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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sum of concave and convex function

Suppose $f$ is the sum of a concave and convex function, i.e. $$f=f_1+f_2$$ where $f_1$ is a concave function and $f_2$ is a convex function. I wonder if $f$ can be written as the following: ...
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1answer
444 views

Convex hull of sets defined by (in)equalities

If you define the convex hull of a set $X$ as the set of all convex combinations of elements of $X$, it becomes difficult to decide if a given element $w$ belongs or not to $conv(X)$ (You have to ...
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521 views

Lovasz Extension Intuition

I am confused by the definition of Lovasz extension. The problem is I don't get the intuition behind the definition. In addition, Lovasz extension can be defined in different ways I don't see that ...
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1answer
155 views

Convex functions

How does one show that $\phi(x)$ convex and twice differentiable implies that $x\phi(\frac{y}{x})$ is convex on the plane $x>0$? Thanks.
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63 views

Jensen-like Inequality

I have the following question: Suppose we have a function $g:\mathbb{Z}_+ \cup \{0\} \rightarrow \mathbb{R}_+$ with the property, $g(\lfloor \frac{x+y}{2} \rfloor)$ + $g(\lceil \frac{x+y}{2} \rceil) ...
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1answer
191 views

Convex conjugate of absolute affine function?

Let $f:\mathbb{R}^n \rightarrow \mathbb{R} \cup \{ \infty \}$ be a convex function. The convex conjugate of $f$, which we call $f^*$ is defined as $f^*(y)=\sup \, \left \{ \langle y,x \rangle -f(x) ...
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3answers
164 views

Euclidean Metric and Convexity

Question: Consider the Euclidean metric space $(\mathbb{R}^n , \Vert\cdot \Vert)$. Let $X\subset \mathbb{R}^n$ and $f\colon X \to \mathbb{R}$. $X$ is said to be a convex set if for every $x,y \in X$ ...
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1answer
81 views

Convex analysis problem

I have the following problem. Let $f:[a,b]\to \mathbb{R}$ be continuously convex. I have to prove that there exists $c\in (a,b)$ such that $$\frac{f(a)-f(b)}{b-a}\in \partial f(c)$$ Firstly, I'm ...
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1answer
298 views

Going in the direction of the gradient

First, a motivating example. Suppose $f(x)$ is convex, differentiable, with a single minimum $x^*$. Then the differential equation $$\dot{x}(t) = -\nabla f(x(t))$$ drives $x(t)$ to $x^*$. Now my ...
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47 views

Example of weakly discontinuous contraction

Can somebody give an example of a projector $P_c$ on the convex closed set C which is not weakly-continuous?
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80 views

Space of solutions to $f(x+y) = f(x) + f(y)$ when $f$ is convex

Let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be a nonlinear convex function, and let $x\in\mathbb{R}^n$ be an arbitrary vector. Define the set $\Omega_f(x)$ as $\Omega_f(x) \triangleq ...
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1answer
228 views

What is the typical method for sampling uniformly in a convex polytope

The polytope in my case is the intersection of the k-plane $Ax=b$ and $\{x>0\}$ where $A$ is the constraint matrix and $b$ is some solution. I'd like to find a method that is fast and efficient for ...
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1answer
328 views

interior points and convexity

Question: Let If $P\subseteq \mathbb{R}^n$ be a convex set. show that $int(P)$ is a convex set. I know that a point $x$ is said to be the interior point of the set $P$ if there is an open ball ...
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1answer
297 views

Jensen integral inequality in multidimensional case

The classical Jensen integral inequality says: Let $\mu$ be a probabilistic measure defined on some $\sigma$-algebra subsets of $\Omega$. If $f\colon \Omega \rightarrow \mathbb R$ be an integrable ...
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1answer
225 views

Concavity and Convexity

A set $X \subseteq \mathbb{R}^n$ is said to be convex if $tx + (1-t)y \in X$ for all $x,y \in X$ and $t \in (0,1)$. Given a convex set $X \subseteq \mathbb{R}^n$, a function $f: X \to \mathbb R$ is ...
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2answers
61 views

Generalizing the Definition of Convexity

The definition of convexity can be given as: Definition: Call a subset of $\mathbb{R} ^ k$, which will be denoted $E$, convex if given two elements of $E$, $\boldsymbol{x}$ and $\boldsymbol{y}$ and ...
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1answer
932 views

Proof of Clarkson's Inequality

Trying to find a proof for Clarkson's inequality, which states that if $2 \leq p < \infty$, then for any $f, g \in L^p$, we have that $$\left|\left|f+g\right|\right|_p^p + ...
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209 views

Show that $2^n>1+n\sqrt{2^{n-1}}$

If $n$ be a positive integer greater than $1$, then prove that $$2^n>1+n\sqrt{2^{n-1}}$$ I found this problem under $AM \ge GM$ chapter. Help me to solve this problem using $AM \ge GM$. ...
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2answers
72 views

Using $AM \ge GM$ which is greater among $1+\dfrac{1}{n}$ and $2^{1/n}$.

