1
vote
0answers
27 views

Legendre transform concave function

Let $f$ be a concave function and define $f^*(y) := \inf_{x}(yx-f(x))$. Is this in any sense related to the Legendre transformation? -If yes, is $f^*$ also concave? Is this transformation invertible ...
1
vote
1answer
32 views

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 ...
0
votes
0answers
19 views

relative interior and the affine map

In the convex analysis book by Hiriart-Urruty &Lemarechal, Proposition 2.1.12 states $ri [A(C)] = A(ri C)$. Where $ri$ is the relative interior and $A: \mathbb{R}^n \to \mathbb{R}^m $ is an ...
1
vote
1answer
49 views

An exercise on convex decreasing function properties

A function f$(x)$ defined for $x\geq0$. It is positive, decreasing, convex and log-convex: $\frac{d^2}{dx^2}\log[f(x)]>0$, $f(0)<1$. Can we prove that $f''(x)x+f'(x)>0$ for sufficiently ...
0
votes
0answers
40 views

$f:D\subset \Bbb R^2 \rightarrow \Bbb R$, where $D$ is a compact and convex set, reaches it maximum at $int(D)$

I'm trying to prove that if $D$ is a compact and convex (for every two elements of $D$, the line that connects them is contained in $D$) then: If $f:D\subset \Bbb R^2 \rightarrow \Bbb R$ and at ...
0
votes
1answer
37 views

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 ...
0
votes
1answer
40 views

Prove that convex function on $[a,b]$ is absolutely continuous

In the book Roberts, Varberg, Convex functions, on pp.9-10 is proved that If $f:(a,b)\rightarrow \mathbb R$ is continuous and convex then $f$ is absolutely continuous on each $[c,d]\subset ...
2
votes
1answer
27 views

A convex function is differentiable at all but countably many points

Let $f:\Bbb R\to\Bbb R$ be a convex function. Then $f$ is differentiable at all but countably many points. It is clear that a convex function can be non-differentiable at countably many points, ...
10
votes
0answers
258 views

Convexity of Matrix Exponential

Consider the function $A: \mathbb{R}^n \rightarrow \mathbb{R}^{n \times n}$ defined as $$ A( x ) := \left[ \begin{matrix} x_1 & a_{1,2} & \cdots & a_{1,n} \\ a_{2,1} & x_2 & ...
1
vote
2answers
88 views

Is every convex function on an open interval continuous?

Let $f:(a,b)\rightarrow \mathbb{R}$. $f$ satisfied the following property: If $\forall x_{1},x_{0},x_{2}\in(a,b)$ and $x_{1}<x_{0}<x_{2};$then$\frac{f(x_{0})-f(x_{1})}{x_{0}-x_{1}}\geq ...
0
votes
0answers
21 views

A Difficult combinatorial optimization problem

Let $\mathcal{J}$ be a closed, bounded, compact, convex set in $\mathbb{R}^L$. (Notations: vector $\mathbf{x}$ is denoted in bold letters and its $i^{th}$ co-ordinate is denoted as $x_i$. ...
0
votes
0answers
14 views

Eliminating variables in convex program

This is a basic convex optimisation question. I have the following problem: $$\max_{\substack{t\le e\\ At\le b}} e^\top t$$ How do I find the optimum $t^*$? I write the KKT conditions, get ...
7
votes
1answer
156 views

Characterization convex function.

Let be $f:[0,1]\rightarrow\mathbb{R}$ a continuous function such that far all $a,b\in [0,1]$ with $a<b$ $$f\left(\frac{a+b}{2}\right)\leq\frac{1}{b-a}\int_a^b f(x)\,dx.$$ How to prove that $f$ is ...
0
votes
1answer
17 views

Questions on conditions for convexity of a real function

We have a function $f:[0,1] \rightarrow \mathbb{R}$. We know it is continuous on $[0,1]$. Aside from a set $S$ of measure $0$, we can compute its right derivative, and can show that this right ...
0
votes
1answer
32 views

what is the dual of the following linear program over a convex set?

Let $\mathbf{x}=[x_0,x_1,\dots,x_N]^T$ be a $(N+1)\times 1$vector. Let $\mathcal{S}$ be a bounded, compact convex set in strictly positive quadrant of $\mathbb{R}^{N+1}$. Consider the following ...
2
votes
0answers
25 views

Convex interpolation between two points with given derivatives

Let's say I have two real values $x_1$ and $x_2$, to each of which I associate $y_i$ and $y'_i$ satisfying $$ (y_2-y_1)(y'_2 - y'_1) \geq 0. \tag{1} $$ I would like to find a polynomial ...
0
votes
0answers
15 views

What is the nature of this one dimensional function?

