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Everything about Lipschitz continuity
Strong form of uniform continuity
In mathematical analysis, Lipschitz continuity is a regularity property of functions between metric spaces that is stronger than uniform continuity, and hence also stronger than continuity. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exists a real number such that, for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is not greater than this real number; the smallest such bound is called the Lipschitz constant of the function. For instance, every function that is defined on an interval and has a bounded first derivative is Lipschitz continuous. Wikipedia →
🧭 Start here: a 6-step reading path
- Read the overview: Lipschitz continuityA 5-minute orientation
- Skim a recent paper: A three-term CGPM-based algorithm without Lipschitz continuity for constrained nonlinear monotone equations with applicationsInfluential work since 2022
- Pick up a classic book: Labyrinth of ThoughtThe most-reprinted book on the subject
- Read the landmark paper: Minimization of functions having Lipschitz continuous first partial derivativesCited 1,664 times
- Go sideways: Hölder conditionA closely related topic
- Go sideways: SmoothnessA closely related topic
📖 Definitions
No dictionary entry for this exact phrase.
🗂️ The subject at a glance
🏆 Landmark papers (most cited)
Minimization of functions having Lipschitz continuous first partial derivatives
Non-Lipschitz continuous stabilizers for nonlinear systems with uncontrollable unstable linearization
Prox-Method with Rate of Convergence O (1/ t ) for Variational Inequalities with Lipschitz Continuous Monotone Operators and Smooth Convex-Concave Saddle Point Problems
Regularisation of neural networks by enforcing Lipschitz continuity
Lipschitz Continuity for Constrained Processes
Lipschitz Continuity of Value Functions in Markovian Decision Processes
On lipschitz continuity of quasiconformalmappings in space
Lipschitz Continuity of the Value Function in Optimal Control
🆕 Influential since 2022
A three-term CGPM-based algorithm without Lipschitz continuity for constrained nonlinear monotone equations with applications
An efficient projection-based algorithm without Lipschitz continuity for large-scale nonlinear pseudo-monotone equations
Adaptive scaling damped BFGS method without gradient Lipschitz continuity
Adaptive proximal algorithms for convex optimization under local Lipschitz continuity of the gradient
Adaptive three-term PRP algorithms without gradient Lipschitz continuity condition for nonconvex functions
📚 Books worth reading
Labyrinth of ThoughtJosé Ferreirós Domínguez · 1999
Global Optimization in Action: Continuous and Lipschitz OptimizationJános D. Pintér · 1995- 📕Lipschitz continuous policy functions for strongly concave optimization problemsLuigi Montrucchio · 1986
Convergence analysis of recurrent neural networksZhang Yi, K.K. Tan · 2003- 📕Representation of Lipschitz continuous functions by potentialsLars Blomqvist · 1972
- 📕On Lipschitz continuity of nonlinear differential operatorsStephen L. Keeling · 1987
- 📕Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficientsMartin Hutzenthaler · 2015
- 📕Local Lipschitz Continuity in the Initial Value and Strong Completeness for Nonlinear Stochastic Differential EquationsSonja Cox, Martin Hutzenthaler · 2024
Ranked by number of editions on Open Library, a rough measure of how lasting a book is.
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