From expansion I see that $1+\dfrac{1}{n} \ge2^{1/n}$.But can't solve it using $AM \ge GM$. Please help.
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3answers
153 views

Prove that $\dfrac{b^2+c^2}{b+c}+\dfrac{c^2+a^2}{c+a}+\dfrac{a^2+b^2}{a+b} \ge a+b+c$

If $a,b,c$ are positive , show that $$\dfrac{b^2+c^2}{b+c}+\dfrac{c^2+a^2}{c+a}+\dfrac{a^2+b^2}{a+b} \ge a+b+c$$ Trial: Here I proceed in this way ...
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153 views

Show that $\frac{x}{1-x}\cdot\frac{y}{1-y}\cdot\frac{z}{1-z} \ge 8$.

If $x,y,z$ are positive proper fractions satisfying $x+y+z=2$, prove that $$\dfrac{x}{1-x}\cdot\dfrac{y}{1-y}\cdot\dfrac{z}{1-z}\ge 8$$ Applying $GM \ge HM$, I get ...
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2answers
218 views

Show that $(1+x)(1+y)(1+z)\ge 8(1-x)(1-y)(1-z)$.

If $x>0,y>0,z>0$ and $x+y+z=1$, prove that $$(1+x)(1+y)(1+z)\ge 8(1-x)(1-y)(1-z)$$ Trial: Here $$(1+x)(1+y)(1+z)\ge 8(1-x)(1-y)(1-z) \\ \implies (1+x)(1+y)(1+z)\ge 8(y+z)(x+z)(x+y)$$ I ...
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1answer
145 views

optimizing a logdet function with respect to a scalar and the Hessian matrix

Given a logdet function $\mathcal{L}(\gamma)$, $$ \mathcal{L}(\gamma) = \log\vert \mathbf{I} + \gamma\mathbf{S} \vert - \mathbf{q}^T(\gamma^{-1}\mathbf{I} + \mathbf{S})^{-1} \mathbf{q}, $$ where ...
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1answer
204 views

strict convexity with a measure theoretic property

Suppose $(x_n)$ is a positive sequence of reals converging to $x$. Furthermore we have a measure space $(E,\mathcal{E},\mu)$ given, with finite measure $\mu$. There are a measurables nonnegative and ...
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1answer
39 views

Does every convex subset of $\mathbb{R}^n$ have normal almost everywhere?

Does every convex subset of $\mathbb{R}^n$ have normal almost everywhere? I want to know if I have any convex subset $M$ of $\mathbb{R}^n$ is it meaningful to talk about integral of second kind ...
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2answers
223 views

Optimize a log det function with respect to a matrix, and the saddle point analysis

Suppose I want to to find the local minima of a logdet function $\mathcal{L}$ with respect to a Matrix $\mathbf{A}$, $$ \mathcal{L} = \log\vert \mathbf{I} + \mathbf{A}\mathbf{S} \vert - ...
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1answer
122 views

Does setting derivative to zero suffice always for minimization of convex functions?

I have this convex function in $X$, given by $Trace(AX^TBX)$ where $A$, $B$ are p.s.d and all entries are real. Now if I had a linear function $l(X)$ that prevents a trivial zero-matrix solution for ...
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1answer
156 views

Equivalent definitions of uniform convexity.

I think I'm dumb, but I can't follow a simple proof from "Functional analysis and infinite dimensional geometry". They show that two different definitions of modulus of convexity of a norm are the ...
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58 views

How to show that $ Ax \le b$ is convex?

For $$ A \in \mathbb{R}^{m \times n}, b \in \mathbb{R}^m, c \in \mathbb{R} $$ one has to show that $$ K:= \{ x \in \mathbb{R}^n: Ax \le b \}$$ is convex. Now I'm aware that by definition, a set ...
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49 views

Distance between convex set and non-convex set?

So in http://en.m.wikipedia.org/wiki/Shapley%E2%80%93Folkman_lemma there is some talk about distance between a mintowksi sum and a convex set. But I couldn't get how distance is being defined. Can ...
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1answer
138 views

A convex function that is bounded on a neighborhood is Lipschitz

Let consider a normed vector space $V$. I want to prove that If $f:V\to \mathbb R$ is a convex function and if for some $x_0 \in V$ the function is bounded on a neighborhood $W$ of $x_0$, then ...
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2answers
51 views

proving that this function does not define a norm on $\mathbb R^2$ since the convexity

This problema use the previous part to conclude something, so I write all the parts. First I have to prove that every norm in $\mathbb R^n$ is a convex function, I did it, it only requires the ...
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1answer
63 views

A reference to learn about duality

I am interested in learning about duality in convex optimization. I am looking for something to read which is: Reasonably short. Fairly self-contained (if it is a chapter in a textbook, I would ...
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687 views

How many points of intersection between an ellipse and an $L_p$-circle?