Let $\mathcal{S}$ be a 2-D convex set whose elements can be represented as $(x,y)\in\mathcal{S}$. Let $p_L$ and $p_U$ be two real constants such that $p_L\leq p_U$. For $p\in[p_L,p_U]$, I define the ...
1
vote
1answer
34 views

Convolution of Strictly Convex Function

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a $C^2$, strictly convex function, and $\theta_\epsilon$ the standard approximation to identity ...
0
votes
1answer
28 views

Subtracting a constant from log-concave function preserves log-concavity, if the difference is positive

I am trying to work out a question from 'Convex Optimization - Boyd' . Specifically, exercise 3.48: Show that if $f : \mathbb R^n \to \mathbb R$ is log-concave and $a > 0$, then the function $g ...
2
votes
1answer
36 views

Affine to linear like conversion of a concave function

Is the following true: $$\log \left( \frac{1}{f(x)+K}\right)\mathrm{is\;concave}\Longleftrightarrow \log \left( \frac{1}{f(x)}\right)\mathrm{is\;concave},$$ where $K\in\mathbb{R} $ and ...
2
votes
1answer
63 views

Point-wise converging convex functions on $[0,1]$

Suppose we have a sequence of continuous convex functions $\{f_n\}$ defined on $[0,1]$ which converge point-wise to a limit $f$ on $[0,1]$, i.e. for all $x \in [0,1]$ $$\lim_n f_n(x) = f(x).$$ Let $G ...
4
votes
3answers
110 views

Is every such function convex or concave?

I was pondering convex functions today, and the following questions naturally posed themselves. Call a function $f : \mathbb{R} \rightarrow \mathbb{R}$ $2$-limited iff for all $m,c \in \mathbb{R},$ ...
0
votes
1answer
41 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 $x_0\in \mathbb{R}$: ...
0
votes
0answers
43 views

separating hyperplane theorem proof

I need help understanding the last part of this proof, the lemma they are refering too is just a lemma about convex sets that shows us that p is unique: I do not see how H* separates C and z. The ...
1
vote
0answers
22 views

Show P is linear if it is convex and positive homogenious functional

it might be too simple but I couldnt show the second part L linear real space $P:L\rightarrow \Bbb R$ is called positive homogenious functional if for every $x\in L$ and $\alpha\ge 0$ , $P(\alpha ...
3
votes
0answers
92 views

Non-trivial compatibility which makes convex functions continuous on $\Bbb R$

Here are the definitions: Let $X$ be a set. Another set $\mathcal C\subseteq \mathcal P(X)$ is called a convexity over $X$ if $\varnothing, X\in\mathcal C$ $\mathcal C$ is closed under arbitrary ...
0
votes
1answer
49 views

Proof of the Hahn-Banach separation theorem

In the proof of the Hahn-Banach separation theorem my notes claim the following: Let $X$ be a normed $\mathbf{R}$ vector space, $A,B\subset X$ be nonempty, disjoint, convex, $A$ compact and $B$ ...
1
vote
1answer
59 views

New variable in a convex optimization problem

Consider the convex optimization program $$ \min_{x \in X } x^\top P x + p^\top x \quad \text{ sub. to: } Ax = b $$ where $X \subset \mathbb{R}^n$ is compact, $P \succ 0$, $A \in \mathbb{R}^{m \times ...
0
votes
1answer
25 views

On level set of concave function

The problem is to show the following: Let $\varphi$ be a closed concave function, and $M=\max_{x \in \mathbb{R}^d} \varphi(x)$. Let $D_r:=\{\varphi\geq r\}$ be the level set. Then given $r \leq s ...
3
votes
1answer
67 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
1answer
23 views

Question about relative interiors and convexity

Suppose that $C\subseteq \mathbb{R}^n$, such that $\operatorname{ri} C\neq \emptyset$ is convex and $\operatorname{cl} C$ is convex. Can we show that ...
4
votes
1answer
56 views

Let $S$ be a closed convex set & $x$ be an extreme pt of $S$ then $S-\{x\}$ is

Let $S$ be a closed convex set and $x$ be an extreme point of $S$, then $S-\{x\}$ is Convex Not Convex May or may not be convex I am thinking that the convexity doesn't fail even if we remove the ...
0
votes
2answers
26 views

Difference quotient inequality with convex functions

A convex function on an interval $ I $ is said to be convex if for every $ 0 < t < 1 $ and $ x,y \in I $ we have that $ f(tx + (1-t)y) \le tf(x) + (1-t)f(y)$. Prove a function is convex if and ...
1
vote
1answer
33 views

Supremum of convex lipschitz functions.

