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Douglas--Rachford algorithm for nonmonotone multioperator inclusion problems

Jan Harold Alcantara, Akiko Takeda

TL;DR

This work extends the Douglas–Rachford algorithm to multioperator inclusion problems by introducing product-space reformulations that reduce the problem to a two-operator setting, enabling convergence analysis under generalized monotonicity when $\sum_{i=1}^m \sigma_i>0$. It develops warped-resolvent machinery and two reformulations (Campoy-style and a flexible weighted version) to handle cases without convex-valuedness and without strict monotonicity assumptions. The authors establish convergence of the DR shadow sequence to fixed points corresponding to zeros of the sum, provide error rates $o(1/\sqrt{k})$, and extend the framework to structured nonconvex optimization, including sums of weakly/strongly convex functions and finite-dimensional problems with Lipschitz gradients. Practical results illustrate how function ordering and the chosen weights affect convergence speed and accuracy, highlighting potential guidelines for algorithm design in nonconvex settings.

Abstract

The Douglas--Rachford algorithm is a classic splitting method for finding a zero of the sum of two maximal monotone operators. It has also been applied to settings that involve one weakly and one strongly monotone operator. In this work, we extend the Douglas--Rachford algorithm to address multioperator inclusion problems involving $m$ ($m\geq 2$) weakly and strongly monotone operators, reformulated as a two-operator inclusion in a product space. By selecting appropriate parameters, we establish the convergence of the algorithm to a fixed point, from which solutions can be extracted. Furthermore, we illustrate its applicability to sum-of-$m$-functions minimization problems characterized by weakly convex and strongly convex functions. For general nonconvex problems in finite-dimensional spaces, comprising Lipschitz continuously differentiable functions and a proper closed function, we provide global subsequential convergence guarantees.

Douglas--Rachford algorithm for nonmonotone multioperator inclusion problems

TL;DR

This work extends the Douglas–Rachford algorithm to multioperator inclusion problems by introducing product-space reformulations that reduce the problem to a two-operator setting, enabling convergence analysis under generalized monotonicity when . It develops warped-resolvent machinery and two reformulations (Campoy-style and a flexible weighted version) to handle cases without convex-valuedness and without strict monotonicity assumptions. The authors establish convergence of the DR shadow sequence to fixed points corresponding to zeros of the sum, provide error rates , and extend the framework to structured nonconvex optimization, including sums of weakly/strongly convex functions and finite-dimensional problems with Lipschitz gradients. Practical results illustrate how function ordering and the chosen weights affect convergence speed and accuracy, highlighting potential guidelines for algorithm design in nonconvex settings.

Abstract

The Douglas--Rachford algorithm is a classic splitting method for finding a zero of the sum of two maximal monotone operators. It has also been applied to settings that involve one weakly and one strongly monotone operator. In this work, we extend the Douglas--Rachford algorithm to address multioperator inclusion problems involving () weakly and strongly monotone operators, reformulated as a two-operator inclusion in a product space. By selecting appropriate parameters, we establish the convergence of the algorithm to a fixed point, from which solutions can be extracted. Furthermore, we illustrate its applicability to sum-of--functions minimization problems characterized by weakly convex and strongly convex functions. For general nonconvex problems in finite-dimensional spaces, comprising Lipschitz continuously differentiable functions and a proper closed function, we provide global subsequential convergence guarantees.
Paper Structure (22 sections, 28 theorems, 114 equations, 2 figures, 1 table, 2 algorithms)

This paper contains 22 sections, 28 theorems, 114 equations, 2 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Contributions of this work
  3. Organization of the paper
  4. Preliminaries
  5. Set-valued operators
  6. Generalized monotone operators
  7. Extended real-valued functions
  8. A general product space reformulation and the Douglas-Rachford algorithm
  9. Campoy's product space reformulation
  10. A product space reformulation without convex-valuedness
  11. Douglas-Rachford Algorithm
  12. Douglas-Rachford algorithm for inclusion problems under generalized monotonicity
  13. Further properties under generalized monotonicity
  14. Convergence results
  15. DR for structured classes of nonconvex optimization problems
  16. ...and 7 more sections

Key Result

Lemma 2.2

Let $A,B:\mathcal{H}\rightrightarrows \mathcal{H}$ be maximal monotone operators. Then the following holds

Figures (2)

  • Figure 1: Convergence of $k{\left\|{\mathrm{Res}(\mathbf{z}^k)}\right\|}_{\infty,F}^2$ to zero for the orderings $(1,2,3,4)$, $(1,2,4,3)$, and $(1,4,3,2)$, with equal weights $\lambda_1=\lambda_2=\lambda_3=\frac{1}{3}$.
  • Figure 2: Heatmaps over $(\lambda_1,\lambda_2)$ (with $\lambda_3=1-\lambda_1-\lambda_2$) for three orderings. Color scales are shared across columns for each metric. Black squares indicate weight triples achieving the minimum.

Theorems & Definitions (61)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Theorem 3.1
  • Example 3.2
  • Theorem 3.3
  • ...and 51 more