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

Jordan Collard, Scott B. Lindstrom

TL;DR

The paper studies the nonconvex feasibility problem of finding a point in the intersection of a smooth plane curve and a line in \mathbb{R}^2, contrasting standard projection methods with extrapolated approaches. It develops and analyzes the Lyapunov surrogate method (LT) for a two-set DR framework and proves local quadratic convergence when the intersection is non-tangential, supported by a detailed distance-ratio analysis. Numerical experiments on circle-line geometry and higher-dimensional semidefinite feasibility problems corroborate the quadratic rate and demonstrate the practical potential of LT over classical DR. The results provide analytical evidence for accelerated convergence in structured nonconvex feasibility problems and motivate extensions to higher dimensions. Overall, the work links Lyapunov-based surrogates with Newton-type behavior to achieve fast convergence in projection-based feasibility tasks.

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.

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

TL;DR

The paper studies the nonconvex feasibility problem of finding a point in the intersection of a smooth plane curve and a line in \mathbb{R}^2, contrasting standard projection methods with extrapolated approaches. It develops and analyzes the Lyapunov surrogate method (LT) for a two-set DR framework and proves local quadratic convergence when the intersection is non-tangential, supported by a detailed distance-ratio analysis. Numerical experiments on circle-line geometry and higher-dimensional semidefinite feasibility problems corroborate the quadratic rate and demonstrate the practical potential of LT over classical DR. The results provide analytical evidence for accelerated convergence in structured nonconvex feasibility problems and motivate extensions to higher dimensions. Overall, the work links Lyapunov-based surrogates with Newton-type behavior to achieve fast convergence in projection-based feasibility tasks.

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 . 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.
Paper Structure (9 sections, 8 theorems, 67 equations, 6 figures)

This paper contains 9 sections, 8 theorems, 67 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Douglas--Rachford Method
  4. Lyapunov Surrogate Method
  5. Quadratic Convergence
  6. Numerical Results
  7. Circle and Line
  8. Semidefinite Feasibility
  9. Conclusion

Key Result

Proposition 1

Let A and B be closed, convex sets, where $A\cap B \neq \emptyset$. Given any $x_0\in \mathcal{H}$, $\{x_{n}\}$ converges weakly to some point $x^* \in FixT$ and $\{P_A(x_n)\}$ is weakly convergent to $P_A(x^*)\in A\cap B$.

Figures (6)

  • Figure 1: A LT update when applied to a circle and line in $\mathbb{R}^2$.
  • Figure 2: Performance profiles comparing the Douglas--Rachford method and Lyapunov surrogate method for a circle and a line in $\mathbb{R}^2$. a Based on number of iterates. b Based on CPU time.
  • Figure 3: Observed convergence rates of the Douglas--Rachford method and Lyapunov surrogate method for a circle and a line in $\mathbb{R}^2$.
  • Figure 4: Observed convergence behaviour of DR, LT, and PLT for Setting 1.
  • Figure 5: Observed convergence behaviour of DR, LT, and PLT for Setting 2.
  • ...and 1 more figures

Theorems & Definitions (18)

  • Proposition 1: Douglas--Rachford fixed point property lions1979splitting
  • Lemma 2
  • proof
  • Theorem 3
  • proof
  • proof
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • ...and 8 more