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ODEs
Published in A. C. Faul, A Concise Introduction to Numerical Analysis, 2018
Lipschitz continuity means that the slopes of all secant lines to the function between possible points v and w are bounded above by a positive constant. Thus a Lipschitz continuous function is limited in how much and how fast it can change. In the theory of differential equations, Lipschitz continuity is the central condition of the Picard–Lindelöf theorem, which guarantees the existence and uniqueness of a solution to an initial value problem.
Elementary Nonlinear Programming Theory
Published in Craig A. Tovey, Linear Optimization and Duality, 2020
Local search from a large number of starting points has long been a popular class of heuristics for global optimization. Suppose f(x) is Lipschitz continuous, meaning that there exists a constant K such that ||f(x)−f(y)||≤K||x−y|| for all x,y). Geometrically, f can't be arbitrarily steep. Perform local search from independent randomly chosen starting points in a bounded domain. Then for all ϵ>0, the probability converges to 1 that a solution whose value is within ϵ of optimal will be found, as the number of searches goes to infinity. That's not a fully satisfying performance guarantee. Moreover, it may be challenging to sample randomly within domains other than balls or hyper-rectangles. For example, sampling uniformly in a polytope can be done in polynomial time by a randomized algorithm, but the algorithm is complicateed, and as of this writing has not been de-randomized.
A generalized mountain pass lemma with a closed subset for locally Lipschitz functionals
Published in Applicable Analysis, 2022
Fengying Li, Bingyu Li, Shiqing Zhang
In [14], Livrea and Marano made the following assumptions: is a locally Lipschitz continuous function. denotes a homotopy-stable family with extended boundary B.There exists a nonempty closed subset F of X such that and, moreover, is a continuous function fulfilling (3), while indicates the metric defined (2).
A Multi-Level Set Approach for Bone Segmentation in Lumbar Ultrasound Images
Published in IETE Journal of Research, 2022
V. Umamaheswari, P.M. Venkatasai, S. Poonguzhali
Here, the multilevel set segmentation algorithm is developed for separating the bone image into the various number of segments for the purpose of easier analysis. The multi-level set segmentation approach is widely used for effective segmentation of the targeted region in an image. The basic expression for this multi-level set approach [17] is given as Here, represents the unit normal to a level curve of at each point and represents the curvature of a level curve. indicates the Lipschitz continuous function. The filtered image and cluster result is used to find the exact intensity region of the bone area. Then, these regions act as a mask in the multilevel set for region growing.
Incremental proximal gradient scheme with penalization for constrained composite convex optimization problems
Published in Optimization, 2021
Let be a function of the form for all , where is a convex function and is a convex differentiable function such that is Lipschitz continuous. Let be a convex differentiable function such that is Lipschitz continuous. In this work, we focus on the problem Let denote the solution set of this problem and assume that is nonempty. In addition, we may assume without loss of generality that .