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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are smoothstep transition function and hermite splines equally in this smoothing problem?
I have trouble understanding the difference in the following problem.
I have the function F(x) which is not differentiable at x=a
$$F(x):=\left\lbrace\begin{array}{cc} 0 & \text{ for $x<a$}\\ G(...
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A few clarifications about multiplication in subgradient calculus
In the subgradient calculus linearity properties, the appropriate side of the addition rule utilizes Minkowski addition of sets. Ordinarily in linearity, a scaling rule agrees with, and is basically ...
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How to prove $D^+u(x) = \operatorname{co} D^*u(x) = \partial^c u(x)$ for semiconcave functions?
Let $u:\Omega \subset \mathbb{R}^N \to \mathbb{R}$ be semiconcave (with a general modulus $\omega$), with $\Omega$ an open set. I would like to prove that
$$
D^+u(x) = \operatorname{co} D^*u(x) = \...
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Are there any results on the numerical method for multi-valued SDEs driven by fractional Brownian motion?
I am studying the numerical solution of multi-valued stochastic differential equations driven by the fractional Brownian motion (fractional white noise).
The multi-valued SDEs can be written as the ...
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Questions about sub-differentials: when is a convex set a sub-differential of convex functions?
I was studying Rockafellar's convex analysis, chapter 24. Their Theorem 24.3 says in one dim functions, a convex set C is a convex function f's sub differential iff C is a complete non-decreasing ...
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Existence of subgradients implies continuity?
Given a function $f:M\to\mathbb R\cup\{\infty\}$ defined on a normed space $M$, a subgradient of $f$ at a point $x\in M$ is a continuous linear form $u\in M^*$ (topological dual of $M$) such that $f(...
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Frechet derivative and subdifferential
In $\mathbb{R}^n$ we have the statement that a convex function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ is differentiable if and only if $\partial f(x) = \{\nabla f(x)\}$. I.e. if and only if the ...
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Fenchel-Rockafellar strong duality by subdifferential
I saw versions of the Fenchel theorem that go like this.
Let $X,Y$ be Banach spaces, $f:X\to\mathbb{R}\cup{\infty},g:Y\to\mathbb{R}\cup{\infty}$ be convex lower semicontinuous functions, $A:X\to Y$ a ...
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Is this non-smooth comparison result for ODEs true?
I am failing to find a reference for the following comparison result for ODEs (if it even holds true).
Suppose we have a non-smooth path $X$, e.g. the path of a stochastic process or solution of an ...
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What is the subdifferential of complex $\ell_1$ norm?
I have a complex least-squares with $\ell_1$ regularization problem. Given the matrix $\mathbf{A}\in\mathbb{C}^{m\times n}$ and the vector $\mathbf{y}\in\mathbb{C}^{m}$,
$$ \arg\min_{\mathbf{x} \in \...
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Conditions for existence of Caratheodary solutions to an ODE
What are some sufficient conditions for existence of a Caratheodary solution for a time-invariant ODE? I am following Cortes' book on Discontinuous Dynamical Systems, and he has defined a notion ...
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Generalization of first-order optimality condition for subgradients
Let $f \colon \mathcal X \subseteq \mathbb R^n \to \mathbb R$ be a convex function. A well-known result in convex optimization is that if $f$ is differentiable, then for any $x^\star \in \mathcal X$, ...
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Local Lipschitzness of $\max(0,\cdot)\colon H^1(\Omega) \to H^1(\Omega)$?
Let $\Omega$ be a bounded smooth domain. We know that $\max(0,\cdot)\colon H^1(\Omega) \to H^1(\Omega)$ is continuous and even satisfies
$$\lVert \max(0,u) \rVert_{H^1(\Omega)} \leq C\lVert u \rVert_{...
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Derivative of composition of convex with non-convex differentiable function
I am looking to see if the following statement is true:
Let $\mathbb{V}$ be a Banach space. If $g:[0,\infty)\to \mathbb{V}$ is nonconvex and differentiable, and $f:\mathbb{V}\to \mathbb{R}$ is convex,...
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Conditions of strong convexity for non-smooth functions
I was reading this paper for some results on the strong convexity for non-smooth functions but I'm not getting this proposition at all:
Lemma II
(i) $f$ is strongly convex with parameter $\mu$.
(ii) ...
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