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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Does every strongly convex function has a stationary point?

Say does every differentiable $\mu$-strongly convex function $f:\mathbb{R}^n\mapsto\mathbb{R}$, with $\mu>0$ have a point where its gradient is $0$? If not so which is the minimum you can impose ...
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Property of Newton step for self-concordant convex functions

Suppose $f(x)$ is a convex and self-concordant function minimized at $x^*$. I have two starting points $\tilde{x}_0$ and $\hat{x}_0$ such that $|\hat{x}_0-x^*| \le |\tilde{x}_0 - x^*|$. We also know ...
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113 views

Segment ordered density conjecture.

I have a set $S\subset\mathbb {R}^2$ with the following property (P) $\forall x,y\in S$, $\forall\mathscr{C}$ a convex set that contains $x$ in its interior, $bd\mathscr{C}\cap [x,y]\subset ...
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Using Taylor's theorem and Lagrange form of the reminder to prove the second order condition for convexity

I try to prove the second order condition for convexity. So far' I've done the following: First, I prove second order => convexity: Let $f$ be a function with positive semi-definite Hessian. Using ...
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1answer
27 views

How to determine if a vector belongs to the conical hull of a set of vectors?

Let $\mathbf{p}_i$ be a finite set of finite-dimensional real vectors with non-negative components with the property that, for any $k$, $\mathbf{p}_k$ cannot be expressed as a linear combination with ...
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22 views

Proofs regarding about all Second Derivative Test cases (Inconclusive & Single Variable)

This is how I would prove f''(c) > 0 that f(c) has local min and I would easily flip the inequalities and state a conclusion for f''(c) < 0 that f(c) has local max. Quick Proof for f''(c) > 0 ...
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Is every convex function differentiable amost every where?

If a function $f: D \subset R^n \rightarrow R$ is convex, then for every $x,y \in D$ and $\alpha \in [0,1]$, $$ f(\alpha x + (1 - \alpha)y) \leq \alpha f(x) + (1 - \alpha)f(y). $$ I konw a convex ...
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Every concave function that is nonnegative on its domain is log-concave?

This is a statment from Wiki. I'm not sure why this is true: If: $f(\theta x+(1-\theta y) \geq \theta f( x) + (1-\theta)f(y)$ And $f(\cdot) \geq 0$ then: $$f(\theta x+(1-\theta y) \geq f( ...
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The polar of a set: Importance and Applications

Given a duality $(X,Y)$, for any subset $A\subset X$, we can define the polar set $A^{\circ}\subset Y$. In this sense, the polar relates sets and sets in the dual space. Since cones are sets, we can ...
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20 views

What's wrong with the following trace optimization?

I'm reading a paper that has used the augmented Lagrange function for optimization. I've tried to derive one subproblem but got a different answer from that in the paper. Could you help check it ...
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Minkowski sum of convex sets in the plane which are not polygons

Can the Minkowski sum of two convex sets in the plane which are not polygons be a polygon? Explicitly my convex set is of the form $ C= \{(x,y) \in \mathbb{R}^2 : x,y \geq 0 \text{ , } ...
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How to express a set as an intersection of halfspaces

I have a set S = {x $\epsilon$ $\mathbb R^n$| $x^Ty \le 1$, $\forall y \epsilon A$} Now, I want to prove that this set is closed and convex. I know that expressing this set as an intersection of ...
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1answer
20 views

Is it right for chain rule in trace function?

The objective function is $$ f(X)=\min_X trace(B^TX^TCXBD) $$ we know the following derivatives from Matrix Cookbook, $$ \frac{\delta{trace(B^TX^TCXB)}}{\delta X}=C^TXBB^T+CXBB^T \\ \frac{\delta ...
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Which functions preserve convexity of the cells of a polyhedral tessellation of $\mathbb{R}^n$?

Let $\mathcal{X}_1, \ldots, \mathcal{X}_m \subseteq \mathbb{R}^n$ be disjoint sets with $\bigcup_{i=1}^m \mathcal{X}_i = \mathbb{R}^n$. Furthermore, let each $\mathcal{X}_i$, $i = 1, \ldots, m$, be an ...
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How do you prove $x^2$ is convex using only the definition of convexity?

I apologize beforehand if this is an extremely easy question, I have very limited experience with proving convexity statements. Also, the reason that I choose such a simple example before tackling ...
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Is Jensen's inequality an iff condition on convex functions?

According to wikipedia this is Jensen's inequality: If X is a random variable and φ is a convex function, then: $$\varphi\left(\mathbb{E}[X]\right) \leq \mathbb{E}\left[\varphi(X)\right].$$ Which ...
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38 views

Proof that $f(x) = x^TMx$ is convex

I have been stuck on this problem for a while. After I use the definition of convexity and some algebra, I end with something like this: $$ \lambda f(x^{(1)}) + (1-\lambda)f(x^{(2)}) + ...
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Convex sets in $\mathbb R^n$: Do they have a particular form ? Does the gradient of a linear convex function $f$ exist on such a set?

