[Paper Review] Some quantitative results on Lipschitz inverse and implicit functions theorems

Phan Phien|arXiv (Cornell University)|Apr 22, 2012

Advanced Differential Equations and Dynamical Systems5 references3 citations

TL;DR

This paper provides quantitative estimates for Clarke's Lipschitz inverse and implicit function theorems by explicitly bounding the neighborhoods of invertibility and the Lipschitz constant of the inverse in terms of the generalized Jacobian at a point. It establishes that the class of mappings satisfying Clarke’s theorem is open under small Lipschitz perturbations, offering constructive bounds for applications in optimization and nonlinear analysis.

ABSTRACT

Let $ f: \mathbb{R} ^ n ightarrow \mathbb{R}^n $ be a Lipschitz mapping with generalized Jacobian at $x_0$, denoted by $\partial f(x_0)$, is of maximal rank. F. H. Clarke (1976) proved that $f$ is locally invertible. In this paper, we give some quantitative assessments for Clarke's theorem on the Lipschitz inverse, and prove that the class of such mappings are open. Moreover, we also present a quantitative form for Lipschitz implicit function theorem.

Motivation & Objective

• To provide explicit quantitative estimates for the neighborhoods of invertibility and the Lipschitz constant of the inverse mapping in Clarke’s inverse function theorem.
• To extend these estimates to a quantitative form of the Lipschitz implicit function theorem.
• To prove that the class of mappings satisfying Clarke’s inverse function theorem is open under small Lipschitz perturbations.
• To offer constructive, computable bounds for the inverse and implicit function theorems in non-smooth settings, addressing a gap in prior qualitative results.

Proposed method

• The paper uses the generalized Jacobian ∂f(x₀) to characterize the local invertibility of Lipschitz mappings f: ℝⁿ → ℝⁿ.
• It applies matrix norm inequalities and the perturbation theory of invertible matrices (via Theorem 2.2) to derive bounds on the inverse mapping’s Lipschitz constant.
• The neighborhoods U and V of x₀ and f(x₀) are explicitly estimated using the infimum of the inverse norm over ∂f(x₀) and the Lipschitz constants of f.
• A quantitative implicit function theorem is derived by reducing the problem to a fixed-point argument in the context of Lipschitz mappings with generalized Jacobians.
• The openness of the class of mappings satisfying Clarke’s theorem is proven by showing that small Lipschitz perturbations of f do not destroy maximal rank of ∂f(x₀).
• The analysis relies on matrix norm properties, subdifferential convex hulls, and the inverse norm formula ‖M⁻¹‖ = 1 / min‖x‖=1 ‖Mx‖.

Experimental results

Research questions

• RQ1 What are explicit, computable bounds for the size of neighborhoods U and V in which a Lipschitz mapping f is locally invertible, given ∂f(x₀)?
• RQ2 How can the Lipschitz constant of the inverse mapping be quantitatively estimated from ∂f(x₀)?
• RQ3 Can a quantitative version of the Lipschitz implicit function theorem be derived using generalized Jacobians?
• RQ4 Under what conditions is the class of mappings satisfying Clarke’s inverse function theorem open under small Lipschitz perturbations?
• RQ5 What happens to local invertibility when the perturbation’s Lipschitz constant exceeds that of the original mapping?

Key findings

• The inverse mapping g has a Lipschitz constant bounded by L(g) = 1/δ, where δ = ½ inf_{M₀∈∂f(x₀)} 1/‖M₀⁻¹‖.
• The neighborhood U of x₀ is estimated as Bⁿ_{(rδ)/(2(K′+L))}(x₀), with r = 1 and δ defined as above.
• The image neighborhood V is given by f(x₀) + (rδ/2)Bⁿ, so V = Bⁿ_{rδ/2}(f(x₀)).
• The class of mappings satisfying Clarke’s theorem is open: if f is perturbed by h with sup_{H∈∂h(x₀)}‖H‖ < 1 / sup_{M₀∈∂f₀(x₀)}‖M₀⁻¹‖, then f+h remains locally invertible.
• In the example with f₀(x,y) = (|x|+2x, |y|+2y), the inverse is locally defined on B²_{1/7}((0,0)) with Lipschitz constant 1.
• When the perturbation h has Lipschitz constant L > K (where K is the lower Lipschitz bound of f₀), the generalized Jacobian may lose maximal rank, and local invertibility fails.