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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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Collecting terms in an example of checking concavity/convexity

$z=x_1^2+x_2^2$ $u=(u_1,u_2)$ $v=(v_1,v_2)$ Height of arc: $f[\theta u+(1-\theta )v]= f[\theta u_1+(1-\theta)v_1,\theta u_2 + (1-\theta)v_2 ]$ $= [\theta u_1+(1-\theta)v_1]^2 + [\theta u_2 + (1-\theta)v_2 ]^2$ Height of line segment: $\theta…
Jordi
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What is the solution for this optimization problem?

I have an optimization problem in the form: $$\max (a-\bar{a})(b-\bar{b}) \qquad \text{subject to} \qquad a+b=1.$$ Here $\bar{a}$ and $\bar{b}$ are known values and both of them are positive. Let $a_{opt}$ and $b_{opt}$ are the solutions or optimal…
Dimitrios
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Prove that there exists some $y \in \mathbb{R}^n$ in the boundary of $C$ satisfying $\{z \in \mathbb{R}^n : (x-y)^T(z-y) = 0\}$.

Let $C \subset R^n$ with $C \neq \emptyset$ be a closed convex set. Consider some $x \in \mathbb{R}^n$ satisfying $x \notin C$. Prove that there exists some $y \in \mathbb{R}^n$ in the boundary of $C$ which implies the existence of a nonempty…
David Smith
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Prove that $A - B$ is closed if $A$ or $B$ is bounded (or both)

Suppose that $A$ and $B$ are two nonempty convex closed sets in $\mathbb{R}^n$, with $A \cap B = \emptyset$. Further, define $A - B = \{a - b \space | \space a\in A, b \in B\}$. Prove that $A - B$ is closed if $A$ or $B$ is bounded (or both). I've…
David Smith
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Unit ball with p norm in $\mathbb{R}^3$ space

I know unit ball for $p$-norm with $p = 2$ is a square, my confusion is how does it look like in $\mathbb{R}^3$ space. In $\mathbb{R}^3$ space it looks like a cuboid, is this correct ?
Nithish
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How would you show that this fraction function is convex and decreasing?

Show that $$ f(\vec{x}) = \frac{1}{x_1 - \frac{1}{x_2 - \frac{1}{x_3 - \frac{1}{x_4}}}} $$ is convex when all denominators are greater than $0$.
user90593
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Is the following fractional function convex?

Is $\displaystyle f(x_1,x_2) = x_1 - \frac{1}{x_2}$ a convex function? What if we restrict the values of this function to the positive reals?
user90593
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Stronger Condition for Strict Convexity?

As I understand, a strictly convex function $f: D \rightarrow \mathbb{R}$ is one that satisfies the property: $\forall x, y \in D, x \neq y, \forall t \in (0,1), f((1-t)x+ty)<(1-t)f(x)+tf(y)$. A mechanical interpretation of this: a convex function…
Tanner
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Properly or strictly separated sets

Let $A=\{ x,y,z: x,y,z\in[0,1] \}$ and $B=\{(x-2)^{2}+(y-2)^{2}+(z-2)^{2}\le 1\}$. Show if the sets $A$ and $B$ can be properly or strictly separated. Does anyone know the solution of this problem?
Laura
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Equivalent characterization of quasi-concavity

For $f: \mathbb R^n \to \mathbb R$ prove that the two statements are equal: For all $x,y \in \mathbb R^n$ and for all $t \in [0,1]$, $f(tx+(1−t)y)\geq \min (f(x),f(y))$ For all $k \in \mathbb R$, $\{ x : f(x) \geq k\}$ is a convex set.
Eddie Yu
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Is $S = \{(x,y) : y = e^x\}$ is convex or not?

If $S = \{(x,y) : y = e^x\}$ was convex, the following relation holds \begin{align} &t y_1 + (1-t) y_2 = e^{t x_1 + (1-t) x_2} \tag{1}\\[2mm] \Longleftrightarrow \quad & t e^{x_1} + (1-t) e^{x_2} - e^{t x_1 + (1-t) x_2} = 0 \tag{2} \end{align} for…
clueless
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Problem with showing convexity of a function

I want to show that $f_n(\zeta) = \frac{1}{n} \log \sum_{w \in W_n} e^{\zeta K_n(w)} P_n (w)$ , with $\zeta \in \mathbb{R}$ is convex. I will not explain what $W_n, K_n$ and $P_n$ are, because this is not necessary according to the question I…
clubkli
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Examples of $C^1$ differentiable convex functions.

Could you please provide examples of convex functions that are differentiable, but their derivatives are not differentiable.
ashim
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(quasi) convexity $\frac{x}{y}$

Hej, I have the function $\frac{x}{y}$ on the domain $R_{++}$. The Hessian matrix is - as I have calculated it - positive semidefinite. But I'm not really sure, if the function is really convex at all on the domain. Thanks for any help.
Masala
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Is the second derivative test for the concavity of variance inconclusive?

For example, we know that $var(x) = \sum\limits_i^n (p_ia_i)^2 - (\sum\limits_i^n (p_ia_i))^2$ Then $\dfrac{\partial Var(x)}{\partial p_k} = \sum\limits_i^n (2p_ia_i) a_k - 2\sum\limits_i^n (p_ia_i)a_k = 0$ But we know that Variance is a concave…
Olórin
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