Convex sets in $\mathbb R^n$: Do they have a particular form ? Does the gradient of a linear convex function $f$ exist on such a set ? Suppose that $S$ is a convex set in $\mathbb R$. Then $S$ is ...
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Legendre Transformation of a Lagrangian in Classical Mechanics

I have some questions about the Legendre Transformation of a Lagrangian in Classical Mechanics to the Hamiltonian: We start with a Lagrangian $L(q,\dot{q})=\frac{\langle \dot{q} , \dot{q}\rangle }{2} ...
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31 views

What is affine hull of conv(A)

Consider the set $A = \{(1,0),(0,1),(-1,0),(0,-1)\}$. The convex hull of $A$, i.e. $conv(A)$, should look like the following: (This is also a $l_1$-norm unit ball.) My question is what is the ...
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In what sense is the Bayesian posterior mean a “convex combination”?

This is related to a previous question that hasn't gotten an answer: Definition of convex combination with matrix-vector multiplication Suppose I want to estimate $x \in \mathbb{R}^n$ from two ...
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86 views

Find function with given properties

Find a smooth function $g: \mathbb{R} \to \mathbb{R}$ that domain $g$ is $\mathbb{R}$ range of $g$ is a subset of $\mathbb{R^+}$ $g$ is concave.
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Arclength comparison of two convex functions

(a) Let $f$ and $g$ be two $C^1([a,b])$ convex functions such $$f(a)=g(a), \ f(b)=g(b)\ \text{ and } \ g(t)\le f(t) \ \text{ for all }t \in [a,b]$$ Then the arclength of the graph of $g$ ...
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Show that $x \mapsto \left( x^{\top} \sigma x , -\mu^{\top}x \right)^{\top}$ transforms a given set into a convex set.

Let's say you have a covariance matrix $\sigma$ and a vector of expected returns $\mu$. Basically $\sigma$ is a symmetric matrix with positive eigenvalues and we can safely assume that $\mu$ is just a ...
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85 views

Is there any version of Jensen's inequality for quasiconvex function

I am looking for some generalization of Jensen's inequality for functions $g:\mathbb{R}^n \rightarrow \mathbb{R}$ where $g(x)$ is quasiconvex (or not convex). We known that for convex functions, ...
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Function with convex polytopes as sub-levelsets

If a function has convex sub-levelsets then it is quasiconvex. What if it has sub-levelsets that are convex polytopes? Obviously, it is still quasiconvex but is there a name for this class of function ...
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244 views

Example of strictly convex space with not strictly smooth dual

I'm trying to find an example of a space $V$ which is strictly convex, but has a dual space $V^*$ which is not strictly smooth. Any help please?
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Prove that $0 < x < y$ implies $\|x\| < \|y\|$ for any norm.

All vectors are real. Prove that $0 < x < y$ (element-wise) implies $\|x\| < \|y\|$ for any norm. This is probably very basic, but I don't seem to get the hang of it. Edit: it turns out this ...
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About the slack variable for hinge-loss SVM

The hinge-loss SVM is defined $$ \min_{w,b} \frac{1}{2}w^T w+\sum_{i=1}^{N}\max\{0,1-y_i(w^Tx_i +b)\} $$ By introducing a slack variable $\xi_i$, the optimization problem is changed to $$ ...
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Topological vector space question

$C[0,1]=$ space of all continuous complex valued function over $[0,1]$. Define metric, $d(f,g)={\int_{0}^{1} \frac {|f(x)-g(x)|}{1+|f(x)-g(x)|}}$, for all $f,g\in C[0,1] .$ Let $(C[0,1],\sigma)$ ...
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If $ S \subseteq \mathbb{R}^n$ is finite, show that conv(S) is a closed set…

If $ S \subseteq \mathbb{R}^n$ is finite, show that conv(S) is a closed set. Is the statement still true if S is not finite? Where conv(S) is the convex hull of S. From what I've read, the convex ...
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Distributed convex optimization problem

Consider the optimization problem $$ \min_{ x_1, \ldots, x_N } \sum_{i=1}^{N} f_i( x_i ) \\ \text{s.t.: } \sum_{i=1}^{N} x_i \in X, \ x_i \in X_i \ \forall i \in \{1, \ldots, N\} $$ where $f_1, ...
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Help with an inequality involving a convex function