Let $f_i:K\to R, i\in I$ be a family of convex, equi-Lipschitz functions on some compact subset $K$ of $\Bbb R^n$. Is it true that $\sup f_i$ is also Lipschitz continuous(assuming that the sup ...
2
votes
0answers
25 views

Convexity in each argument and directional derivative

Let $f(x,y)$ be a continuous function, convex in each argument separately. Does this imply the existence of one-sided directional derivatives in any direction? For example, does there exist (finite or ...
0
votes
0answers
38 views

Sufficient conditions for convexity using the right derivative

We have a function $f:[0,1] \rightarrow \mathbb{R}$ that is continuous on $[0,1]$ with a non-decreasing right derivative everywhere in $(0,1)$. Is this definitely sufficient to show that $f$ is convex ...
0
votes
0answers
26 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
1answer
46 views

A question about norm for bounded linear transformations

Let $H$, $K$ be Banach spaces, and let $A: H \rightarrow K$ be a bounded linear transformation. Its norm is defined by: \begin{equation} \|A\| = sup\{\|Ah\|_K: \|h\|_H \le 1\} \end{equation} How to ...
2
votes
0answers
123 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
22 views

Robust LP question using box uncertainty model

I am trying to solve this robust LP problem by writing it as a QP $$\min_x x^TSx : \mu \leq r^T x , Ax \leq b$$ Under Box uncertainty model: $$R = \{r : \| r - \hat{r}\|_\infty \leq \rho\}$$ Here ...
1
vote
1answer
34 views

Proving upperbound using convexity

The original question is to prove $$\frac{1}{n}\sum_{i=1}^n x_i \leq \log{(\frac{1}{n}\sum_{i=1}^n e^{x_i})} \leq \max_{1 \leq i \leq} x_i$$ I show that $$x_{max} = \max_{1 \leq i \leq} x_i$$ ...
1
vote
1answer
22 views

Convexity and Jensen's Inequality for simple functions

Suppose $\varphi$ is convex on $(a,b)$. I want to show that for any $n$ points $x_1,\dots,x_n \in (a,b)$ and nonnegative numbers $\theta_1,\dots,\theta_n$ such that $\sum_{k=1}^n \theta_k = 1$ we are ...
0
votes
1answer
25 views

Decreasing Function Projected onto Simplex

Consider $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$, defined as $f(x) := a x + b$, where $a<0$ and $b \in \mathbb{R}_{\leq 0}^{n}$. Note that $f$ is decreasing: $$ x \geq y \Longrightarrow f(x) ...
1
vote
2answers
71 views

Minimization of log-sum-exponential function subject to constraints.

I would like to minimize the following function: $f(x)=log(e^{-x_1}+..+e^{-x_n})$ Subject to: $\sum_{i=1}^{n}{x_i}=1$ $0 \leq x_i \leq 1$ So far I have discovered the following: If all the ...
0
votes
1answer
31 views

How to prove that $f$ is convex function

Let $f:(a,b) \mapsto \mathbb{R}$ be continuous function such that $f(\frac{x+y}{2})\leq \frac{1}{2}f(x) + \frac{1}{2}f(y) \;\; \forall x,y \in (a,b)$ Show that $f$ is convex function. Please give ...
0
votes
2answers
42 views

Notion of a concave function and proving ln is concave

I've just checked that the definition is right, a function is convex if: $(1-t)f(x_1)+tf(x_2)\ge f((1-t)x_1+tx_2)$ which is odd because this is ... well I was taught (very young age) that concave ...
0
votes
0answers
44 views

Minimization of product function subject to constraints

I want to minimize the following function: $\prod_{i=1}^{n}{x_i}$ Subject to the following constraints: $\sum_{i=1}^{n}{x_i}=1.1+(n-1)(0.1)$ and $0.1 \leq x_i \leq 1.1$ How should I go about it? ...
1
vote
1answer
41 views

Show that for $f,g: [0, + \infty) \to \mathbb{R}$ convex functions of Class $C^2$ their product $fg$ is convex

Problem: Let $f,g: [0, + \infty[ \to \mathbb{R}$ be two convex functions of Class $C^2$. Assume that $$ f(0) \geq 0, g(0) \geq 0 \text{ and } f'(0) \geq 0, g'(0) \geq 0 \tag{!}$$ Show that $fg$ ...
1
vote
1answer
42 views

convex closed and unclosed functions and (lower semi)continuity

I'm grappling a bit with lower semicontinuity and convex functions. Let me consider convex functions as functions to $\mathbb{R}$ and not to $\mathbb{R}\cup \{\pm\infty\}$. By Rockafellar's book ...
1
vote
1answer
30 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 ...