Year: 2025
Type: preprint
Abstract: We develop efficient algorithms for optimizing piecewise smooth (PWS) functions when their smooth regions are \emph{unknown} to the algorithm. For convex PWS functions satisfying a quadratic growth (QG) condition, we demonstrate that the bundle-level (BL)-type method~\cite{lemarechal1995new} achieves global linear convergence---to our knowledge, the first such result for any algorithm for this problem class. We extend the method to solve weakly convex PWS problems, achieving the same order of complexity as in smooth non-convex optimization. Furthermore, we introduce a verifiable and accurate termination criterion for PWS optimization. Like the gradient norm in smooth optimization, this certificate tightly characterizes the optimality gap under the QG condition and can be evaluated without knowledge of any problem parameters. We develop a search subroutine for this certificate and embed it within a guess-and-check framework, resulting in almost parameter-free algorithms for both convex QG and weakly convex PWS problems.
Source: arXiv (Cornell University)
Authors: Zhang, Zhe, Sra, Suvrit
Institution: Purdue University West Lafayette
Language: English
Subfield: Numerical Analysis
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4416411517
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