Quadratic Convergence of a Projection Method for a Plane Curve Feasibility Problem

TL;DR

This paper proves quadratic convergence of a projection method for finding intersections of smooth curves and lines in 2D, demonstrating the effectiveness of extrapolated methods in structured nonconvex feasibility problems.

Contribution

It establishes quadratic convergence for a projection algorithm in a nonconvex plane curve feasibility problem under certain conditions, extending understanding of accelerated methods.

Findings

Quadratic convergence proven under specific geometric conditions.

Numerical experiments confirm theoretical convergence rates.

Results suggest potential for higher-dimensional extensions.

Abstract

Under conditions that prevent tangential intersection, we prove quadratic convergence of a projection algorithm for the feasibility problem of finding a point in the intersection of a smooth curve and line in $\mathbb{R}^2$. This nonconvex problem has been studied in the literature for both Douglas-Rachford algorithm (DR) and circumcentered reflection method (CRM), because it is prototypical of inverse problems in signal processing and image recovery. This result highlights the potential of extrapolated methods to meaningfully accelerate convergence in structured feasibility problems. Numerical experiments confirm the theoretical findings. Our work lays the foundations for extending such results to higher dimensional problems.