Consider an ellipse $E$ in the plane, centered at the origin. (In my case, the minor axis points into the nonnegative quadrant.) Let S be an "$L_p$-circle": $S = \{(x,y) : |x|^p + |y|^p = 1\}$, ...
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2answers
95 views

Is it a convex function?

Let $f(.)$ be a function. If $f(X)$ is a convex function of $X$, where $X$ is a matrix. Is $f(AXB)$ also a convex function of $X$? ($A$ and $B$ are fixed matrices).
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1answer
72 views

convex function, inequality

If $f: R^n\rightarrow R$ is convex and $f(\alpha x)=\alpha f(x), \alpha \geq 0$, show that: a) $f(x+y)\leq f(x)+f(y)$ for all $x,y\in R^n$ b) $f(0)\geq 0$ c) $f(-x)\geq -f(x)$ for all $x\in R^n$ d) ...
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1answer
69 views

Proof of convexity of a function

I have to prove that function $J(x)=e^{x^3+x^2+1}$ is convex on $[0,\infty]$. I used a Theorem which says: **$U\subset R^n$ is convex set with non-empty interior and $J\in C^2(U)$. Function $J$ is ...
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1answer
269 views

Why it is sufficient to show $|f'(z)-1|<1$?

According to an article entitled "On the Univalency of Certain Analytic Functions" by Wang et al. (2006), we have to show that $|f'(z)-1|<1$ in order to find the radius of univalency for the class ...
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1answer
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Is $f(x,y) = x^2y + x y^2$ (quasi-) concave or convex?

I should analyze whether the function $$f(x,y) = x^2y + x y^2 \text{ where } x,y > 0$$ is (quasi-) concave or convex. Thus, as usual, I set up the Hessian as $$ D^2f(x,y) = \left( ...
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Is the following function of several variables concave?

Suppose that $w_1,\dots,w_p\in(0,1)$ satisfy the condition $\sum_i w_i=1$, and let $$F(w_1,\dots,w_p) = \frac{1-\sum_i w_i^2}{\sum_i(1-w_i)^2 \left[\sum_i \frac{w_i}{1-w_i}\right]^2}.$$ Is function ...
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1answer
297 views

Characterisation of linearly separable points of a hypercube

Essentially, linearly separable points are just those corners that can be cut off with just one slice as marked out by a hyperplane. E.g. for a cube, the following 4 points (red) are not linearly ...
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1answer
85 views

Is $f(X)=-tr(AXBX^T)$ convex?

Given $A,B \in \mathbb{R}^{N \times N}$ and they are non-negative matrix. Is $f(X)=-tr(AXBX^T)$ convex when $X$ is also non-negative? If yes, how can I show that?
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78 views

Convex function Inequality (3 - point version)

I was reading this article on inequalities (which some of you may find useful) here. On page 7, I came across this question by Titu Andreescu, which I shall reproduce here: Question: Let f be a ...
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1answer
45 views

convexity of two linear spaces connected by a convex equality constraint

If there are two sets of linear constraints in different variables, Ax <= b with x_l <= x <= x_u and Cy <= d with y_l <= y <= y_u, and a set of equality constraints of a specific ...
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1answer
179 views

Bound on Expectation of a convex function of a Random variable

My friend asked me the following question, which I at first thought was simple and straightforward: If $X$ is an integrable random variable and $g$ is a convex function(all real valued), then is it ...
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46 views

Determining the value of m for an m-convex set that is also non-convex

I'm looking within my PhD at atm at decomposing a random non-convex subset of the Euclidean Plane into a union of n convex sets, particularly hoping that the these sets (that from the overall ...
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2answers
325 views

Can one define the derivative of a function using tangent cones? Does such a notion already exist?

I'm interested in finding an analogue of a derivative that applies to functions which are defined more general subsets of $\mathbb{R}^n$ than open subsets. In particularly, I'm looking at functions ...
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1answer
99 views

Projections onto closed and convex sets

I have to prove that if $A$ is convex and closed set, then $z=P_A(x)$ for all $z\in A$ if and only if $\langle x-z, z-y\rangle \geq 0$ for all $y\in A$ I have following proof which is not much ...
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2answers
70 views

determining whether a set is convex

I came across this exercise question from a course on optimisation. It only discussed basic aspects of convex functions. The question asks: if the solutionn set of the following inequalities convex. ...
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1answer
177 views

How many hyperplanes does it take to separate $n$ points in $\mathbb{R}^m$?

By 'separate', I mean that each point lies in its own little region/cell. For instance, it takes a minimum of $P = 4$ lines to separate $n = 7$ points in $\mathbb{R}^2$ ($m=2$), assuming that no 3 ...