Lipschitzlike

Lipschitzlike is a regularity notion for set-valued maps used in variational analysis and optimization. It describes when the solution or image of a parameterized problem changes in a controlled, Lipschitz manner under small perturbations, generalizing the idea of Lipschitz continuity to multivalued mappings.

Formally, let F be a set-valued map between metric spaces X and Y, and suppose y0 is in F(x0). F is Lipschitzlike at x0 for y0 if there exist neighborhoods U of x0, V of y0, and a constant L ≥ 0 such that for all x in U, F(x) ∩ V ⊆ F(x0) + L ||x − x0|| B_Y, where B_Y is the unit ball in Y. Equivalently, the distance between F(x) near y0 and F(x0) is controlled linearly in ||x − x0||, often formulated via the excess metric: e(F(x) ∩ V, F(x0)) ≤ L ||x − x0|| and e(F(x0) ∩ V, F(x)) ≤ L ||x − x0|| for x in U. The property is also known as the Aubin property.

In the special case where F is single-valued, Lipschitzlike reduces to ordinary Lipschitz continuity near x0. The concept is central in studying stability of solutions to generalized equations, variational inequalities, and optimization problems, where one seeks to ensure that small parameter changes produce only proportionally small changes in the solution set.

Examples include the projection mapping onto a convex set, which is nonexpansive and hence Lipschitzlike with constant 1, and linear mappings F(x) = {Ax + b}, which are Lipschitzlike with constant equal to the operator norm of A. Related notions include metric regularity and the broader Aubin property for set-valued maps.