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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Eigenvectors of a quadratic form and iterative descent

I am interesting in using eigenvectors of a quadratic form to perform iterative steps to get the function value to a certain point. While other methods may be more common, my quadratic form is not ...
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38 views

Is exponential function strongly convex?

Assume $x \in \mathbb{R}$. In the wiki page, one property of strongly convex functions $f(x)$ is that it satisfies: $f''(x)\geq m > 0~\forall x$ with with parameter $m > 0$. Given $f(x) ...
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Derivative of a multivariable function

I have a fairly basic question that has perplexed me for a few hours now. I am trying to evaluate the derivative of a function $g(t):\mathbb{R}\rightarrow\mathbb{R}$ defined as \begin{equation} g(t) ...
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what is the convex hull of the rank k psd matrix

Given the set $\{X|0\preceq X , rank(X)=k\}$. What is the convex hull (convex envelope) of this nonconvex set? If we further require $X=VV^T$, where $V^TV=I$, $V$ has the size $n\times k$. Then the ...
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Properties of upper level sets of convex functions

Let $h_1$ and $h_2$ be two differentiable convex functions on $\mathbb R^n$ and $\bar x\in \mathbb R^n$ such that $h_1(\bar x)=h_2(\bar x)=0$ and $\nabla h_1(\bar x)=\lambda \nabla h_2(\bar x)$ for ...
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54 views

Convex function with non-symmetric Hessian

Let $U$ be an open convex subset of $\mathbb R^n$ and $f:U\to\mathbb R$ a convex function on it. It is a well-known fact that if the second partial derivatives exist everywhere on $U$ and are all ...
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Is this function of Semi-definite matrices, convex?

Is the following function is convex. Consider the following function $$ f(X)=\langle X,D\rangle-c \cdot \sqrt{\langle X,E\rangle}$$ where $X,D,E$ are all semi-definite matrices. Is this function ...
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24 views

Gradient and Hessian of a function with Matrix Variables

What is the Gradient and Hessian of this function? $$ f(X)=\langle X,D\rangle-c \cdot \sqrt{\langle X,E\rangle}$$ where $X,D,E$ are all semi-definite matrices. Where Gradient becomes zero?
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Why is $(T + N_X)(x) \subset T(x)$ when $Dom T \subset X$?

I'm trying to show that given a maximal monotone operator $T$ and a closed convex set $X$ with $Dom T \subset X$ then for a given $x \in Dom T$ it holds $(T + N_X)(x) \subset T(x)$ where ...
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If $\Omega$ is convex, then $K_{\Omega}$ is convex?

Let $\Omega\subset\mathbb{R}^n$ and $$K_{\Omega}=\{\lambda x|\lambda\ge0,x\in\Omega\}$$ Is it true that if $\Omega$ is convex, then $K_{\Omega}$ is also convex. Let $\gamma\in(0,1)$ and $z,t\in ...
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27 views

Convex set-valued functions

Suppose $f$ is a set valued function such that $f(x)\subset \mathcal{S}$, $x\in\mathcal S$, and $\mathcal{S}\subset \mathbb{R}^n$. I want to understand what does it mean to say that $f$ is a convex ...
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783 views

Convex conjugate of $\ell_1$ and $\ell_2$ norm

Given a function $f$, we can define a function $f^{*}$, called the convex conjugate (also known as the Fenchel conjugate) of $f$ as follows: $$ f^*(\vec{z})= \sup_{\vec x \in \mathbb ...
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40 views

Closure of the interior of the epigraph

Suppose $f:E\to(-\infty,\infty]$, where $E$ is a Banach space, is lower semi-continuous, convex, and the interior of $epi(f)\neq\emptyset$. Show that $\overline{int(epi(f))} = epi(f)$ \begin{equation} ...
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Prove that the intersection of half-spaces is a half-space

Well I want to prove that Let $K\subset\mathbb{R}^n$ a convex closed and non-empt set such that $\mathbb{R}^n\setminus K\neq\emptyset$ is convex. Show that $K$ is a closed halfspace Can you help ...
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Volume of a compact set, not necessarily convex

Looking through my lecture notes, I came across the notion that if a set $X\subset \mathbb{R}^n$ is compact and convex and $vol(X)=2^n$, then by choosing an $0<\epsilon <1$, then $X\subsetneq ...
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Minimum volume covering ellipse

Given a convex polygon in the plane, consider the smallest-area ellipse which contains this polygon. This is the "minimal volume covering ellipsoid" or "minimal volume enclosing ellipsoid" or in short ...
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Feeling behind Krein-Milman theorem

We said that the Krein Milman theorem is valid in a LCS $X$ for non-empty convex and compact sets $K$ and it tells us that: 1.) ex(K) $\neq \emptyset$ 2.) $K = \overline{co}(ex(K))$. 3.) If $K = ...
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Convex set with empty interior is nowhere dense?

