tiltstable

Tiltstable (also written tilt-stable or tilt-stability) is a concept in optimization and variational analysis describing the robustness of a local minimizer when the objective function is subject to small linear perturbations. Intuitively, a tiltstable minimizer remains the minimizer of a nearby tilted objective, and the selection of the minimizer under perturbation changes in a controlled, predictable way.

Formally, let f be a lower semicontinuous function on a finite-dimensional space, and for a small perturbation p in the dual space consider the tilted problem minimize f(x) − ⟨p, x⟩. A local minimizer x of f is tilt-stable if there exists a neighborhood of p = 0 and a Lipschitz continuous map p ↦ x(p) defined for p near 0 such that x(p) minimizes f(x) − ⟨p, x⟩ locally, with x(0) = x. The map x(p) is called a tilt-stability (or tilt-stable) selection.

Characterizations of tiltstable minimizers depend on smoothness and constraint structure. In smooth settings (f is C2 near x), tilt stability is tied to second-order conditions and nondegeneracy, often reducing to a positive definite Hessian on the relevant subspace. In nonsmooth or constrained problems, tilt stability is closely related to strong metric regularity of the subdifferential ∂f at (x, 0), and it implies uniqueness and Lipschitz continuity of the local minimizer under small tilts.

Applications of tiltstable concepts include sensitivity analysis of optimization models, parametric programming, and the design of algorithms that exploit stable dependence of solutions on perturbations. The notion is a central tool in variational analysis for understanding robust optimality beyond classical convex or smooth frameworks.