Smooth optimization has superlinear convergence: near the minimum, each step reduces the error by a factor that itself shrinks to zero, producing acceleration. Newton's method squares the number of correct digits per step. But nonsmooth optimization — where the function has kinks, corners, or worse — is stuck at linear convergence. Each step reduces the error by a fixed fraction, never accelerating. This limitation seemed intrinsic: the kinks destroy the curvature information that superlinear methods exploit.
Gebken and Ulbrich (arXiv: 2603.23236) break this barrier. Their method achieves superlinear convergence for nonsmooth functions using higher-order cutting-plane models.
Standard cutting-plane methods approximate a nonsmooth function by the maximum of several first-order expansions (tangent planes) centered at previously visited points. The approximation is piecewise linear — a polyhedral underestimator. As more points are visited, the approximation improves, but only linearly because each plane captures only first-order information.
The innovation: replace first-order planes with higher-order expansions. For lower-C^2 functions (functions that are locally expressible as maxima of smooth functions), these expansions exist and provide local curvature information even though the overall function is nonsmooth. The trust-region bundle method built from these higher-order models converges superlinearly.
The through-claim: nonsmoothness is a global property that doesn't prevent local acceleration. A function can have kinks everywhere and still be locally smooth around any particular kink — the kink is a transition between smooth pieces, each of which has curvature. Higher-order cutting planes use the curvature of the smooth pieces, even at the kink. The nonsmooth function is smooth enough for superlinear convergence; the previous methods just weren't looking hard enough at the local structure.
Gebken & Ulbrich, 2603.23236. Optimization / nonsmooth analysis / bundle methods / superlinear convergence / cutting-plane methods.