# Questions tagged [non-smooth-analysis]

The theory that develops differential calculus for functions that are not differentiable in the usual sense.

67 questions
Filter by
Sorted by
Tagged with
30 views

### If a function is convex, then its subgradient set is non-empty

Theorem: A function $f: \textbf{dom}(f) \to \mathbb R$ is convex if and only if $\textbf{dom}(f)$ is convex and $\partial f(\textbf x)$ is not empty for all $\textbf x \in \textbf{dom}(f)$ I know the ...
103 views

### Strict inclusion in subdifferential sum rule $\partial{f(x)}+\partial{g(x)}\subseteq{\partial(f+g)(x)}$.

I wish to find an example to show that the inclusion in the subdifferential sum rule $$\partial{f(x)}+\partial{g(x)}\subseteq{\partial(f+g)(x)}$$ is strict. However, I have a problem understanding the ...
• 85
12 views

• 397
29 views

### Gradient flows when there are two directions of steepest descent

Loosely speaking, a gradient flow $$\dot{x}(t) = - \nabla E(x(t)),\quad x(0) = x_0$$ says that the trajectory of $x$ is evolving in the direction of steepest descent of some functional $E$ (usually ...
41 views

### Could non-smooth time-limited functions been Analytical?

Could non-smooth time-limited functions been Analytical? Please read the scenarios first I was reading about analytic functions definitions on Wiki and looks like some of its properties where ...
• 829
41 views

Suppose I have a function $h$ and a function $f$. The function $f$ is non-differentiable and thus, does not have a gradient. However, the function $h$ belongs to the subgradient of $f$ at every $x$, i....
• 35
85 views

### Can L-smooth (L>0) convex function to be non-differentiable?

As we know, a function $f:\mathbb{R}^n\to \mathbb{R}$ is called L-smooth (with a finite $L>0$), if $x\mapsto \frac{L}{2}\|x\|^2 - f(x)$ is convex. This definition does not restrict $f$ to be ...
• 101
20 views

### Continuity property of sub-differentials

Assume that $g\colon\mathbb{R}^n\to\left(-\infty,\infty\right]$ is continuous over its domain, convex, closed and proper. For some $x,y\in\mathbb{R}^n$, assume that $x\in\partial g\left(y\right)$. ...
• 986
71 views

### Lower bound of a strongly convex function

Let $f,h\colon\mathbb R^n\to\mathbb R$ be two strongly convex functions such that $f\ge h$ and $f\left( x^*\right)=h\left( x^*\right)$, where $x^*$ is a joint unique minimizer for both. Assume that $f$...
• 986
1 vote
57 views

### Clakre-subdifferential of a continuously differentiable function

Assume $f:\mathbb{R}^{n} \to \mathbb{R}$ is continuously differentiable. Let $x \in \mathbb{R}^{n}$ be a point. Show that it holds: $$\partial_{C} f(x) = \{ \nabla f(x)\}\,.$$ In the following I ...
21 views

### Applications of the Lipschitz mean value theorem?

I am reading the paper: Analysis and Optimization of Lipschitz Continuous Mappings by B.H. Pourciau. I just got to the theorem of the Lipschitz Mean Value, and I wasn't able to decipher its ...
• 21
1 vote
86 views

### Prove that (Clarke's) generalized directional derivative for locally Lipschitz function is subadditive

I'm reading the proof on link(doi: 10.1.1.145.6632), lemma 2.6, on subadditivity of (Clarke's) generalized directional derivative for locally Lipschitz functions. The generalized directional ...
1 vote
140 views

• 622
85 views

### Subdifferential of a function

I am a beginner in functional analysis. Recently, I came across the term "subdifferential" of a convex function. I would like to know the notion of subdifferential of a convex function in ...
• 1,228
52 views

### Finding the gradient of $𝑓(𝑥,𝑦)=\max\{|𝑥|,|𝑦|\}$ for $|x|=|y|$

We have the function $f : \mathbb{R}^2 \to \mathbb{R}$ given by $$f(x,y) := \max \left\{ |x|, |y| \right\}$$ Let $C := \left\{ (x,y) \mid |x|=|y| \right\}$. Given point $(x,y) \in C$, how can I find ...
85 views

### Finding the gradient of $f(x,y)=\max \{|x|,|y| \}$

We have the function $f : \mathbb{R}^2 \to \mathbb{R}$ given by $$f(x,y) := \max \left\{ |x|, |y| \right\}$$ Let $C := \left\{ (x,y) \mid |x|=|y| \right\}$. Given point $(x,y) \notin C$, how can I ...
1 vote
46 views

42 views

### Which is stronger, Kurdyka-Łojasiewicz property or regular? If the comparison is not proper, what are their range of applications?

(Kurdyka-Eojasiewicz property and exponent). We say that a proper closed function $h: \mathbb{X} \rightarrow \overline{\mathbb{R}}$ satisfies the Kurdyka-Eojasiewicz $(K L)$ property at $\hat{x} \in$ ...
• 41
67 views

1 vote
34 views

### Sub-differential of a convex function along a particular direction

Take a convex function $f: \mathbb{R}^n \rightarrow \mathbb{R}$. Choose an arbitrary direction $d \in \mathbb{R}^n$ and consider the restriction of $f$ to the line through $x \in \mathbb{R}^n$ in the ...
1 vote
I have two functions: one of them $h\colon\mathbb{R}^n\to\mathcal{S}$ is smooth, but not necessarily convex, and the other $g\colon\mathcal{S}\to\mathcal{S}$ is convex, non-expansive, and not ...