Questions tagged [non-smooth-analysis]
The theory that develops differential calculus for functions that are not differentiable in the usual sense.
67
questions
0
votes
1
answer
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 ...
- 419
4
votes
2
answers
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
0
votes
0
answers
12
views
Generic conditions under which $\sigma \mapsto (1/\sigma^2)\|\mathbb E_z[(f(x+\sigma z)-f(x))z]\|^2$ is non-decreasing function, for every $x \in R^n$
Let $f:\mathbb R^n \to \mathbb R$ be a continuously-differentiable function. For every $\sigma>0$, define $f_\sigma,g_\sigma:\mathbb R^n \to \mathbb R$ by $f_\sigma(x) := E_z[f(x+z)]$, and $g_\...
- 8,279
0
votes
1
answer
27
views
Are Hadamard derivatives strict derivatives according to Clarke?
I have a function $f$ that is Hardamard differentiable but it is unclear to me if that also implies that $f$ is strictly differentiable according to Clarke (see Optimization and Nonsmooth Analysis).
...
0
votes
0
answers
29
views
compute the subgradient set of $\max(x,0)$ using Danskin
I'd like to compute the subgradient set of $f(x):=\max(x,0)$ using Danskin's theorem, similar to what is done here.
We have
\begin{align}
f(x) = \max(x,0) = \max\{\phi(x,z): z\in Z\}
\end{align}
where ...
- 2,345
0
votes
0
answers
18
views
Approximating expansion for non-smooth or nonlinear functions?
Are there some examples or research trends to find approximating expansions to nonlinear or non-smooth functions that have some nice properties from the Taylor expansion - e.g. possibility to use some ...
- 1,181
1
vote
0
answers
15
views
Limit Derivative of solution to linear complementarity problem at discontinuity
Consider the linear complementarity problem with $\mathbf{M}$ a $P$-matrix and a vector $\mathbf{q}$ with unknown $\mathbf{z}$:
Find $\mathbf{z}$ such that $\mathbf{M}\mathbf{z}+\mathbf{q} \geq \...
- 397
0
votes
0
answers
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 ...
- 1
0
votes
1
answer
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
0
votes
1
answer
41
views
Subgradient everywhere terminology
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
3
votes
1
answer
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
0
votes
0
answers
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
3
votes
1
answer
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
1
answer
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 ...
- 515
0
votes
0
answers
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
1
answer
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
0
answers
140
views
Calculate subdifferential of a function
Calculate the subdifferential of a function $f: \mathbb{R}^3 \to \mathbb{R}$ defined by $$f(x) = \vert x_1 - x_2\vert + \vert x_2 - x_3\vert$$
My attempt
Applying the sum rule we have $$\partial f(x) ...
- 557
1
vote
0
answers
53
views
Directional derivative of absolute value function lower semicontinuous?
Consider the absolute value function $f(x) = |x|$.
This function is convex and therefore directionally differentiable, i.e. for every point $x \in \mathbb{R}$ and every direction $d \in \mathbb{R}$ ...
- 515
0
votes
0
answers
135
views
Upper Semicontinuity of Clarke-Subdifferential
Assume $f: \mathbb{R}^{n} \to \mathbb{R}$ is locally Lipschitz-continuous.
Let $\partial_{C}f(x)$ denote the Clarke-subdifferential of $f$ at $x \in \mathbb{R}^{n}$.
Because $f$ is assumed to be ...
- 515
2
votes
0
answers
86
views
Nonsmooth Optimization: Clarke's directional derivative and Clarke-stationarity
Assume $f: \mathbb{R}^{n} \to \mathbb{R}$ is locally Lipschitz-continuous.
Let $f^{\circ}(x;s)$ denote Clarke's generalized directional derivative of $f$ at the point $x \in \mathbb{R}^{n}$ in the ...
- 515
1
vote
1
answer
32
views
Some basic subdifferential computations
I'm trying to understand a bit of nonsmooth analysis, but I'm struggling even to compute a simple example. Any help would be awesome!
Could you please confirm how do the subdifferentials of these ...
- 402
1
vote
0
answers
36
views
How to find a subdifferential of $(x_1, x_2) \mapsto |x_1 + x_2| + |x_1 - x_2|$?
I know that function $x \mapsto |x|$ has the following subdifferential
$$\partial f(x) =
\begin{cases}
1 &, x>0 \\
[-1,1] &, x=0 \\
-1 &, x<0.
