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Efficient Douglas-Rachford Methods on Hadamard Manifolds with Applications to the Heron Problems

D. R. Sahu, Shikher Sharma, Pankaj Gautam

Abstract

Our interest lies in developing some efficient methods for minimizing the sum of two geodesically convex functions on Hadamard manifolds, with the aim to enhance the convergence of the Douglas-Rachford algorithm in Hadamard manifolds. Specifically, we propose two types of algorithms: inertial and non-inertial algorithms. The convergence analysis of both algorithms is provided under suitable assumptions on algorithmic parameters and the geodesic convexity of the objective functions. This convergence analysis is based on fixed-point theory for nonexpansive operators. We also study the convergence rates of these two methods. Additionally, we introduce parallel Douglas-Rachford type algorithms for minimizing functionals containing multiple summands with applications to the generalized Heron problem on Hadamard manifolds. To demonstrate the effectiveness of the proposed algorithms, we present some numerical experiments for the generalized Heron problems.

Efficient Douglas-Rachford Methods on Hadamard Manifolds with Applications to the Heron Problems

Abstract

Our interest lies in developing some efficient methods for minimizing the sum of two geodesically convex functions on Hadamard manifolds, with the aim to enhance the convergence of the Douglas-Rachford algorithm in Hadamard manifolds. Specifically, we propose two types of algorithms: inertial and non-inertial algorithms. The convergence analysis of both algorithms is provided under suitable assumptions on algorithmic parameters and the geodesic convexity of the objective functions. This convergence analysis is based on fixed-point theory for nonexpansive operators. We also study the convergence rates of these two methods. Additionally, we introduce parallel Douglas-Rachford type algorithms for minimizing functionals containing multiple summands with applications to the generalized Heron problem on Hadamard manifolds. To demonstrate the effectiveness of the proposed algorithms, we present some numerical experiments for the generalized Heron problems.

Paper Structure

This paper contains 10 sections, 25 theorems, 134 equations, 9 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Efficient Douglas-Rachford Algorithms on Hadamard Manifolds
  4. Numerical Experiments
  5. Applications
  6. Minimization of functional containing multiple summands
  7. The generalized Heron problem in Hadamard manifolds
  8. Conclusion
  9. Proof of Proposition \ref{['Newpr2']}
  10. Proof of Proposition \ref{['Newpr3']}

Key Result

Lemma 1

Let $\{a_n\}$ and $\{b_n\}$ be sequences of positive real numbers such that $\sum_{n=1}^{\infty} a_n b_n < \infty.$ Suppose that $\sum_{n=1}^{\infty} a_n = \infty$ and that the sequence $\{b_n\}$ is decreasing. Then $b_n = o\!\left(\frac{1}{\sum_{i=1}^{n} a_i}\right).$

Figures (9)

  • Figure 1: Selection of comparison points
  • Figure 2: Comparison of $E_n$ with respect to number of iterations for the Rosenbrock Problem
  • Figure 3: Generalized Heron problems for two points with a ball constraint.
  • Figure 4: Comparison of Alg(PDRA), Alg(In-M), and Alg($p$-Acc) based on the relationship between the number of iterations and $Er(n)$ for Example \ref{['DiscConsEx']}.
  • Figure 5: The generalized Heron problem for four points with a ball constraint
  • ...and 4 more figures

Theorems & Definitions (47)

  • Lemma 1: Dong2015comments
  • Definition 2: SahuNFAO
  • Lemma 3: Sahusoft2020
  • Lemma 4: Alvarez2004
  • Proposition 5: Sakai
  • Remark 6: Martin2
  • Proposition 7: Sakai
  • Proposition 8: ShikherCNSNS2024
  • Proposition 9
  • Proposition 10: ShikherCNSNS2024
  • ...and 37 more