Questions tagged [non-smooth-analysis]

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

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53 views

Functions similar to $f(x) = x^2 \sin{\frac{1}{x}}$

I am analysing this function: $f(x) = x^2 \sin{\frac{1}{x}}$ The specific feature of this function that I am interested in is the increasing smoothness as you move away from zero. Are there similar ...
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78 views

Norm of subgradients bouded

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be locally Lipschitz at $\bar{x} \in \mathbb{R}^n$ with constant $L$. Let $(x^{k})\subset\mathbb{R}^{n}$ be a sequence with $x^{k} \rightarrow \bar{x}$. ...
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49 views

Proof that generalized directional derivative is upper semicontinuous

In "Nonsmooth Optimization" by Mäkela and Neittaanmäki the definition of the generalized directional derivative is given as follows: Definition 3.1.1 (Clarke). Let $f: \mathbf{R}^{n} \...
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30 views

Generalized Directional Derivative

I have a question regarding the concept of generalized directional derivative and how to compute it for 'fairly complex' scenarios. I report here the definition I am using: $$ V^o (z; d) := \limsup _{...
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17 views

What is lower C1 regularity?

In the context of nonsmooth analysis and optimization, what is lower C1 regularity? I have found a definition of subsmoothness/ lower-C1 but cannot find anything about lower C1 regularity. Many thanks....
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21 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 $\hat{x} \in$ ...
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1answer
34 views

Equivalence of the two expression for dual averaging

I have a question about two versions of the algorithm for dual averaging. The first one is $$x_{t+1}=x_t+\nabla f(w_t)\to w_{t+1}=P_C\left(-\frac{1}{\sqrt{t+1}}x_{t+1}\right)=\arg\min_{w\in C}\left\|w+...
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1answer
20 views

Limit point in the subdifferential

Let $f : \mathbb{R} \to \mathbb{R}$ be a convex and l.s.c. function. Take a point $x \in \mathbb{R}$ and consider the subdifferential of $f$ at $x$ denoted by $\partial f (x)$. Take any sequence $\{...
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28 views

How to interpret $\limsup_{t\rightarrow 0^+,~y\rightarrow x}$?

I know how to interpret the expression below for a fixed y but not when $y\rightarrow x$ simultaneously with $t→0^+$ $$f^\circ(x,v)=\limsup_{t\rightarrow 0^+,~y\rightarrow x}\dfrac{f(y+tv)-f(y)}{t}$$
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Proving convexity of non-smooth function using smooth approximation

Suppose we are given a positive semi-definite matrix $C \in \mathbb{R}^{n \times n}$ and suppose we define the function $\sigma_{s}(\boldsymbol{x}) = [\max \lbrace x_1, s \rbrace, \cdots, \max \lbrace ...
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10 views

Fixed point of “min” function with linear arguments

What are the conditions on the matrix $\mathbf{A}$ such that, for any real vector $\mathbf{f} \in \mathbb{R}^N$, the problem \begin{equation} \mathbf{u} = \min(\mathbf{1},\mathbf{A}\mathbf{u}-\mathbf{...
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25 views

How does a function's smoothness relate to the smoothness of its inverse?

Given $f$ is invertable, a classic example of a smooth function having a non-smooth inverse is $f(x) = x^3$ since $x^{1/3}$ is non-differentiable at 0. The converse of this is that $ f(x) = x^{1/3} $ ...
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1answer
128 views

Prove that the cone $S = \{(x,y,z) \in \Bbb R^3 ~|~x^2+y^2-z^2=0 \}$ is not a smooth surface

Prove that the cone $S = \{(x,y,z) \in \Bbb R^3 ~|~x^2+y^2-z^2=0 \}$ is not a smooth surface. The book I am reading - Vector calculus by Peter Baxandall gives the following hint : (Let $S \subseteq \...
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2answers
78 views

What are the subdifferentials $\partial f(0)$ and $\partial f(1)$?

Let $ f: \mathbb{R} \to \mathbb{R} $ given by \begin{equation*} f(x) = \left\{ \begin{array}{rl} x \log x -x & \text{if } x \geq 0\\ \infty & \text{if else}\\ \end{array} \right. \end{equation*...
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1answer
56 views

Expression for the Clarke subdifferential of a weakly convex function

Let $\gamma\in\left]0,+\infty\right[$, let $f$ be a proper, convex, lower semicontinuous function from a real Hilbert space $\mathcal{X}$ to $\left]-\infty,+\infty\right]$, and set $g=f-\frac{\gamma}{...
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72 views

About Subdifferential of matrix norm

We have the definition of subdifferential of convex function $f : X \to R$ (where $X$ is complex Banach space) at a point $x \in X$ is set of linear function $v^*\in X^*$ such that $$f (y) − f (x) \...
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24 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 ...
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40 views

Chain rule for subdifferentials of nonconvex functions

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 ...
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If $g$ is smooth and $f$ is convex non-smooth, then $\partial f(g(x))\cdot Dg(x)\subseteq \partial (f\circ g)(x))$?