Suppose $C\subseteq\mathbb R^n$ is a convex set and $C^o=\varnothing$. Is it necessarily true that $(\overline C)^o=\varnothing$? In general, is this true if $\mathbb R^n$ is replaced by a topological ...
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Smallest diameter for convex polytopes with a given volume

Let $K\subset\mathbb R^n$ be a full-dimensional, bounded, convex polytope formed by the intersection of $m$ half-spaces. Is there a nice lower bound for the diameter $D(K) = \sup_{x,y\in K}\|x-y\|_2$ ...
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Singularities in convex functions of more than one variable

I've heard that a convex function of a single variable is continuous in the interior of its domain, and is differentiable everywhere with the possible exception of a countable number of points. (I ...
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Strict convexity of a non-differentiable multivariate function

Suppose $F: \mathbb{R}^N \mapsto \mathbb{R}$ is differentiable. In order to check for the convexity of $F$, we can restrict it to a line. Thus $F$ is convex iff the function $g: \mathbb{R} \mapsto ...
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Convex Optimization: minimize over unknown convex set starting in center

Essentially I am trying to develop an algorithm to minimize a function over a convex set that I don't know explicitly. However, I have a starting point "deepest in the set" (i.e. with largest norm ...
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63 views

Existence and uniqueness of the minimizer of Moreau-Yosida approximation

Let $f:H\to\mathbb{R}$, where $H$ is a Hilbert space, be a function that is bounded below, convex ($f(tx+(1-t)y)\leq tf(x)+(1-t)f(y) \text{ for all } x,y\in H \text{ and } 0\leq t\leq 1$), and lower ...
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Proof that this set is convex

I need a help with prooving that a given set is a convex set: $\{ x \in R^n | Ax \leq b, Cx = d \}$ I know the definition of convexity: $X \in R^n$ is a convex set if $\forall \alpha \in R, 0 ...
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Concavity near the boundary

I am trying to understand a paper, and have come across an "it is easy to see ..." but i don't find it easy at all to see. As far as I can understand, I see the problem thus: We have a convex bounded ...
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Every face of compact convex set is closed?

Well, this is my doubt: Let $\vec{E}$ be a n.v.s. and $K\subset \vec{E}$ a compact convex set. Then every face of $K$ is closed. Any hint in order to prove it is welcome. Thanks in advance!
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Equations for interior of Platonic solids

It is well-known that for Platonic solids: The interior of cube a.k.a. hexahedron can be described with inequality $\max\{|x|,|y|,|z|\}<a$. The interior of octahedron is $|x|+|y|+|z|<a$. But ...
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Is the sub-level set of a concave function convex?

Suppose I have a set defined as follows: \begin{align*} S=\{ x: f(x) \le c\} \end{align*} where $f(x)$ is continuous and concave function defined over some $x \in K$ where $K$ is compact and convex. ...
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Test Convex Hull of Vectors

My mathematical background is generally not so great so please pardon me if my question appears silly. I am trying to test the convex hull of 3 vectors for an intersection with coordinate axes as ...
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Convex, non-negative function starting from 0

Let $h(x)$ be defined for $x \ge 0$. Suppose that $h(x)$ is non-negative and convex with $h(0) = 0$. I need to show that for $x_2 \ge x_1$ $h(x_2)\ge h(x_1)$. I need to do this from the definition of ...
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Trying to show that $(c_0, \| \cdot \|_s)$ is strictly convex, where $\| x \|_s = \underset{i = 1}{\overset{\infty}{\sum}} \frac{1}{2^i} | x_i |$

I'm trying to show that $ (c_0, \| \cdot \|_s) $ is a strictly convex space, where $$ \| x \|_s = \underset{i = 1}{\overset{\infty}{\sum}} \frac{1}{2^i} | x_i |,$$ $ x = (x_1, x_2, ..., x_i, ...) \in ...
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Example for a Schur-convex function that is not convex

Let $x \succ y $ be the majorization pre-order on real vectors. (Wikipedia link) We say a function from real vectors to the reals is Schur convex if $x\succ y$ implies $f(x) ≥ f(y)$. With the result ...
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$f:\Bbb R\to\Bbb R$ increasing and convex $\Rightarrow f(x_0)\le f(x)-c(x-x_0)$

Let $f:\Bbb R\to\Bbb R$ such that $f',f''\ge0$ on the whole real line. Then for every $x_0$ fixed, $\exists\; c\in\Bbb R$ s.t. $$ f(x_0)\le f(x)-c(x-x_0)\;\;,\;\;\forall x\in\Bbb R. $$ Now ...
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Calculating the Convex hull of a specific set in $\mathbb{R}^3$

I have to calculate the convex hull of $A=\{(-4k,k^2+2,2k^2-2k)|k\ge 2\}\cup \{(k,k^2/4-1,-k^2/4-k)|l\le -2\}$. I am aware with the Fenchel-Bunt theorem, so I just have to consider every (closed) ...
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Matrix convexity!

