0
votes
0answers
24 views

Common subdifferentials of convex function

Let $f: \mathbb R^n \rightarrow \mathbb R$ be a convex function. By a subdifference of $f$ in $x\in \mathbb R^n$ we mean an $h\in \mathbb R^n$ such that $f(x) \geq f(p)+<x-p,h>$ for all ...
1
vote
1answer
39 views

Prove that $f(x) > g(x)$ where both functions are convex and have the same value and slope at $0$

Let $f: [-a,a] \to \mathbb{R}$ and $g: [-a,a] \to \mathbb{R}$ be two non-negative, convex and smooth functions. We further know $f(0) = g(0)=0$ and $f'(0) = g'(0)=0$. I'd like to show $$f(x) \ge ...
1
vote
0answers
33 views

A proof for Jensen’s inequality

I’m trying to prove a version of Jensen’s inequality, but I end up with the wrong result. I’d appreciate any help or comments. The theorem states: let $\varphi :{{R}^{k}}\to R$ be convex. Then for ...
0
votes
1answer
38 views

Two Strictly Convex Functions with Contact of Order 1

Let $f,g: \mathbb{R}\rightarrow \mathbb{R}$ be two strictly convex functions, where $f$ is differentiable, $g$ is smooth, and $f\geq g$. Suppose that for some $x_0\in \mathbb{R}$: ...
1
vote
1answer
17 views

Differentiable Strictly Convex Function on Interval

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a differentiable, strictly convex function. Let $I\subset \mathbb{R}$ be a closed, bounded interval such that $f'(x) \neq 0$ on $I$. Is $f$ strongly ...
0
votes
1answer
30 views

Strictly Convex and Differentiable Implies

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be strictly convex and differentiable. Is $f$ strongly convex when restricted to a closed and bounded interval $[a,b]$? This is true if $f$ is smooth but am ...
1
vote
0answers
30 views

(Strictly) concave/convex function: Increasing / decreasing slope triangle?

I have the following (possibly quick) question. In a paper I am working with, the following conclusions are drawn which I have a hard time to understand. Since they are given without proof, I assume ...
1
vote
1answer
25 views

Difference in Definitions of Quasiconvexity

So I've seen a few different definitions of quasiconvexity of a function and I cannot, after a bit of working, figure out how all of them are related: Let $X$ be a convex subset of a real vector ...
3
votes
1answer
61 views

How to prove Godunova's inequality?

Let $\phi$ be a positive and convex function on $(0,\infty)$. Then $$\int_0^\infty \phi\left(\frac{1}{x}\int_0^x g(t)\,dt\right)\frac{dx}{x} \leq \int_0^\infty \phi(g(x))\frac{dx}{x}$$ The ...
0
votes
0answers
28 views

Proof of sufficient condition of existence of Lagrange multipliers

Consider the optimization problem $$ (P) \quad \inf\{ f(x) : g_i(x) \le 0, i = 1,\ldots, m, x \in \mathbb R^m \} $$ where $f, g_i : \mathbb R^n \to (-\infty, \infty]$ are convex and $0 \ne ...
0
votes
1answer
33 views

Sufficient condition on convex function such that $f(x) > -\infty$ for all $x$.

Let $f : \mathbb R^n \to [-\infty, \infty]$ convex and let $f(\overline x) > -\infty$ for $\overline x \in \mbox{int}(\mbox{dom}(f)$. Show that $f(x) > -\infty$ for all $x \in \mathbb R$. ...
0
votes
0answers
21 views

Polar set and adjoint operator

I'm trying to understand the following statement: A bounded operator between Banach spaces $u:X\rightarrow Y$ satisfies: $$\textrm{inf}\{\;||u(B_X \cap S)||:S\subset X,\,\textrm{codim}S=k \; ...
0
votes
0answers
24 views

Dual convex pairs

I am currently trying to understand a certain proof. The author uses the term dual convex pair for a pair $(\phi,\psi)$ of convex functions defined on subsets $X,Y$ of $\mathbb R^n$ satisfying: $$ ...
2
votes
0answers
119 views

Convex subsets, Normed spaces, Separating hyperplane

I'm trying to understand the proof of a theorem taken from a textbook. Theorem: If $S_1,S_2$ are disjoint non-empty convex subsets of a real vector space $X$ (which may be infinite dimensional) then ...
0
votes
0answers
29 views

What Projections preserve Pseudocontractiveness?

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a pseudocontraction, i.e., $$ \left\| f(x) - f(y) \right\|^2 \leq \left\| x-y\right\|^2 + \left\| f(x) - f(y) - (x-y) \right\|^2 $$ for all $x,y \in ...
1
vote
1answer
29 views

Equality constrained Quadratic Program

Consider the QP $$ x^* = \arg \min_{\displaystyle x \in \mathbb{R}^n{\geq 0}} \ \frac{1}{2} x^\top P x + q^\top x \ \text{ sub. to: } A x = b, $$ where $P \succ 0$. Without the non-negativity ...
2
votes
1answer
59 views

Existence of global minimum

Could someone help me with this problem? Let $C$, $D$ convex and closed sets such that the intersection is empty. I want to show that the function $f: \mathbb{R^n} \to \mathbb{R}$ defined by $f(x) = ...
5
votes
1answer
117 views

A bounded subset in $\mathbb R^2$ which is “nowhere convex”?