Let $g\colon\mathbb{R}^n\to\mathbb{E}$ be a smooth function and $f\colon\mathbb{E}\to\mathbb{E}$ be a convex function (for my purposes, a projection onto a convex set), where $\mathbb{E}$ is an $m$-...
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35 views

Find all points where $\textbf{r}$ (parametric surface) is not smooth

$$ \begin{align*} f(x) = \begin{cases} x^2 & x > 0 \\ 0 & x \le 0 \end{cases} \end{align*} $$ A parametric surface is defined as $\textbf{r}(u,v) = \langle u, f(v), f(-v) \rangle $ Find ...
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9 views

Subdifferential of Sum

Let $f$ be class $C^{\infty}(\mathbb{R}^d,\mathbb{R}^k)$ functions and let $a<b$ define $1_{[a,b]^d}$. What is a subgradient of $$ f1_{[a,b]^d}? $$ Intuitively it seems like it should be $1_{[a,b]...
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2answers
101 views

Subdifferential of $f = \max \{f_1(x), f_2(x) \}$

Let $f_1,f_2$ be convex function and let $f(x)=\max\{f_1(x), f_2(x)\}$. It is clear to me that if $f_1(x) = f_2(x)$, then $[\nabla f_1(x), \nabla f_2(x)] \subseteq \partial f(x)$, but why do we also ...
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1answer
41 views

Computing subderivative for two variable function

I need to compute a suberivaqtive for the following function $$f\big((x, y) \big) = |x-2y+1| + |x-4y-3| + |2x-y+2|.$$ I also need to find its global extrema. I know that a subderivarive of function ...
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78 views

Uses of nonsmooth analysis in mathematical research

To give some context: I am aware of the uses of Convex Analysis (and its applications in Convex Optimization), I have been studying (for a while) the developments of Nonsmooth Analysis (and its ...
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79 views

Compute the Clarke subdifferential of a function

The Clarke subdifferential of $f(x)=|x|$ at $x=0$ is a set $[-1, 1]$, and $\{sign(x)\}$ otherwise, just like the ordinary subdifferential because of convex. Now for \begin{equation} g(x)= \begin{cases}...
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255 views

Is the Clarke Subdifferential always defined for Lipschitz continuous functions?

According to various websites, for some function $f:X\to R$ we can define a map $$ D(x,v):= \lim_{y\to x} \sup_{t\searrow0} \frac{f(y+tv)-f(y)}{t} $$ and the Clarke Subdifferential (Def 1) is $$ \...
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2answers
191 views

Subdifferential

I compute the subdifferential of the convex function $f(x)= \displaystyle\max_{1\le i \le N} \{f_{i} + <v_{i},x-z_{i}> \} $. The result is: $conv(\{v_{i}, i \in I(x)\})$ where $I(x) = \{i \in \{...
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1answer
152 views

A question about properties of convex subdifferential

Let $X$ be a reflexive Banach space. Let $$\mathcal{P}_{fc}(X)=\{A\subset X\, |\, A\,\text{is nonempty, closed, convex\}}.$$ Let $F:X\to \mathcal{P}_{fc}(X^*)$ be an operator. Consider a convex and ...
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1answer
135 views

Directional subderivatives: definition and computation

For a continuous vector field $f$ and some vector $v$ in a Banach space, the (lower) directional subderivative can be defiend as follows: $$ Df_v(x) \triangleq \liminf_{ \substack{ t \rightarrow 0 \\...
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1answer
442 views

Subdifferential - equivalent definitions?

I've been reading an article on Clarke critical values of subanalytic Lipschitz functions. There I've come across the following definition(s) of subdifferential: $$f: U \to \mathbb{R}^n, \ \ \ \ \...
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1answer
3k views

Definition of subgradient of $1$-norm

I have an example which seems to contradict the definition of the subgradient of the L1 norm. Obviously I made a mistake, but I can't see where. We start with the definition of a subgradient (from ...
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1answer
904 views

Clarke's generalized gradient formula computed on functions defined on open sets

In the book [1], Clarke et al. define the generalized gradient for a Lipschitz function $f:\mathbb{R}^n\to\mathbb{R}$ as follows. 8.1. Theorem (Generalized Gradient Formula). Let $x\in\mathbb{R}^n$,...
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1answer
2k views

Subderivative of $ ||Au||_{L^{\infty}} $ to Compute Proximal Operator / Prox Operator

I am looking for ways to compute the subderivative of $ ||Au||_{L^{\infty}} $, as I want to solve the minimization problem of \begin{equation} \min\limits_u \quad \lambda ||Au||_{L^{\infty}} + \frac{...
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1answer
353 views

Characterizing subdifferential of nuclear norm of $X^T X$

I am interested in characterizing the subdifferential of $f=|X^TX|_*$, i.e., the nuclear norm of $X^T X$. Two ways I am looking at it right now. Certainly, the singular values of $X^T X$ are the ...
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2answers
2k views

Computing subgradients, a basic example.

A vector $v \in \mathbb{R}^n$ is a subgradient of a convex function at $x$ if $$f(y) \geq f(x) + <v, y - x>$$ for all $y \in dom(f).$ The set of all sub gradients is known as the sub ...
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1answer
2k views

Subdifferential of the sum

Let $C \subset \mathbb R^n$ a nonempty subset. Let us define the indicator function of $C$ $$ I_C(x) = \begin{cases} 0 & x \in C \\ +\infty & x \notin C \end{cases}. $$ Let us consider, in ...
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2answers
9k views

Deriving the sub-differential of the nuclear norm

Let $$f(K) = \| K \|_*$$ be the nuclear norm (sum of the singular values) of $K=U\Sigma V^T$. How can one compute the subdifferential $\partial f$? This may be a basic question, I'm trying to work my ...
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1answer
355 views

How to find the subdifferential of $|x|$?

I want to compute the subdifferential of $ f $ on $ \mathbb{R} \setminus \{ 0 \} $ when $ f(x) = |x| $. How do I do this?