\end{cases}$$
for $x \...
- 51
0
votes
0
answers
93
views
Subgradient of Frobenius norm and ReLu function with matrix variable
It is easy to compute the sub-differential of ReLU function $\sigma(x)=\max\{x,0\}$ for $x\in\mathbb{R}$. There is also a post (subdifferential of ReLU function composition with affine function) ...
- 626
6
votes
1
answer
256
views
Evaluating $\mathrm{\int_0^1 ?(x)dx}$
This function interestingly shown as ?(x) is dubbed the Minkowski Question Mark Function. It looks very similar to x. Wolfram Alpha can even plot the derivative of this apparently smooth function. ...
- 5,129
0
votes
1
answer
40
views
Double derivative of piecewise continuous function
The following function is defined for $x \in (-\infty,\infty)$
\begin{equation}
f(x) = \begin{cases} x & x<1 \\
1 & x \geq 1
\end{cases}
\end{equation}0
The function is obviously continuous....
- 397
0
votes
2
answers
91
views
Calculate the subgradient at $\pm \sqrt{2}$ for $1/2 x^2$
Let $f(x) = (1/2) x^2, x \in [-\sqrt{2}, \sqrt{2}]$ and $f(x) = +\infty$ elsewhere
I would like to compute the subgradient of this function at the boundary $\pm\sqrt2$. However, I am not sure how to ...
- 757
0
votes
1
answer
44
views
Find a subgradient of a real function
Let $f(x)=\chi_{\{-1\}}(x)+I_{[-1,1]}(x)$, where $\chi_{A}(x)=1$ when $x\in A$ and $\chi_{A}(x)=0$ when $x\notin A$, and $I$ is the indicator funcion (which is zero in its domain and infinity ...
- 493
0
votes
1
answer
39
views
Subdifferential of $f(x) = |c^{T} x|$
I want to find a subdifferential of function $f(x) = |c^Tx|$, where $x \in \mathbb{R}^n$.
I know that if $h(x) = f(Ax + b)$ then $\partial{h(x)} = A^T\partial{f(Ax+b)}$, which is exactly my case. ...
- 207
1
vote
1
answer
59
views
How to find the subgradient of $x \mapsto \max \left( (x+1)^2, (x-3)^2 \right) $ at $x=1$?
I need help finding the subgradient of the following function at the point $x = 1$.
$$ \max \left( (x+1)^2, (x-3)^2 \right) $$
I think it's $$ [-4,4] $$ since that is the range between the left-hand ...
- 13
1
vote
0
answers
37
views
Condition under which the Clarke's subdifferential is locally Lipschitzian
Given a locally Lipschitz continuous function $f: \mathbb{R}^n \rightarrow \mathbb{R}$. The Clarke's subdifferential set is the set given by
$$
\partial f (x) = {\text{cl co}}\left\lbrace \lim_{k\in\...
- 622
0
votes
1
answer
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
0
votes
0
answers
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 ...
user780847
0
votes
1
answer
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 ...
user780847
1
vote
0
answers
46
views
Find the subdifferential of a given function
Find the subdifferential of the following function $$f(w)=\frac{1}{2}\left \| w \right \|^2_2+C\cdot \sum_{i=1}^{m} \max(0,1-\alpha _i \cdot x_i^Tw)$$ where $C>0$, $w, x_i \in \Bbb R^n$ and $\...
- 481
-2
votes
1
answer
72
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 ...
- 11
0
votes
1
answer
155
views
Subdifferential of locally Lipschitz function locally bounded
Let $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$ be locally Lipschitz near $x \in \mathbb{R}^{n}$ with constant $L>0$, i.e. there exist $\varepsilon > 0$ and $L>0$ such that
\begin{equation}
|f(...
- 515
4
votes
1
answer
120
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} \...
- 515
4
votes
0
answers
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
0
votes
1
answer
67
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+...
- 47
-1
votes
1
answer
23
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 $\{...
- 1,364
0
votes
0
answers
66
views
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 ...
- 549
1
vote
0
answers
36
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} $ ...
2
votes
1
answer
417
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 \...
- 8,434
1
vote
2
answers
149
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*...
- 25
3
votes
1
answer
184
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}{...
- 3,406
1
vote
0
answers
126
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) \...
- 109
1
vote
0
answers
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
0
answers
222
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 ...
- 402
0
votes
0
answers
24
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]...
- 1
0
votes
2
answers
659
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 ...
- 1