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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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 ...
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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 ...
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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_\...
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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). ...
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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 ...
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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 ...
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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 \...
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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 ...
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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 ...
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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....
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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 ...
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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 $x\in\partial g\left(y\right)$. ...
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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 $f$...
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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 ...
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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 ...
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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 ...
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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) ...
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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 $d \in \mathbb{R}$ ...
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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 ...
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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 ...
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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 ...
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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 \...
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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) ...
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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. ...
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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....
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  • 397
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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 ...
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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 ...
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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. ...
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1 answer
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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 ...
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1 vote
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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\...
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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 ...
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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 ...
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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 ...
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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 $\...
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1 answer
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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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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(...
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4 votes
1 answer
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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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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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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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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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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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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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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 \...
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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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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}{...
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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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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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1 vote
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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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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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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 ...
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