with you. As this is a new question, without any ...
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$L_{p}(\mathbb{T})$ is not uniformly convex if $p \in \{1,\infty\}$.

How can I prove that $L_{p}(\mathbb{T})$ is not uniformly convex if $p \in \{1,\infty\}$. Here $\mathbb{T} = \mathbb{R}/\mathbb{Z}$
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Convex hull is the minimal convex set containing $X$

How one can prove that convex hull is the minimal convex set containing $X$? We need to show that for each convex set $M$ if $X\subseteq M$ then $conv(X)\subseteq M$. I am thinking of proof by ...
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Property of convex functions

I am trying to show that if a function $f$ defined in $\mathbb R^n$ is differentiable and convex then $f(y)-f(x)\ge \nabla f(x)(y-x).$ for each $x,y\in\mathbb R^n$ Using differentiability of $f$ I ...
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Submodularity of the product of two non-negative, monotone increasing submodular functions

I'm trying to prove the submodularity of the product of two non-negative, monotone increasing submodular functions Formally, we have $f$ and $g$ are submodular functions, that is, ...
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About the convexity of Ky Fan's norm

As we know, the Ky Fan norm is convex, and so is the Ky Fan k-norm. My question is, does this imply that the difference between them is a non-convex function, since it results from "difference between ...
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A query about convexity in $L^p$ spaces

It defines the set $H^{p}_{\varepsilon}=\lbrace f \in L^{p}(0,1):\Vert f\Vert _{p}=(\int \vert f\vert^{p}dm)^{\frac{1}{p}}<\varepsilon\rbrace$ with respect to the measuring space ...
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$ \text{int}(A) = \text{int(cl}(A))$, where $A$ is convex

How can one prove that $ \text{int}(A) = \text{int(cl}(A))$, where $ A \subseteq \mathbb{R}^n$ is convex?
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$ \text{cl}(\text{int}(A)) = \text{cl}(A)$ when A is convex

How can one prove that $ \text{cl}(\text{int}(A)) = \text{cl}(A)$, where $ A \subseteq \mathbb{R}$ is convex? I know that if $A$ is convex, $\text{int(A)}$ and $\text{cl(A)}$ are convex too. I need ...
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105 views

Is this function convex

Is function $$f(x, y) = \left(\frac{x}{y} - a\right)^2 \left(\frac{y}{x} - \frac{1}{a}\right)^2$$ convex on the domain $$\{(x,y): x, y \in \mathbb{R}, x >0, y >0 \}\quad?$$ Now I think that it ...
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Is it possible for a Strongly Convex function to be unbounded below?

Let $X$ be a non-reflexive Banach space and $f:X\rightarrow\mathbb{R}$ a $C^1$ function that is Strongly Convex, i.e. $$f(u)-f(v)\geq\langle f'(v),u-v\rangle+c\|u-v\|^2$$ where $c>0$ is constant. ...
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Convex relaxation for the complement of Lorentz cone

Is it possible to obtain a convex relaxation for $$ \{ (x,t): t \le \|x\|_2\} \in \mathbb{R}^{d+1} $$ where $x \in \mathbb{R}^d$ and $\|x\|_2$ is the usual Euclidean norm, by moving to higher ...
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A real differentiable function is convex if and only if its derivative is monotonically increasing

I'm working on a problem in baby Rudin, Chapter 5 Exercise 14 reads: Let $f$ be a differentiable real function defined in $(a,b)$. Prove that $f$ is convex if and only if $f'$ is ...
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Find a tight upper bound of the following expectation.

I am stuck in finding a tight upper bound (as tight as possible) of the following expectation $$E\left [ (1-a\cdot b^{X})^{m} \right ]$$ where $X\sim B(n-1,p)$ is a binomial random variable.In ...
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non convex optimisation

\begin{eqnarray} {\textbf{maximise}} \hspace{2mm} Ar^{-(a+b)} + Br^{-(a+b+c)}-C \nonumber \end{eqnarray} such that, \begin{eqnarray} c= l(h-m_{0}) \nonumber\\ m_{1} \leq h \leq m_{2} \nonumber\\ ...