Let $F : \mathbb S^1 \to \mathbb R^2$ represents a simple closed curve $C$ in $\mathbb R^2$. The Jordan curve theorem says that the curves bounds a interior domain $\Omega$ and $\partial \Omega= C$. ...
0
votes
1answer
29 views

A characterization of differentiability of a convex function

Let $\phi : \mathbb R^n \to \mathbb R$ be a convex function. For all point $x\in \mathbb R^n$, define the subdifferential as $$\partial \phi(x) = \{ y\in \mathbb R^n | \ \phi(z) \geq \phi(x) + ...
1
vote
1answer
87 views

Tangent Cone is a cone?

I first give two definitions. Def1: A set $S$ is a cone if $x \in S, \lambda \geq 0 \implies \lambda x \in S$. Def2: Let $S$ be any set (we may assume $\mathbb{R}^n$ with the usual Euclidean ...
0
votes
1answer
35 views

Proof of Projection property of convex set

suppose a set S$\subseteq$ $R^{m*n}$ is convex.Prove that T={$x_1$ $\in$$R^{m}$ :($x_1$,$x_2$) $\in$ S } is convex. This is projection property of convex.Can somebody tell how to prove it.
3
votes
1answer
68 views

Hyperplane which can separate two closed convex set

Define: $$E=:\ell^1(R)=\{x=(x_n)_n: \|x\|_1=\sum_{n=1}^{\infty}|x_n|<\infty\}$$ We have two closed convex sets $X,Y$ as subsets of normed vector space $\ell^1(R)$ with $X\cap Y=\emptyset$ and $Y$ ...
2
votes
1answer
96 views

Relation between convex set and convex function

Let $E$ be an normed vector space and $A\subset E$ be a closed nonempty set. Define $$\phi(x)=\operatorname{dist}(x,A)=\inf_{a\in A}\|x-a\|$$ Prove that if $\phi$ is convex then $A$ is convex. ...
0
votes
1answer
52 views

when convex diverging functions are monotone when divided by $x$

here is a calculus question that someone asked me to help him wuth and I have no answer for him. any help or ideas? Given $f:(0,\infty) \rightarrow \mathbb{R}$ is a convex function, and ...
3
votes
2answers
71 views

Conjugate convex functions property

$E$ is normed vector space.Let $f\in E^*$ in a bounded linear functional from $E$ to $C$ and fix $x\in E$. We have $$\forall y\in E;\ \ \ \ f(y-x)\leq \frac{1}{2}\|y\|^2-\frac{1}{2}\|x\|^2$$ And I ...
3
votes
1answer
83 views

Does $\phi\circ g$ convex imply $\phi\circ f\circ g$ convex?

Assume that $\phi$ is an increasing function and $g$ is a decreasing function with $\phi\circ g$ convex. If $f$ is an increasing convex function does this imply $\phi\circ f\circ g$ is a convex ...
0
votes
1answer
44 views

Proving $x<y<z \implies \frac{f(y)-f(x)}{y-x} \le \frac{f(z)-f(y)}{z-y}$

Suppose $f$ is convex on $I$ and $(x,y,z)\in I^3$: How to prove that: $$x<y<z \implies \frac{f(y)-f(x)}{y-x} \le \frac{f(z)-f(y)}{z-y}$$
0
votes
1answer
90 views

Unlike Jensen's Inequality, can we upper bound $\log \sum_{i}{u_i \exp(x_i)}$?

According to Jensen's Inequality, it is not hard to derive the lower bound for $\log \sum_{i}{u_i \exp(x_i)}$ due to the convexity of $\log(\cdot)$: $\log \sum_{i}{u_i \exp(x_i)} \geq \sum_{i}{u_i ...
1
vote
0answers
43 views

Prove that $\{(x, y): x\in ri (dom f), y >f(x)\}\subset ri (epi f)$

$f: R^n \to (-\infty, \infty]$ is a convex function. Prove that $ri (epi f)=\{(x, y): x\in ri (dom f), y >f(x)\}$ I have used definition of convex functions to prove that the right-side is ...
1
vote
2answers
73 views

Is a continuously differentiable function convex if all its partial second derivatives are non-negative?

