Mathematics

Questions tagged [non-smooth-analysis]

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

113 questions
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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 $f$…
Lepidopterist
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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.
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10
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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 says:…
CKM
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9
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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
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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
  • 14,281
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
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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
  • 618
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 $$\min_{\vec x} f(\vec x) \text{ such that } g(\vec x) = 0$$…
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
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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
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0 answers

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 applications in Nonsmooth Optimization) as traced by…
ex.nihil
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5
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1 answer

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: $$f: U \to \mathbb{R}^n, \ \ \ \ \emptyset \neq U \subset \mathbb{R}^n$$ $f$ is locally…
Hagrid
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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
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