Given $M\in\Bbb Z_{\geq0,\leq b}^{n\times n}$, if $\mathsf{rank}(M-Q_i)=\mathsf{rank}(Q_i)$ where $i\in\{1,2\}$ with $Q_i\in\Bbb R_{\geq0}^{n\times n}$, then if $\forall ...
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On the weak* compactness of subdifferentials

Let $X$ be a normed vector space over $\mathbb R$ and $X'$ its dual space (the set of norm-continuous linear functionals on $X$). Let $f:X\to\mathbb R$ be a convex function. Consider the ...
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$f$ convex, $\lim_{x\to\infty}\frac{f(x)}{x}=0$, then $f$ is constant

Let $f$ be a convex function of $\Bbb R$ and suppose $\lim\limits_{x\to\pm\infty}\frac{f(x)}{x}=0$. How we can prove that $f$ is constant function?
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45 views

Convexity of a certain function connected to the norm

Suppose that we are given two vectors $x,y$ in a normed space $X$. Can we prove in general that the function $$t\mapsto \|x-ty\|$$ is convex? It is certainly the case if the normed space has ...
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Convex Constraint on Sine Wave Simularity

So lets say you have a vector X = [x1 x2 x3 ..... xn] You want to optimize a cost function over X. However you want to constrain the vector X to look like a sine wave. Say you can parameterize a ...
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Is multiplication of monotonically decreasing convex functions convex?

I'm aware that if $h(x)$ and $f(x)$ are convex functions, $g(x) = h(x)f(x)$ may not necessarily be convex. I'm curious whether $g(x)$ is convex if both $h(x)$ and $f(x)$ are also monotonically ...
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Any example of strongly convex functions whose gradients are Lipschitz continuous in $\mathbb{R}^N$

Let $f:\mathbb{R}^N\to\mathbb{R}$ be strongly convex and its gradient is Lipschitz continuous, i.e., for some $l>0$ and $L>0$ we have $$f(y)\geq f(x)+\nabla f(x)^T(y-x)+\frac{l}{2}||y-x||^2,$$ ...
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What is the definition of convexity from $f : \mathbb{R}^2 \rightarrow \mathbb{R}$?

$f(\lambda x + (1-\lambda y) \leq \lambda f(x) + (1- \lambda) f(y)$. This is the definition of convexity I am used to. If $f$ is a convex function, then $f : \mathbb{R} \rightarrow \mathbb{R}$. What ...
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Infinite dimensional convex cone

For every infinite set $I$, the closed convex cone $S:=\{f\in \mathbb{R}^{(I)}:f\geq 0\}$ in $\mathbb{R}^{(I)}$, equipped with the finest locally convex topology, has empty interior. How do I ...
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Proving a function is convex

From the Defintion of convex: Theorem to be proven: If $f$ is differentiable and $f'$ is increasing, then $f$ is convex. Use Proof by Contradiction. Consider, $I = (a, b)$ with $a < x < ...
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How to proof that a straight line can split a convex to at most two regions?

I am self-studying the book "Concrete Mathematics". The authors state the statement: "A straight line can split a convex region into at most two new regions, which will also be convex" 1) How can one ...
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Multivariate Normal Density Concavity

For this variance compunent model $Y$~$N(X\beta, \Omega)$, where $\Omega=\sum_{i=1}^m\sigma_i^2V_i$, the log likelihood function is $(\beta, \sigma_1^1, ..., \sigma_m^2)=C+\frac12\log ...
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36 views

Convexity and equality in Jensen inequality

Theorem 3.3 from W. Rudin, Real and complex analysis, says: Let $\mu$ be a probabilistic measure on a $\sigma$-algebra of subsets of a given set $\Omega$. If a function $f:X \rightarrow \mathbb R$ ...
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31 views

Does having positive second derivative at a point imply convexity in some neighborhood?

Suppose that I have a real valued function of a single variable $f(x)$ which is twice differentiable in some open interval $I$. Then, I know from calculus that if $f''(x) >0 $ on $I$, then $f$ is ...
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31 views

is this a convex optimization problem?

Can someone clarify is this a convex optimization problem or not. $min \| X-UV\|_{F}\quad $ s.t $ \quad U \geq ,V\geq0$ . If not , what makes the problem non-convex?
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Is closure of convex hull of C equal to convex hull of closure.

If $C$ is a set in a topological vector space (or in particular a metric space), can we say that $\text{cl}(\text{conv}(C)) = \text{conv}(\text{cl}(C))$, where cl$(\cdot)$ represents closure and ...