Lipschitz-like property relative to a set and the generalized Mordukhovich criterion

Meng, Kaiwen; Li, M. H.; Yao, Wang; Yang, Xiaoqi

doi:10.1007/s10107-020-01568-0

Cited by 9 publications

(8 citation statements)

References 27 publications

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“…However, these works only provided sufficiency for the relative Lipschitz-like property, which could still fail when the reference point lies on the boundary. In Meng et al [59], a new tool, the projectional coderivative, was introduced and both sufficiency and necessity were provided for characterizing the Lipschitz-like property relative to a closed and convex set. For other stability properties relative to a set, see Van Ngai and Théra [83], Ioffe [39], Arutyunov and…”

Section: Literature Reviewmentioning

confidence: 99%

“…To characterize the relative Lipschitz-like property, in [59], a new tool, the projectional coderivative, is introduced.…”

Section: Preliminariesmentioning

confidence: 99%

“…The projectional coderivatives initiated by [59] (see Definition 1.3.6), was invented to capture the Lipschitz-like property of a multifunction relative to a closed and convex set in full. Such a definition is derived via the sufficient and necessary condition for the property (see [59,Theorem 2.3]) with existence of κ > 0:…”

Section: Projectional Coderivatives and Chain Rulesmentioning

confidence: 99%

“…As in the last section, we derived a fixed-point expression for projectional coderivative of a mapping relative to a smooth manifold (see Proposition 2.1.5). Considering that the generalized Mordukhovich criterion in [59] are for closed and convex sets, in this section, we extend the criterion to the setting of a smooth manifold.…”

Section: Lipschitz-like Property Relative To a Smooth Manifoldmentioning

confidence: 99%

“…Although the Lipschitz-like property has been widely studied, the Lipschitz-like property relative to a set is greatly understudied. Not until recently, a new tool in variational analysis, projectional coderivative, has been introduced in Meng et al [59] and employed to establish a generalized Mordukhovich criterion to characterize the property relative to a closed and convex set. This verifiable condition also pins down the associated graphical modulus of the set-valued mapping relative to the closed and convex set with the outer norm of projectional coderivatives.…”

Section: Introductionmentioning

confidence: 99%

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Relative Lipschitz-like Property of Parametric Systems via Projectional Coderivatives

Yao

Yang

2023

SIAM J. Optim.

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In this thesis, fundamental properties of a newly introduced tool, projectional coderivatives, are illustrated. Some examples of calculation are also presented. When the set we refer to is a smooth manifold, the projectional coderivative can be simplified as a fixed-point expression. Therefore, we extend the generalized Mordukhovich criterion to such a setting. Moreover, chain rules and sum rules are developed for projectional coderivatives. Different levels of constraint qualifications are incorporated to generate upper estimates accordingly and all these upper estimates converge under the setting of smooth manifolds. By applying the sum rule to parametric systems, we obtain the upper estimate of the projectional coderivative of the solution mapping, which is also an implicit mapping, making it possible to analyse the relative Lipschitz-like property via projectional coderivatives. The difference between Foremost, I would like to express my heartfelt gratitude to my supervisor, Prof.Yang Xiaoqi for his continuous support and guidance of my PhD study, as well as his unwavering patience and overwhelming encouragement in working with me for the past five years. Without his constant help, this dissertation would not have been possible. I'm greatly indebted to him. I would like to thank Dr. Meng Kaiwen and Dr. Li Minghua for sharing their insightful ideas and providing detailed and constructive suggestions to improve this dissertation. The enjoyable but also rigid discussions we had have inspired and empowered me on research skills and the logic of presentation. I benefited tremendously from collaborations with them. Many thanks also to Prof. Zhang Kai and Dr. Hu Yaohua for their kind help in exploring research topics in my early PhD life. I'm also grateful to my undergraduate advisor, Dr. Cai Tao, for introducing me to academic research. His optimism, kind patience and constant encouragement have empowered me in pursuit of challenge. I benefited a lot from the scientific and quantitative analytic skills I learned from him. I would like to express my sincerest thanks to my friends and colleagues, Ms.

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Section: Literature Reviewmentioning

confidence: 99%

“…To characterize the relative Lipschitz-like property, in [59], a new tool, the projectional coderivative, is introduced.…”

Section: Preliminariesmentioning

confidence: 99%

Section: Projectional Coderivatives and Chain Rulesmentioning

confidence: 99%

Section: Lipschitz-like Property Relative To a Smooth Manifoldmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Relative Lipschitz-like Property of Parametric Systems via Projectional Coderivatives

Yao

Yang

2023

SIAM J. Optim.

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Special Issue: Continuous Optimization and Stability Analysis

2021

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Continuous optimization is a vital research area with many active branches such as linear and nonlinear optimization, stochastic optimization, vector optimization, semi-infinite optimization, etc. Stability analysis of optimization problems is of crucial interest in applications, and makes optimization an ideal testing bench for many disciplines such as convex analysis, set-valued analysis, variational analysis, and wellposedness, among others.The present special issue offers some recent developments in this wide area and aims to honor Prof. Marco Antonio López Cerdá on the occasion of his 70th birthday, as well as to show recognition by his colleagues to his contribution to this field. His achievements include some monographs, several surveys and more than 150 scientific articles on optimality, duality, stability and algorithms in semi-infinite programming, Lipschitz-type properties, error bounds, subdifferential calculus, and others such as game theory, robustness, stationarity and regularity.Marco A. López was born in Alcoy (Alicante, Spain) in 1949 and has spent most of his academic career (since 1985) as a full professor of statistics and operations research at the University of Alicante, where nowadays he is Emeritus Professor. Marco is a corresponding fellow of the Spanish Royal Academy of Sciences, an Honorary Research Fellow of the Federation University (Australia), and a Doctor Honoris Causa at the University of Limoges (France). He has also led important research projects, theoretical and applied. From 2008 to 2012 he was the coodinator of i-MATH Consolider B J.

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MultiSQP-GS: a sequential quadratic programming algorithm via gradient sampling for nonsmooth constrained multiobjective optimization

Rashidi,

Soleimani-damaneh

2024

Comput Optim Appl

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Lipschitz-like property relative to a set and the generalized Mordukhovich criterion

Cited by 9 publications

References 27 publications

Relative Lipschitz-like Property of Parametric Systems via Projectional Coderivatives

Relative Lipschitz-like Property of Parametric Systems via Projectional Coderivatives

Special Issue: Continuous Optimization and Stability Analysis

MultiSQP-GS: a sequential quadratic programming algorithm via gradient sampling for nonsmooth constrained multiobjective optimization

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