[Paper Review] Lipschitz Properties of Nonsmooth Functions and Set-Valued Mappings via Generalized Differentiation and Applications

Nguyen Mau Nam, Gerardo Lafferriere|arXiv (Cornell University)|Feb 7, 2013

Optimization and Variational Analysis12 references3 citations

TL;DR

This paper establishes necessary and sufficient conditions for Lipschitz continuity of nonsmooth functions—specifically the minimal time function and scalarization function—using generalized differentiation tools, particularly Mordukhovich's subdifferential and coderivative criteria. The key contribution is a characterization of Lipschitz behavior via the triviality of the singular subdifferential and the transversality of the normal cone to the direction vector at the boundary point.

ABSTRACT

In this paper, we revisit the Mordukhovich's subdifferential criterion for Lipschitz continuity of nonsmooth functions and coderivative criterion for the Aubin/Lipschitz-like property of set-valued mappings in finite dimensions. The criteria are useful and beautiful results in modern variational analysis showing the state of the art of the field. As an application, we establish necessary and sufficient conditions for Lipschitz continuity of the minimal time function and the scalarization function, that play an important role in many aspects of nonsmooth analysis and optimization.

Motivation & Objective

• To revisit and refine Mordukhovich’s subdifferential and coderivative criteria for Lipschitz properties in finite-dimensional variational analysis.
• To establish necessary and sufficient conditions for Lipschitz continuity of the minimal time function and scalarization function.
• To bridge theoretical generalized differentiation with practical applications in nonsmooth optimization and sensitivity analysis.
• To extend the Aubin/Lipschitz-like property characterization using limiting normal cones for set-valued mappings.
• To provide a complete characterization of Lipschitz continuity via the singular subdifferential and normal cone transversality.

Proposed method

• Utilizes Mordukhovich’s limiting normal cone and subdifferential theory to analyze Lipschitz behavior of nonsmooth functions.
• Applies the coderivative criterion: the Aubin property holds around $(\bar{x},\bar{y})$ iff $ (u,0) \in N((\bar{x},\bar{y});\text{gph}\,\mathcal{F}) \Rightarrow u = 0 $.
• Employs the minimal time function $ \varphi_d(x;\Omega) = \inf\{ t \geq 0 \mid x + td \in \Omega \} $ as a central object of study.
• Uses the metric projection and distance function to derive subdifferential representations via limiting normal cones.
• Applies the singular subdifferential $ \partial^\infty \varphi_d(\bar{x};\Omega) = \{0\} $ as a necessary and sufficient condition for Lipschitz continuity.
• Establishes a limiting normal cone representation: $ \partial \varphi_d(\bar{x};\Omega) = \{ w \in \mathbb{R}^n \mid \langle w, -d \rangle = 1, \, w \in N(\tilde{x};\Omega) \} $, where $ \tilde{x} = \bar{x} + t d $.

Experimental results

Research questions

• RQ1 Under what conditions is the minimal time function $ \varphi_d(\cdot;\Omega) $ Lipschitz continuous around a point $ \bar{x} \in \text{dom}\, \varphi_d(\cdot;\Omega) $?
• RQ2 How can the Aubin property of set-valued mappings be characterized using the Mordukhovich coderivative criterion?
• RQ3 What is the relationship between the singular subdifferential of the minimal time function and its Lipschitz continuity?
• RQ4 When does the normal cone $ N(\tilde{x};\Omega) \cap \{d\}^\perp = \{0\} $ imply Lipschitz continuity of $ \varphi_d(\cdot;\Omega) $?
• RQ5 Can the coderivative criterion be applied to derive necessary and sufficient conditions for Lipschitz continuity of scalarization functions?

Key findings

• The minimal time function $ \varphi_d(\cdot;\Omega) $ is Lipschitz continuous around $ \bar{x} $ if and only if $ \partial^\infty \varphi_d(\bar{x};\Omega) = \{0\} $, which is equivalent to $ N(\tilde{x};\Omega) \cap \{d\}^\perp = \{0\} $, where $ \tilde{x} = \bar{x} + t d $, $ t = \varphi_d(\bar{x};\Omega) $.
• The limiting subdifferential of the minimal time function satisfies $ \partial \varphi_d(\bar{x};\Omega) = \{ w \in \mathbb{R}^n \mid \langle w, -d \rangle = 1, \, w \in N(\tilde{x};\Omega) \} $, under regularity assumptions.
• The singular subdifferential $ \partial^\infty \varphi_d(\bar{x};\Omega) = \{0\} $ is a necessary and sufficient condition for the Lipschitz continuity of $ \varphi_d(\cdot;\Omega) $ around $ \bar{x} $.
• If $ \Omega $ is normally regular at $ \tilde{x} $, then $ N(\tilde{x};\Omega) \cap \{d\}^\perp = \{0\} $ is both necessary and sufficient for Lipschitz continuity of $ \varphi_d(\cdot;\Omega) $.
• The coderivative criterion for the Aubin property is equivalent to the condition that $ (u,0) \in N((\bar{x},\bar{y});\text{gph}\,\mathcal{F}) \Rightarrow u = 0 $, using the Mordukhovich normal cone.
• The scalarization function inherits Lipschitz properties under similar normal cone and subdifferential conditions, generalizing results from the minimal time function.