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.

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The generalized Pythagorean property of Bregman divergence

So suppose a function $f$ is strictly convex and continuously differentiable. The Bregman divergence associated with $f$ is \begin{equation} D_f(x,y) = f(x) - f(y) - \nabla f(y)^{T}(x-y), \forall x,y\...
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Minimization of $\log \det$ plus $\| \cdot \|_1$

Given fat matrices ${\bf A} \in \mathbb{R}^{m\times n}$ (where $m < n$) and ${\bf B}\in \mathbb{R}^{p\times n}$ (where $p < n$ and $\mbox{rank}({\bf B}) = p$), and $m \times m$ symmetric ...
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A condition for global convexity

Let us consider four real numbers $-\infty<a < b < c < d<\infty$. Let us suppose that a function $f:(a,d) \to \mathbb R$ is convex on $(a,c)$ and $(b,d)$. Is it true that $f$ is convex ...
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If $p$ is a real parameter, is $\min \{c^T x|Ax≤pb+d\}$ a concave function of $p$? is $\max \{c^T x|Ax≤pb+d\}$ a convex function of $p$? [closed]

Title says it all. I suspect they are both true, but I'm unsure how to prove it.
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Can we tell if ANY Function is Convex or Non-Convex?

Reading the mathematical definition of convexity (https://en.wikipedia.org/wiki/Convex_function), it seems that there is a relatively clear definition as to what makes a function "convex": ...
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Real functions with the property: $\ f(x_1)f(x_2) = f\left( \frac{x_1+x_2}{2} \right)^2$ for all $\ x_1,\ x_2\in\mathbb{R}.\$

Suppose $\ f:\mathbb{R}\to\mathbb{R}\$ has the property:$\ f(x_1)f(x_2) = f\left( \frac{x_1+x_2}{2} \right)^2\$ for all $\ x_1,\ x_2\in\mathbb{R}$. I made some educated guesses and stumbled upon the ...
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Do we have any way of knowing if natural phenomena in the real world follow the "Lipschitz Condition"?

Recently, I keep coming across terms containing "Lipschitz" pertaining to statistical models and machine learning. This includes terms such as "p-lipschitz (rho), lipschitz convexity, ...
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Strictly convex function and second derivative [closed]

Let $f(x)=x^2+e^x$ since $f''(x)=2+e^x>0$ always, can I say that the function is strictly convex?
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Proving convexity/concavity of a function

The given function is $$f(P) = y^T(APA^T)^{-1}y$$ It is given that $P$ is a diagonal matrix made of elements of the vector $p$, and the elements of $P$ are strictly positive. My attempt : Since $P$ is ...
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How to prove that Squared Error Loss is convex

I have the squared loss given by: $L(z;\textbf{y}) = \frac{1}{2}\left\| z-\textbf{y} \right\|^{2}_{2} = \frac{1}{2}(z-\textbf{y})^{T}(z-\textbf{y})$ where $z=\textbf{W}^{T}\phi(x)+\textbf{b}$. I need ...
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The monotonicity or convexity of an ODE regard to its coefficient

I am trying get some ideas on how to prove that the solution of the following ODE is monotone or convex in the constant $k$: $$f(x)-x(f'(x))^2+k(x-x^2)f'(x)+(x-x^2)f''(x)=0$$ where $k\in(0,1)$ and the ...
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$\{(x,y)\in \mathbb{R}^2: |x|+|y|^{1/2}<1\}$ is convex

How to prove that $A=\{(x,y)\in \mathbb{R}^2: |x|+|y|^{1/2}<1\}$ is convex? I tried using the definition but couldn’t go far, since the second component involves square root(tried squaring, that ...
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General optimization problem

I've tried many efforts on this optimization problem but don't find any idea how to continue: Let $n \geq 2$ be an integer, and $J: \mathbf{R}^{n} \longrightarrow \mathbf{R}$ a continuous, coercive ...
I recently read about convexity with respect two functions $u, v$. I tried to look up for some definition, but I failed. That is why I am asking it here. What does it mean a function $f$ is convex ...