The Fun is Finite: Douglas-Rachford and Sudoku Puzzle -- Finite Termination and Local Linear Convergence

TL;DR

This paper analyzes the Douglas-Rachford method's convergence properties for non-convex problems like Sudoku and s-queens, proving finite termination and specific convergence rates.

Contribution

It provides the first local convergence analysis showing finite termination and linear convergence rates for non-convex feasibility problems, including Sudoku and s-queens.

Findings

Douglas-Rachford converges finitely for s-queens.

Linear convergence rate of √5/5 for Sudoku.

Numerical results support theoretical convergence claims.

Abstract

In recent years, the Douglas-Rachford splitting method has been shown to be effective at solving many non-convex optimization problems. In this paper we present a local convergence analysis for non-convex feasibility problems and show that both finite termination and local linear convergence are obtained. For a generalization of the Sudoku puzzle, we prove that the local linear rate of convergence of Douglas-Rachford is exactly $\frac{\sqrt{5}}{5}$ and independent of puzzle size. For the $s$-queens problem we prove that Douglas-Rachford converges after a finite number of iterations. Numerical results on solving Sudoku puzzles and $s$-queens puzzles are provided to support our theoretical findings.