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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When is upper/lower Dini derivative finite?

Consider a function $f:\mathbb{R}\rightarrow \mathbb{R}$. Lower right Dini derivative is defined as $D_+f(x)=\liminf_{t\rightarrow 0+}\cfrac{f(x+th)-f(x)}{h}$ (Also, the Wiki link for the definition ...
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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) = \...
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Directional derivative of maximum function

For convex functions $f_i: \mathbb{R}^n \to \mathbb{R}$, $1 \le i \le m$, let $f:\mathbb{R}^n \to \mathbb{R}$ be defined via $f(x) = \max_{i \in [m]} f_i(x)$. I want to prove that the directional ...
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Dynamics of Loss in Homogeneous, Non-Smooth Models Using Clarke Subdifferential

tl;dr: Seeking insights on the application of Clarke subdifferential for analyzing the optimization differential inclusion with smooth objective and homogeneous model. I'm interested in its validity, ...
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Subdifferential of convex functions is nonempty

I read the counterexample in this post, and I am wondering if it is possible to fix the statement in this way: Given a convex function $f:\mathbb R^d\rightarrow\mathbb R\cup\{\infty\}$ and $x\in\...
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Let $f$ be convex, and $g$ be a convex surrogate of $f$. Does a Lipschitz-smooth $g-f$ imply $f$ and $g$ have the same subgradients?

Let $f,g : \mathbb{R}^N \to \mathbb{R}$ be convex functions, but not necessarily differentiable. Suppose $g$ is a 'majorant' or 'surrogate' of $f$ at $\xi$ with the following properties: $g(x) \geq f(...
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Nonsmooth analysis: Need help clarifying a step in the proof that $co D^\ast u(x) = \partial f(x)$

I am reading the book optimization and nonsmooth analysis by Frank.H Clarke and there's a step I'm stuck in the proof of theorem 2.5.1. So I'll write the definitions first. We work in $\mathcal{R}^n$ ...
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Directional derivative of proximal mapping of a convex function

Let $f:\mathbb{R}^n\rightarrow\overline{\mathbb{R}}$ be a proper closed convex function that is locally Lipschitz continuous on its domain $D(f)$. Define the proximal mapping of $f$ to be $$\textbf{...
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The equivalent definition of tangent cone in normed space

I am trying to prove that $v \in T_{S}(x)$ if and only if $\lim \inf_{t \downarrow 0}d_{S}(x+tv)/t=0$ and $d_S$ is the distance function of $S$ and S is a subset of the normed space $X$. For the ...
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The tangent cone at a interior point

Let $S$ be a subset of $X$ and $X$ is a normed space. The tangent cone to $S$ at a point $x \in S$, denoted $T_S(x)$, defined by $$ T_S(x)=\{v:\exists \ t_k \downarrow 0, \exists\ v_k \to v \text{ ...
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Is a single-valued (convex) subdifferential sufficient to show differentiability?

Definitions (we could assume $f$ is convex for this problem too, if needbe): For a function $f\colon\mathbb{R}^n\to\left]-\infty,+\infty\right]$, the subdifferential at a point $x\in\mathbb{R}^n$, $\...
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For a Lipschitz function, is the gradient (when defined) orthogonal to level set?

For a differentiable function f, we know that the gradient is orthogonal to the level sets of f. What if f is Lipschitz continuous? we know that its gradient exists almost everywhere. Is the gradient (...
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Fundamental Theorem of Calculus for non-differentiable convex functions

Suppose $C\subset\mathbb{R}^n$ is a convex set and $f:C\to\mathbb{R}$ is a convex function. I wonder if the following statement is true. Suppose $g:C\to\mathbb{R}^n$ satisfies $g(x)\in\partial f(x)$ ...
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Proof of Theorem 6.14 in "Variational analysis (Rockafellar, Wets)"

I am currently reading "Variational analysis" by Rockafellar and Wets, and I am trying to understand the proof of Theorem 6.14 about normal cones to sets with constraint structure. But, I ...
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Time derivative a nonsmooth convex function. Chain rule.

