The spectral barrier method to solve analytic convex optimization problems in function spaces
Year: 2025
Type: article
Abstract: Spectral methods approximate the solutions of variational problems, boundary value problems and partial differential equations with high degree polynomials. Such methods are especially well-suited to problems where the solution is holomorphic. We focus on convex optimization problems such as the p -Laplacian, which has long been considered hard to solve. We solve these problems by the barrier method. Theoretically, the barrier method requires the use of “short t -steps.” Our new spectral barrier (SPB) method uses “long t -steps.” By computing these long steps on progressively higher degree polynomial spaces, we ensure that the overall method converges to a tolerance $${\mathop {\textrm{tol}}\limits }>0$$ tol > 0 in $$\hat{O}(\log {\mathop {\textrm{tol}}\limits }^{-1})$$ O ^ ( log tol - 1 ) Newton iterations, where the hat indicates that we neglect very slow growing functions like $$\log \log $$ log log and $$\log ^*$$ log ∗ , and provided the problem is reverse Hölder regular. We confirm this theoretical performance estimate with numerical experiments.
Source: Numerische Mathematik
Author: Sébastien Loisel
Institution: Heriot-Watt University
Language: English
Subfield: Computational Mechanics
Field: Engineering
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,190
OpenAlex ID: W4416392018