Let $a< f(x) < b $, $x \in \Omega $, $\mu(\Omega )=1 $, and set $t=\int f d \mu $. Then $a < t < b $. Suppose $\phi $ is a convex function on $(a,b) $ then by definition of convexity ...
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Proof that the intersection of any finite number of convex sets is a convex set

How to prove that the intersection of any finite number of convex sets is a convex set? I have no idea.
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61 views

A function is convex and concave, show that it has the form $f(x)=ax+b$

A function is convex and concave, it is called affine function. That is the function: $$f(tx+(1-t)y)=tf(x)+(1-t)f(y),\, \, t\in (0,1) $$ Force $y=0$(suppose $0$ is in the domain of $f(x)$), we ...
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union and difference of convex set

suppose X,Y are two convex sets x1, x2 in X and y1, y2 in Y defn of X and Y being convex: tx1+(1-t)x2 in X ty1+(1-t)y2 in Y it is clear that: 1) X+Y is convex. 2) X intersection Y is convex 3) ...
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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 ...
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49 views

Is this function convex or concave

Consider the following function: $$f:(0, \infty)^2 \rightarrow \mathbb{R}: (\phi,\psi) \rightarrow \frac{\phi}{\psi}$$ Is this function convex or concave? (Or neither?) I tried by calculating the ...
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Is the algebraic interior relatively open in a closed convex set?

I am struggling with proving or disproving the following: Let $X$ be a locally convex space and let $C\subset X$ be closed convex with non-empty algebraic interior which we denote by $C^i$ (recall ...
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finding a counter example to Caratheodory's convex hull theorem for infinite dimentional space

Recently I learned in class Caratheodory's convex hull theorem which suggests that in a finite dimentional (n) space every x can be written as a convex-combination of maximum (n+1) vectors. I was ...
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Jensen's inequality: proof by using linear functions

Here's an extract from Stochastic Calculus for Finance Volume 1 by Shreve. I don't understand the statement that says a convex function is the maximum of all linear functions that lie below ...
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Origin of the term `quermassintegral'.

What is the origin of the term `quermassintegral'? I think this is a german word. What would be its literal translation in English? The definition of quermassintegrals from wikipedia: Let ...
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Equivalence of semi concavity of function $g$ and convexity of function $x\mapsto \frac c 2 |x|^2 - g(x)$

$g\in C^2(\mathbb R^n)$ is called semi concave, if there exists $c>0$ such that for all $x,y\in\mathbb R^n$ the following holds: $$g(x+y) - 2g(x) + g(x-y) \leq c|y|^2$$ Now, in Evans "Partial ...
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Interior of sum of sets equals sum of interior of summands

I'd like to have the answer to the following question. If $X_1,X_2\subseteq \mathbb{R}^n$ are convex and compact sets of dimension $n$, does the following hold: ...
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How to find the convex hull of a given set?

$A=\{(0,0),(0,1),(1,0)\}$ $B=\mathbb{Q}^2$ $C=\{(x,\sqrt{x})\in \mathbb{R}^2:x\ge0\}$ I have to find Conv(A), Conv(B) and Conv(C). My attempt Conv(A) is the boundary (correction: obviously it ...
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If $f$ is continuous and $\,f\big(\frac{1}2(x+y)\big) \le \frac{1}{2}\big(\,f(x)+f(y)\big)$, then $f$ is convex

Let $\,\,f :\mathbb R \to \mathbb R$ be a continuous function such that $$ f\Big(\dfrac{x+y}2\Big) \le \dfrac{1}{2}\big(\,f(x)+f(y)\big) ,\,\, \text{for all}\,\, x,y \in \mathbb R, $$ then how do we ...
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How to construct a suitable example in matrix convex

To show that the function $X \to X^{3}$ is generally not matrix convex of order 2 on $S_{+}^{2}$. I cannot find an example and even don't know how to construct one.
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2answers
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Conditional expectation of a random vector taking values in convex sets

on a probability space $(\Omega, \mathcal{F},\mathbb{P})$ i have a random vector $X\in L^1_{\mathbb{P}}(\mathbb{R}^d)$ (integrable with values in $\mathbb{R}^n$), such that $\mathcal{P}-a.s.$ $$X\in ...
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1answer
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What is the sub-differential of the separable sum $R(w) = \sum^{d}_{j=1} |w_j|$?

Recall the definition of a sub-differential: $$\partial F(w_0) = \{ v : \forall w, F(w)-F(w_0) \geq v \cdot (w - w_0)\} $$ Intuitively, for any w in the domain of the function one can draw a plane ...
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discussing the existence of a convex function

If $g$ is a positive function on $[0,1]$ such that $g(x)$ tends to $\infty$ as $x$ tends to $0$, then there is a convex function $h$ on $[0,1]$ such that $ h \leq g$ and $h(x)$ tends to $\infty$ as ...