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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Type of convex function?

I want a convex function $f:\mathbb{R} \to \mathbb{R}$ with the following property: given points $x,d \in \mathbb{R}$, and $\alpha \in (0,1)$, we have $$f(x + \alpha d) \geq \alpha f(x + d).$$ Is ...
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Approximating a set of convex quadratic inequalities by a convex polytope

I have a convex set of the form $$Z = \{x|x^TQ_ix+b_i^Tx+c_i\le0,i=1,\ldots,m\}$$ where each $Q_i\succeq0$, that I wish to approximate by a set of the form $$\hat Z = \{x|Ax\le b\}$$ We can further ...
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81 views

Convex decomposition of a vector

Let $(a_i)_{i=1}^n$ be a probability vector, that is, $a_i\geq 0$ and $\sum_i a_i=1$ and let $(U_{ij})_{i,j=1}^n$ be a unistochastic matrix, that is, the pointwise square of a unitary matrix. Now ...
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314 views

Solution set of a quadratic inequality

Let C $\subseteq$ $\Re^n$ be the solution set of a quadrtatic inequality, C = $\{x \in \Re^n | x^TAx +b^Tx + c \leq 0\}$. $A \in \Re$, b $\in \Re^n$ and c $\in \Re$. We want to show: That C is ...
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49 views

Prove a certain matrix is positive semidefinte.

Consider a stochastic matrix $P$, i.e. real, non-negative, square, rows sum to one. Consider $\Xi$ to be a diagonal matrix with a principal left eigenvector of $P$ on the main diagonal and zeros ...
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55 views

Analytical solution to the first PCA direction

It is known that the first PCA direction for a dataset of $n$ points is the unit vector with max variance after projecting the points onto this vector. I wonder whether there are some analytical ...
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33 views

convexity of piece wise function

I have a piece-wise function defined on the real line. The pieces are connected continuously, and the second derivative of each piece is strictly positive. Does this means that the function is convex? ...
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164 views

Looking for numerical methods for finding roots of convex vector function ${\bf f}({\bf x})={\bf 0}$

Consider the function ${\bf f}:\mathbb{R}^n\to\mathbb{R}^m$ defined as ${\bf f} = (f_1,f_2,\ldots,f_m)$ where each $f_i:\mathbb{R}^n\to\mathbb{R}$ is twice-continuously differentiable convex in ${\bf ...
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27 views

Properties of functions having the form $g(x,t) = t f(\frac{x}{t})$

I have been frequently coming across the function $g(x,t) = t f(\frac{x}{t})$ in my course on convex optimization. A friend of mine mentioned that it is the perspective function, but the book on ...
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578 views

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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How to prove the convexity of $f$ if the strict epigraph of $f$ is convex

I have trouble to prove the equivalence of the following two definitions of convex function. For convenience, I list them as follows: Def-A. Let $f : I\to \mathbb{R}$ be a function, where $I$ is (and ...
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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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48 views

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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342 views

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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2answers
137 views

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( x)^{\...
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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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132 views

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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57 views

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{ , } \sqrt{y}+\...
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119 views

Uniquness and Exisstence of One Theorem

I need a short and nice Proof for Uniqueness and Existence of the following theorem: Suppose (H, <0,0> ) is a Hilbert space, and M is a closed convex set and $x \in H$, then there is a unique ...
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37 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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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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333 views

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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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 an ...
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55 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)}) + \lambda(1-\...
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62 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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102 views

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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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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1answer
369 views

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 $$ \min_{w,b,\...
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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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212 views

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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2k views

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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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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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 conv$(...
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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 \overline{...
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119 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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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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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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101 views

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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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 it. ...
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60 views

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 $K\...
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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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489 views

Does there exist regular planar pentagon?

How to prove or disprove that the boundary of any convex body in $\mathbb{R}^3$ (treated as a surface) includes 5 points which form a regular planar pentagon?
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Why a set of positive definite matrices define a half space?

A half-space is a set of the form $\{x|a^Tx \leq b\}$. Also it is stated that the set $\{X\in S^n | z^TXz \geq 0 \}$, with $S^n$ denote the set of symmetric $n\times n$, is a half space$^1$, Can we ...
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43 views

Second order cone with quadratic interpretation

Could you please help me to understand how the second part of the equation (quadratic form) derived form the first one? The basic definition of the second-order cone is: $C = \big\{(x,t) \in \mathbb{...
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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 $x$...
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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 ...