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SUBDERIVATIVE

Subderivative

Generalization of derivatives to real-valued functions

the subderivative (or subgradient) generalizes the derivative to convex functions which are not necessarily differentiable. The set of subderivatives at

Subderivative

Sign function

Function returning minus 1, zero or plus 1

many subderivatives at zero, with just one of them taking the value sgn ⁡ ( 0 ) = 0 {\displaystyle \operatorname {sgn}(0)=0} . A subderivative value

Sign function

Sign_function

Convex hull

Smallest convex set containing a given set

Convex function Invex function Legendre transformation Semi-continuity Subderivative Main results (list) Carathéodory's theorem Ekeland's variational principle

Convex hull

Convex_hull

Jensen's inequality

Theorem of convex functions

at each real number x {\displaystyle x} we have a nonempty set of subderivatives, which may be thought of as lines touching the graph of φ {\displaystyle

Jensen's inequality

Jensen's_inequality

Weak derivative

Generalisation of the derivative of a function

for problems of differential equations and in functional analysis. Subderivative Weyl's lemma (Laplace equation) Evans, Lawrence C. (1998). Partial differential

Weak derivative

Weak_derivative

Karush–Kuhn–Tucker conditions

Concept in mathematical optimization

{\displaystyle h_{j}\colon \mathbb {R} ^{n}\rightarrow \mathbb {R} } have subderivatives at a point x ∗ ∈ R n {\displaystyle x^{*}\in \mathbb {R} ^{n}} . If

Karush–Kuhn–Tucker conditions

Karush–Kuhn–Tucker_conditions

Clarke generalized derivative

Types generalized of derivatives

Subgradient method — Class of optimization methods for nonsmooth functions. Subderivative Clarke, F. H. (1975). "Generalized gradients and applications". Transactions

Clarke generalized derivative

Clarke_generalized_derivative

Hamilton–Jacobi–Bellman equation

Optimality condition in optimal control theory

required, in which conventional derivatives are replaced by (set-valued) subderivatives. Consider the following problem in deterministic optimal control over

Hamilton–Jacobi–Bellman equation

Hamilton–Jacobi–Bellman_equation

Moreau's theorem

semi-continuous extended real-valued functional on H. Let A stand for ∂φ, the subderivative of φ; for α > 0 let Jα denote the resolvent: J α = ( i d + α A ) − 1

Moreau's theorem

Moreau's_theorem

Glossary of real and complex analysis

{\displaystyle C(X,\mathbb {R} )} . subanalytic subanalytic. subderivative subderivative. subharmonic A twice continuously differentiable function f {\displaystyle

Glossary of real and complex analysis

Glossary_of_real_and_complex_analysis

Convex function

Real function with secant line between points above the graph itself

Logarithmically convex function Pseudoconvex function Quasiconvex function Subderivative of a convex function "Lecture Notes 2" (PDF). www.stat.cmu.edu. Retrieved

Convex function

Convex_function

Regularization (mathematics)

Technique to make a model more generalizable and transferable

differentiable due to the kink at x = 0. Subgradient methods which rely on the subderivative can be used to solve L 1 {\displaystyle L_{1}} regularized learning

Regularization (mathematics)

Regularization_(mathematics)

Arithmetic derivative

Function defined on integers in number theory

Let S {\displaystyle S} be a nonempty set of primes. The arithmetic subderivative of x {\displaystyle x} with respect to S {\displaystyle S} is defined

Arithmetic derivative

Arithmetic_derivative

Variational analysis

convex functions by generalizing the notion of derivative to that of subderivative. Further generalization of the notion of the derivative such as the

Variational analysis

Variational_analysis

Subgradient method

Concept in convex optimization mathematics

Subgradient methods are convex optimization methods which use subderivatives. Originally developed by Naum Z. Shor and others in the 1960s and 1970s, subgradient

Subgradient method

Subgradient_method

Generalizations of the derivative

Fundamental construction of differential calculus

case is the variational derivative in the calculus of variations. The subderivative and subgradient are generalizations of the derivative to convex functions

Generalizations of the derivative

Generalizations_of_the_derivative

Convexity in economics

Significant topic in economics

Wikipedia's quality standards. The specific problem is: Relationship between subderivatives and non‑convexity remains cryptic. Please help improve this section

Convexity in economics

Convexity_in_economics

List of numerical analysis topics

function f such that f(tx + (1 − t)y) ≤ max(f(x), f(y)) for t ∈ [0,1] Subderivative Geodesic convexity — convexity for functions defined on a Riemannian

List of numerical analysis topics

List_of_numerical_analysis_topics