I'm having trouble understanding the relevant Wikipedia article which begins with a convex set $X$ and then uses functions of single variables for succeeding examples; the MathWorld article seems to ...
1
vote
1answer
88 views

Prove that $ri (epi f)=\{(x, y): x\in ri (dom f), y >f(x)\}$

$f: R^n \to (-\infty, \infty]$ is a convex function. Prove that $ri (epi f)=\{(x, y): x\in ri (dom f), y >f(x)\}$ I have used definition of convex functions to prove that the right-side is ...
2
votes
2answers
374 views

Show that $f(x)=||x||^p, p\ge 1$ is convex function on $\mathbb{R}^n$

Show that $f(x)=||x||^p, p\ge 1$ is convex function on $\mathbb{R}^n$. I have tried to use Holder's inequality, but I still cannot solve this problem. Could you help me with this problem? Thank you ...
7
votes
1answer
168 views

$\max\min$ problem

Suppose there is a real valued function $f(x,y)$ where $x,y$ are vector variables $x\in\mathbb{R}^n$ $y\in\mathbb{R}^m$. Moreover $f$ is strictly/concave in $x$ and strictly/convex in $y$. $f$ is also ...
3
votes
2answers
801 views

Second derivative positive $\implies$ convex

In proof of the following theorem; If $f$ has a second derivative that is non-negative (positive) over an interval then $f$ is convex (strictly convex). $f$ is in real number space., the book I ...
1
vote
0answers
61 views

function defined as an integral involving Bessel functions

i need to analyze a function of the form $$F(x,y) = \int_0^{1} e^{-(1+s)\alpha x}\sinh((1-s)\beta y) I_0(\sqrt{(x^2-y^2)s}) ds $$ Where $I_0$ is the modified Bessel function. $x>y$ always. ...
1
vote
1answer
99 views

Convex functions and Hahn-Banach application

Let $Z$ be a convex subset of a real vector space, and $f:Z \to \mathbb{R}^m$ be such that every component $f_i:Z \to \mathbb{R}$ is a convex function. Let $S:\mathbb{R}^m \to \mathbb{R}$ be defined ...
0
votes
2answers
107 views

Is the function of two strictly concave functions also concave?

This may be a trivial question to most, but here we go: I have two strictly concave functions, say $f(x)$ and $g(x)$. From this can I say that a function of those two functions, $h[f(x), g(x)]$, is ...
3
votes
0answers
82 views

Averages of a Function

Let $(X,M,\mu)$ a measure space and $f:X \rightarrow \mathbb{C}$ a function in $L^{\infty}(\mu)$. Define $A_f$ as the set of all averages \begin{equation} \frac{1}{\mu(E)} \int_{E} f d \mu ...
2
votes
1answer
107 views

equivalent condition for moment generating function

Consider a random variable $x$ with pdf $f(x)$, and have $x \ge 0$. The moment generating function is defined as $M(t)=\int^{\infty}_{-\infty}e^{-tx}f(x)dx$ (noted that we change the sign of $t$ ...
0
votes
1answer
95 views

convexity and lower semi-continuity for weak convergence

My question is a general one, whose answer can probably be found in any decent convex analysis book. I unfortunately don't have any at hand right now, so here it is: Let's consider a "reasonable" ...
3
votes
2answers
920 views

How to prove convexity?

Let us consider the function $$I(p):= \frac {\Gamma(2-p)\Gamma(3p)}{(p\Gamma(p))^2} $$ on the interval $(0,1),$ where $\Gamma(x)$ denotes the gamma function. How to prove its convexity there?
0
votes
1answer
104 views

How to prove the property of convex function in higher dimension

Suppose $f:\mathbb{R}^n\mapsto \mathbb{R}$ is convex, could anyone tell me how to prove the following fact? (1) If $f\in C^1$, then for any $u,v$ $$ f(v)\geqslant \langle\nabla f(u),v-u\rangle $$ ...
2
votes
3answers
226 views

A robust convex optimization problem

Consider a function $ f: \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R} $ such that $\forall x \in \mathbb{R}^n$ the map $f(x,\cdot)$ is convex, and $\forall y\in \mathbb{R}^m$ the map ...
1
vote
1answer
39 views

Robust feasibility with halfspace?

Consider a convex function $f: \mathbb{R}^n \rightarrow \mathbb{R}$, such that for all $x \in \mathbb{R}^n$ we have $$ a_1^\top x + b_1 \leq f(x) \leq a_2^\top x + b_2 $$ for some given $a_1, a_2 ...
6
votes
2answers
273 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 ...
0
votes
1answer
66 views

Pointed Convex cone: one-to-one correspondence extreme rays - extreme points

Hoi, let $V$ be a finite dimensional real vector space with inner product $\left\langle .\right\rangle$. Let $\Gamma\subset V$ and $\Gamma \neq \left\{0\right\}$ a pointed convex cone. (Pointed means ...
2
votes
1answer
257 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 ...
3
votes
1answer
93 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 ...
0
votes
1answer
28 views

Directions of decrease for a convex functions

Suppose $f(x,y)$ is a convex function and $$ f(x+\Delta x, y) < f(x,y), ~~~ f (x, y + \Delta y) < f(x,y)$$ Does this imply $$ f(x+\Delta x, y + \Delta y) < f(x,y)$$? I am guessing the ...
2
votes
1answer
134 views

Weak convexity and continuity

For any open interval $(a, b)\subset {\mathbb R}\,$, define a weakly convex function $f:(a, b) \rightarrow {\mathbb R}$ as one for which $$f(q\;x_0 + (1 - q)\;x_1) \leq q\;f(x_0) + (1-q)\;f(x_1)$$ ...