The theory that develops differential calculus for functions that are not differentiable in the usual sense.
Questions tagged [non-smooth-analysis]
110 questions
14
votes
2 answers
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 way through a paper in which minimizing $f$ over…
Lepidopterist
- 3,047
11
votes
2 answers
Chain rule for subdifferential
I have two functions $g(x)$ and $h(z)$ where $h:\mathbb R^n\to \mathbb R$ and $g: \mathbb R\to \mathbb R$. Both are convex, neither are smooth. How can I apply the chain rule to find $\partial (g\circ h)$? (in terms of $\partial g$ and $\partial…
Y. S.
- 1,876
10
votes
2 answers
How equality in Fenchel-Young inequality characterizes subdifferential?
I am not able to see why equality in the Fenchel-Young inequality
characterizes subgradients.
As per Fenchel-Young inequality:
\begin{equation}
f(x)+f^*(u) \geq \langle x,u \rangle
\end{equation}
while the definition of subdifferential set…
CKM
- 1,642
9
votes
3 answers
Gradient of $A \mapsto \sigma_i (A)$
Let $ A $ be an $m \times n$ matrix of rank $ k \le \min(m,n) $. Then we decompose $ A = USV^T $, where:
$U$ is $m \times k$ is a semi-orthogonal matrix.
$S$ is $k \times k$ diagonal matrix , of which its diagonal entries are called singular…
Firas Abd El Gani
- 2,136
8
votes
1 answer
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. Here are some details for what I want to find…
Тyma Gaidash
- 13,576
8
votes
2 answers
Subgradient of the $\ell_0$ "norm"
I am trying to characterize the sub-gradient of $\ell_0$ "norm"
$$f(x) := \|x\|_0 := \sum_{i=1}^n 1\{{x_i \neq 0}\}$$
At first, since it satisfies the triangle inequality, I thought that the $\ell_0$ "norm" is convex and non-smooth. Then, I tried to…
mr noname
- 167
7
votes
1 answer
When do two functions have the same subdifferentials?
For two functions $f$ and $g$, if $\nabla f(x) = \nabla g(x)$, $f = g + c$ for some constant $c$. Does the same hold if the gradient is replaced by the (convex) subdifferential, ie $\partial f(x) = \partial g(x)$ for all $x$ ?
And, as a stronger…
P. Camilleri
- 588
6
votes
2 answers
Generalizing Lagrange multipliers to use the subdifferential
Background:
This is a followup to the question Lagrange multipliers with non-smooth constraints. Lagrange multipliers can be used for constrained optimization problems of the form
$$\min_{\vec x} f(\vec x) \text{ such that } g(\vec x) = 0$$
Briefly,…
dan_x
- 285
6
votes
1 answer
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 $f: \mathbf{R}^{n} \rightarrow \mathbf{R}$ be locally Lipschitz at a point $x \in…
mathsstudentTUD
- 567
- 2
- 10
5
votes
2 answers
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 way this inclusion is stated.
This inclusion in…
NBS
- 103
5
votes
3 answers
Is the subdifferential always convex and closed set?
Two properties of the subdifferential set are stated as follows:
Given a function $f : \mathbb{R}^n → \mathbb{R}$,
(i) the subdifferential set $\partial f(x)$ is always convex and closed, even if $f$ is nonconvex.
(ii) $\partial f(x)$ can be…
user85842
- 101
4
votes
2 answers
Lagrange multipliers with non-smooth constraints
I read in a textbook a passing comment that Lagrange multipliers are not applicable if there are points of non-differentiability in the constraints (even if the constraints are continuous). For example, in the following problem:
$\min_{\boldsymbol…
dan_x
- 285
4
votes
1 answer
Relationship between Clarke-subdifferential $\partial_{C}f(\, \cdot \,)$ and Bouligand-subdifferential $\partial_{B}f(\, \cdot \,)$
Let $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$ be locally Lipschitz sontinuous.
(1) The Clarke-subdifferential $\partial_{C} f(x) \subset \mathbb{R}^{n}$ of $f$ in $x \in \mathbb{R}^{n}$ is defined by
\begin{equation*}…
mathsstudentTUD
- 567
- 2
- 10
4
votes
1 answer
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 differentiable. I would like to know: "is there any…
kaienfr
- 301
4
votes
0 answers
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 $\hat{x} \in$ dom $\partial h$ if there are $a \in(0, \infty],…
wwliu
- 305