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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
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Maximum of a convex function over a polyhedron

Show that the maximum of a convex function $f$ over the polyhedron $\mathcal{P} = \textbf{conv}\{v_1,\dots,v_k\}$ is achieved at one of its vertices, i.e., $ \sup_{x \in \mathcal{P}} f(x) = \max_{i=1,\dots,k} f(v_i)$. I am trying to prove this…
HatCucumber
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Adjoint of nondifferentiable convex function is convex or not.

Let f be a real-valued function over $ \Omega=(0,+\infty)$. Firstly, I assume that f is twice differentiable $(*)$. Under the condition $(*)$ it's easy to see the following proposition $\textbf{Proposition}$ If f is convex on $\Omega$ then the…
No name
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Convex set is closed

I have some points $x_1,\dots,x_n$ in $\mathbb{R}^j$. I define the set $C(x_1,\dots,x_n)= \left\{\sum_{i=1}^{n} \lambda_i x_i : \lambda_1+\dots+ \lambda_n = 1, \lambda_i \ge 0 \right\}$. The closure of a set is given by all its cluster points. We…
user397268
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Is it true that convex function on bounded subset of $R^n$ is bounded below?

If $\Omega$ is a bounded convex subset of $\mathbb{R}^n$ and $f$ is a convex function on $\Omega$, can we say $f$ is lower bounded on $\Omega$? If it is not true, can any one provided a counter example?
Lin Xuelei
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If $f(x,y)$ is convex in $x$ (for any fixed $y$) and convex in $y$ (for any fixed $x$), is it convex?

Let $f(x,y)$ be a function $X\times Y\rightarrow\mathbb R$, where $X\subset \mathbb R^n$ and $Y\subset \mathbb R^m$, and $x\in X,y\in Y$. Suppose that $f(x,y)$ is convex in $x$ for any fixed $y$, and convex in $y$ for any fixed $x$. Does it follow…
a06e
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Is $f(x,y)=y\left(2^{\frac{x}{y}}-1\right)$ a strictly convex function when $x\ge0,y\ge0$?

Is function $$f(x,y) = y\left(2^{\frac{x}{y}}-1\right)$$ strictly convex when $x\ge0,y\ge0$? I can show its Hessian matrix is positive semidefinite, but it is only a sufficient condition for the strictly convex. Any help is appreciated.
Dave
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what happens for points far away from sublevel sets of a proper convex function

I met problems about sublevel sets of proper convex functions. Suppose $f: R^n \Rightarrow \overline{R}$ is a proper convex function, and $\alpha > \inf f$. Denote $S_{\alpha}$ the $\alpha$-sublevel set of $f$, i.e., $S_{\alpha} = \{ x: f(x) \le…
Huan Zhang
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Rockafellar's Theorem 20.1

I am trying to understand the proof of Theorem 20.1 from Rockafellar's classic book "Convex Analysis". My issue is the argument: ..., and hence $$\text{dom(}g_1)\cap\text{ri(dom}(g_2))\not=\varnothing.$$ This implies that, for the affine hull…
Ariel Serranoni
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$f(tz)\rightarrow \infty$ as $t\rightarrow \infty$ when $f$ is convex

Suppose $f:\mathbb R ^n \rightarrow \mathbb R$ is convex and not constant. How can I show that $f(tz)\rightarrow \infty$ as $t\rightarrow \infty$ for some $z\in \mathbb R ^n$? This seems very intuitive because $f$ is defined on the entire space,…
MasterJ
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Let $ \ A \ $ be a positive definite matrix . Prove that $ \ \frac{1}{2} \left\langle Ax-b \ , \ A^{-1}(Ax-b)\right\rangle \ $ is strictly convex

Let $ \ A \ $ be a positive definite matrix . Prove that $ \ \frac{1}{2} \left\langle Ax-b \ , \ A^{-1}(Ax-b)\right\rangle \ $ is strictly convex, where $ \ \left\langle u,v \right\rangle \ $ means dot product between vectors $ \ u \ $ and $…
MAS
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Is this $f(x)$ convex?

I want to check if the function $f(x)$ is convex, where $$ f(x)=\bigl|a|x|-x\bigr|^2.$$ There are several possibilities to check if the function is convex: The second derivative: Not possible, because it is only subdifferential. Geometry: too…
jester
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What's the function $n_{\chi[0,1/n]}$?

On one paper I saw the function: $$f_n=n_{\chi[0,1/n]}$$ What is this function? I read that it's $n$ multiplied by the characteristic function on set $[0,1/n]$. But since $f_n$ in this case is supposed to defined for $f \in L^1([0,1])$ s.t.…
mavavilj
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Convexity of composition: log and a sum of reciprocals

Any suggestions on proving the convexity of $f (\boldsymbol{x}) = \log \left ( 1 + \left ( \sum_i \frac {1}{\sqrt{x_i}} \right )^2 \right )$ where $\boldsymbol{x} = \left [x_1~x_2~\dots~x_L \right ]$ and $x_i > 0, \forall i$? A straightforward…
Zelig
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Proof of convexity from definition ($x^Tx$)

I have to prove that function $f(x) = x^Tx, x \in R^n$ is convex from definition. Definition: Function $f: R^n \rightarrow R$ is convex over set $X \subseteq dom(f)$ if $X$ is convex and the following holds: $x,y \in X, 0 \leq \alpha \leq 1…
Smajl
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Convexity of certain set

I need to show that $S = \{(x,y) \in \mathbb{R}^2 : y \geqslant x^2\}$ is a convex set, but I'm having a bit of trouble. Simply applying the definition of convexity has got me nowhere.
johnny
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