From Nonsmooth Minima to Smooth Branches via Heat Kernel Regularization

TL;DR

This paper investigates heat kernel regularization to analyze nonsmooth optimization problems, demonstrating that it restores Hessian nondegeneracy and enables continuation of minimizer branches near singular points.

Contribution

It provides a rigorous second-order framework showing how heat regularization ensures local solvability and Hessian positivity in nonsmooth optimization.

Findings

Regularized objectives admit global minimizers localized at the heat scale O(√t).

Asymptotic Hessian behavior depends on the local profile, remaining positive definite for quadratic cases.

Regularized Hessian stays nondegenerate for small t, enabling continuation even at nonsmooth minimizers.

Abstract

Many optimization problems in science and engineering involve objective functions that are nonsmooth at their minimizers. A common strategy is to trace a branch of minimizers of a regularized objective as the smoothing scale tends to zero; however, for nonsmooth functions, it is generally unclear whether such a branch can be continued and whether the associated continuation equation remains locally solvable. We study heat-kernel regularization and the resulting continuation equation along a local minimizing branch connected to a minimizer of the original objective. Under a global growth condition and a local leading-order description of the form $|x|^a$ with $1 \le a \le 2$, we first show that the regularized objective admits global minimizers and that any such minimizing branch localizes at the natural heat scale $O(\sqrt{t})$. We then prove that the asymptotic behavior of the regularized Hessian is determined by the local profile of the original objective: it remains uniformly positive definite in the quadratic case $a=2$, while in the subquadratic regime $1 \le a < 2$ its smallest eigenvalue grows at the controlled rate $t^{(a-2)/2}$. Consequently, the regularized Hessian remains asymptotically nondegenerate for all sufficiently small $t>0$, and the continuation equation remains locally solvable, even when the original objective does not admit a classical Hessian at the minimizer. Our results provide a rigorous second-order framework for continuation-based analysis in nonsmooth optimization by showing how heat regularization restores nondegeneracy near singular minimizers.