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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.

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maximum and minimum singular values

I was studying the Stephen Boyd's textbook on convex optimization and have a question. The book says the following: The singular values of $A$, $\sigma_1 \geq \sigma_2 \geq ... \geq \sigma_n$ are the square roots of the eigenvalues of $\lambda_1…
DSKim
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Is there a notion of 'pre-convex' sets?

A set $A$ in $\mathbb{R}^n$ is convex if for any two points $p,q \in A$ and real $\lambda \in [0,1]$, the point $\lambda p + (1-\lambda)q$ is also in $A$. There are many beautiful theorems about convex sets, perhaps the most important of which is:…
Ben McMillan
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Convex sets proof

Prove the following theorem: Let $V$ be a linear space and $D$ a convex set. Let $x_1,\ldots,x_k$ be $k$ points in $D$. Let $a_1,\ldots,a_k$ be non-negative scalars such that $\sum\limits_{i=1}^n a_i=1$. Then the so called convex combination…
dreamer
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How to prove that if convex $A \subset \mathbb{R}^n$ and $A+A=A$ then $0 \in cl(A)$?

How to prove that if convex $A \subset \mathbb{R}^n$ and $A + A = A$ then $0 \in cl(A)$? For a example of $A$ which holds a condition but $0 \notin A$ consider $(0,\infty)$.
Ashot
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Continuous boundary of a convex set

Is the boundary of a compact convex set in Rn continuous? Seems like the answer should obviously be yes, but I cannot find any such result in the literature. Can somebody provide a reference (or a proof)? Thanks.
user59820
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Boyd & Vandenberghe, question 2.31(d) — stuck on simple problem regarding interior of a dual cone

Crossposted at MathOverflow In Boyd & Vandenberghe's Convex Optimization, question 2.31(d) asks to prove that the interior of the dual cone $K^*$ is equal to $$\text{int } K^* = \{ z \mid z^\top x > 0, \,\forall x \in K \} \tag{1}$$ Recall that the…
ted
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Hausdorff distance via support function

Let $A$ and $B$ be convex compact sets in $\mathbb{R}^n$. Define $$ h_{+}(A,B) = \inf \left\{ \varepsilon > 0 \mid A \subseteq B+\mathbb{B}_{\varepsilon} \right\} $$ where $\mathbb{B}_{\varepsilon}$ is an $\varepsilon$-ball centered at origin.…
Appliqué
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Proving $\{(x_1, x_2) \in \mathbb{R}^2_{+} \mid x_1 x_2 \geq 1\}$ is convex

This is so called a hyperbolic set $$\{(x_1, x_2) \in \mathbb{R}^2_{+} \mid x_1 x_2 \geq 1\}$$ We proceed to prove that it is convex by showing that a convex combination of points (a line segment) will lie in the set Suppose $x = (x_1, x_2)$, $y =…
Fraïssé
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If points in a convex set $C$ escape to infinity roughly in direction $v$ then an infinite ray in that direction exists

Let $C\subseteq \mathbb R^n$ a convex set. Assume there is a sequence $\{c_k\}_{k\in\mathbb N}$ with $c_k\in C$, $|c_k|\to\infty$ such that $v:=\lim \frac 1{|c_k|}c_k$ exists. Does this imply that there exists $a\in C$ with $a+[0,\infty)\cdot…
Hagen von Eitzen
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If $(\nabla f(x)-\nabla f(y))\cdot(x-y)\geq m(x-y)\cdot(x-y)$, why is $f$ convex?

I was reading on wikipedia that a strongly convex function is also strictly convex. I say that a function $f\colon\mathbb{R}^n\to\mathbb{R}$ is convex if $$ f(\lambda x+(1-\lambda)y)\leq\lambda f(x)+(1-\lambda)f(y) $$ for all $x,y\in\mathbb{R}^n$,…
YN Chew
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Prove that the square distance to convex set is a convex function.

I find the following link Gradient of squared distance to a convex set. But I don't understand how prove that the function is convex. My attempt was use the $$\langle f^{\prime}(y)-f^{\prime}(x), y-x\rangle \geq 0$$ and try use that $\|,\|^{2}$ is…
Zacarias89.
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How can I prove that $f(x) = \left(1 + \frac1x \right)^x$ is concave in $\mathbb{R}^+$?

$$f(x) = \left(1 + \frac1x \right)^x$$ In general, attempting to demonstrate that $$f(ax + (1-a)y) \ge af(x) + (1-a)f(y)$$ leads to algebraically difficult expressions with no clear resolution. I am aware that we can attempt to show $f''(x) \le 0$.…
Marcus Emilsson
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Convexity-preserving approximation methods

Are there well-known or somewhere-well-studied methods of interpolation that will produce convex approximants when fed data that could conceivably fit a convex function? A lot of stuff (e.g. Lagrange-style polynomial interpolation, piecewise-smooth…
leslie townes
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How can I prove that a function $f(x,y)= \frac{x^2}{y}$ is convex for $y \gt 0$?

How can I prove that a function $f(x,y)= \frac{x^2}{y}$ is convex for $ y \gt 0$? I take the Hessian matrix of $\displaystyle \frac{x^2}{y}$, and I got: $$H = \displaystyle\pmatrix{\frac{2}{y} & -\frac{2x}{y^2} \\-\frac{2x}{y^2} &…
Cheung
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Is the function $f(x) = |x|$ convex?

I am asking because some Internet pages contradict. In Wikipedia, the definition I found of a convex function is: "Let $X$ be a convex set in a real vector space and let $f: X \to R$ be a function. • $f$ is called convex if: For all $x_1\ne x_2 \in…
Ronald Becerra
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