Let us consider a convex function $f : \mathbb{R}^n \rightarrow \mathbb{R}$. Let us consider that is composed with an absolutely continuous function of a real variable $t$, $ x : \mathbb{R} \...
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Subdifferential and affine transformation

I'm having trouble proving this theorem. $\mathbf{Theorem.}$ Suppose $f:\mathbb R^m\to \mathbb R$ is a convex function and $A\in \mathbb R^{m\times n}, b \in \mathbb R^n$. If we let the mapping $h:\...
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Subgradient of the spectral norm

I am working on developing a numerical algorithm that needs to use a subgradient of $\|\cdot\|_2$ (matrix norm) at each iteration. According to Characterization of the Subdifferential of Some Matrix ...
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Optimality check for non-differentiable convex function

I have a doubt on how to check if a given point of a convex, but nondifferentiable, function is a minimum/maximum. For instance, let's say that I have the following function: $$ f \left( x \right) = ...
TheLearner's user avatar
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Clarke Subdifferential for a function of two variables.

I am looking of an example where equality don't hold in the below relation: let $f: \mathbb{R} \times \mathbb{R} \mapsto \mathbb{R} $ be locally lipschitz and regular at $ x = (x_1 , x_2)^T \in \...
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Integral of Dirac Delta Derivative Times Non-Smooth Function

I know that if we have some function f(y) that is smooth and has compact support, we get the following by using integration by parts. $\int f(y) \delta '(y-x) dy = -\int f'(y) \delta (y-x) dy =-f'(x)$ ...
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Subdifferentiablity of a convex functional

On a general Banach space, for a functional $F:X\to\mathbb{R}_{\infty}$, we define the Frechet subdifferential and subdifferential by the following: \begin{align*} \mbox{Frechet differential: }\...
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Derivation of subgradient of a matrix's nuclear norm

I was going through the derivation of subgradient of the nuclear norm of a matrix from an old homework of a Convex Optimization course (CMU Convex Optimization Homework 2 - Problem 2). The setup is ...
Phat Tran's user avatar
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Proximal point, Moreau envelope, grad. of Moreau env. when domain is constrained / subset of $\mathbb{R}^d$

Let $f:\mathcal{K}\rightarrow \mathbb{R}$ be a convex function, with $\mathcal{K}\subset \mathbb{R}^d$ a convex set, the domain of $f$ is constrained/a strict subset of $\mathbb{R}^d$. How do we ...
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Defining the Subgradient with Polar of the Tagent Cone on the Epigraph, or the Convex Hull

To start, I would like to state both definitions for sub-gradient and where they came from. Definition 1 Let $f$ be $\mathbb E \mapsto \mathbb {\bar R}$ with $x\in \text{dom}(f)$ then $v\in \mathbb E$...
Alto Lagato's user avatar
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Subgradient method for nonconvex nonsmooth function

Gradient descent or stochastic gradient descent are frequently used to find stationary points (and in some cases even to local minimum) of a nonconvex function. I was wondering if the same can be said ...
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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 ...
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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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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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Find the subdifferential for $\max\left(x^2,|x|\right)$

We define the function $f:\mathbb R \rightarrow \mathbb R$ as the following: $$f\left(x\right)=\max\left(x^2,|x|\right)$$ Find the subdifferential $\partial f\left(x\right)$ for all $x\in \mathbb R$. ...
Tarek Badr's user avatar
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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) ...
ohana's user avatar
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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 ...
Myth's user avatar
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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 \...
Math Brother's user avatar
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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. ...
Тyma Gaidash's user avatar
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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....
Snifkes's user avatar
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2 answers
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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 ...
Curaçao Hajek's user avatar
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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 ...
Diego Vargas's user avatar
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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. ...
Roman Dryndik's user avatar
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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 ...
DockingBlade's user avatar
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\...
R. W. Prado's user avatar