Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [convex-analysis]

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.

Convex analysis is the study of properties of convex sets and convex functions.

Formally, a set $S$ is convex if, for all $x, y \in S$ and $t \in [0,1]$, $tx + (1-t)y \in S$. Intuitively, this says that for any two points in $S$, the line segment connecting those points is also in $S$.

Formally, a function $f: S \to \mathbb{R}$ defined on a convex set $S$ is convex, if for all $x,y \in S$ and $t \in [0,1]$, $f(tx + (1-t)y) \leq t f(x) + (1-t) f(y)$. Intuitively, this says that, for any two points on the graph of $f$, the line segment connecting those points lies above the graph of $f$.

Some of the more important results in convex analysis include Carathéodory's Theorem, Jensen's Inequality, Minkowski's Theorem, and the supporting hyperplane theorem.

9641 questions
3
votes
3 answers

Show that the set of all minimum points of a convex function over a convex set is convex

I am given the following problem, but can't figure it out. Let $f\colon\mathcal{C}\rightarrow\mathbb{R}$ denote a convex function defined on the convex set $\mathcal{C}$. A (global) minimum of $f$ is an $x^*\in\mathcal{C}$ with $f(x^*)\leq f(x)$ for…
user1058
3
votes
1 answer

Is this function Strictly convex or not?

We have a function $f(u)= u^{T}N^TNu$ where $u$ is a $n$-dimensional vector and $N$ is a $n\times n$ matrix. Is this a strictly convex function in $u$? I know that if the hessian of $f(u)$ with respect to $u$, which is $N^TN$, is a positive…
m0_as
  • 1,075
3
votes
1 answer

A question about Caratheodory's Theorem of Convex Sets

As I understand it, Caratheodory's Theorem of Convex sets essentially states If $Q$ is a set in a vector space of dimension $n$ and x lies in the convex hull of $Q$, then x can be written as a convex combination of no more than $n+1$ points in $Q$.…
PurdueCHE
  • 41
3
votes
2 answers

Supporting hyperplane of a convex set

Let $\Omega$ be a bounded convex set in $\mathbb{R}^n$, and let $\partial \Omega$ denote its boundary. Fix a point $p$ in $\Omega$, and let $c$ denote the point on $\partial \Omega$ that is closest to $p$. Then, intuitively it seems that a…
user74261
  • 2,286
3
votes
2 answers

Convex combination in compact convex sets.

Let's $K\subset\mathbb{R}^n$ compact. If $K$ is convex then who to prove that any point of $K$ is convex combination of one or two extremal points of $K$? Intuitively, for any closed ball that is true.
Elias Costa
  • 14,658
3
votes
1 answer

Upper and lower bound on Hessian

Let $x \mapsto f(x) \in \mathcal{C}^2$ be convex, i.e. $\forall x \in \mathbb{R}^n$, $\nabla^2f(x) \succeq 0$. Let $A \in \mathbb{R}^{m \times n}$ and suppose we have $M I_n \succeq \nabla^2f(x) + A^\top A \succeq m I_n$, where $M \geq m > 0$. Is it…
User129599
  • 303
3
votes
1 answer

Is exponential function strongly convex?

Assume $x \in \mathbb{R}$. In the wiki page, one property of strongly convex functions $f(x)$ is that it satisfies: $f''(x)\geq m > 0~\forall x$ with with parameter $m > 0$. Given $f(x) =e^x$, since $lim_{x\to -\infty} f''(x) = 0$ does this mean…
Adam I.
  • 359
3
votes
2 answers

lipschitz property of the derivative of a convex function

Let $f\in C^1(\mathbb R^n\to \mathbb R)$ be a convex function. Suppose the equation $$f(x+\Delta x)-f(x)-\langle f'(x),\Delta x\rangle\leq A|\Delta x|^2$$ holds for some constant $A>0$, any $x\in \mathbb R^n$ and any sufficiently small $\Delta x\in…
Lion
  • 137
3
votes
1 answer

Property of log-concave function

In S.Boyd's lecture: And in his vedio, he said: You are allowed one positive eigenvalue in the Hessian of log-concave function. http://web.stanford.edu/class/ee364a/videos/video04.html (at 1:03:30) I am confused about this? Why it is true?…
sleeve chen
  • 8,281
3
votes
1 answer

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 convex if A $\succeq$ 0. For the first part we use the…
Thoth
  • 863
3
votes
3 answers

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 stated as an implication reads as follows: If Convex…
Charlie Parker
  • 6,546
2
votes
0 answers

tangent cone of a projection

I am stuck with what looks like a basic problem from convex analysis and I don't seem to have the answer. Any pointers or insights would be of great help. Suppose $K$ is a closed convex set in $\mathbb{R}^n$ and $R \in \mathbb{R}^{m \times n}$ is a…
Ankur
  • 409
2
votes
3 answers

Why doesn't the definition of the interior of a set depend on the dimension of the set

I have just started with a course on convex optimization and have been introduced to the concept of the interior of a set. I have a fairly basic question. I am still trying to understand this topic, so please forgive me if this is a stupid or…
Karthik Upadhya
  • 734
  • 7
  • 15
2
votes
3 answers

Quasi-Concavity and Quasi-Convexity

My book states that: $f$ is a quasiconcave function on $U$ if for all $x,y \in U $ and $t \in [0,1]$: $f(x) \geq f(y) \implies f(tx + (1 - t)y) \geq f(y)$ $f$ is a quasiconvex function on $U$ if for all $x,y \in U $ and $t \in [0,1]$: $f(y)…
Peter Pan
  • 31
2
votes
0 answers

Extending the notion of convex hull in $\Bbb R^n$

The super convex hull of a set $A \subseteq \Bbb R^n$, is the set of all $\sum_{i=1}^{\infty}\lambda_i x_i$ such that $\lambda_i \geq 0$ and $\sum_{i=1}^{\infty}\lambda_i =1$, which is denoted by cco(A). It is easily verified that $co(A) \subseteq…
Arman
  • 821
  • 4
  • 13
1 2 3
…
69 70