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Monotonicity of Pairs of Operators and Generalized Inertial Proximal Method

Ba Khiet Le, Zakaria Mazgouri, Michel Théra

TL;DR

The paper tackles non-monotone inclusions of the form $0 \in F(x)$ by exploiting the monotonicity of operator pairs $(F,v)$ and warped resolvents. It develops (i) constructive approaches to obtain monotone pairs in linear/quadratic settings and (ii) a Generalized Inertial Proximal Point Algorithm (GIPPA) with inertial terms that converges weakly, strongly, or linearly under mild assumptions; and it provides numerical experiments illustrating improved performance over GPPA. The results extend monotone operator theory to non-monotone inclusions and suggest robust operator-splitting schemes based on pair monotonicity for large-scale optimization. The approach unifies kernel design and inertial proximal methods, enabling scalable solutions to nonconvex or nonmonotone variational problems with practical impact.

Abstract

Monotonicity of pairs of operators is an extension of monotonicity of operators, which plays an important role in solving non-monotone inclusions. One of challenging problems in this new tool is how to design the associated mappings to obtain the monotone pairs. In this paper, we solve this problem and propose a Generalized Inertial Proximal Point Algorithm (GIPPA) using warped resolvents under the monotonicity of pairs. The weak, strong and linear convergence of the algorithm under some mild assumptions are established. We also provide numerical examples illustrating the implementability and effectiveness of the proposed method.

Monotonicity of Pairs of Operators and Generalized Inertial Proximal Method

TL;DR

The paper tackles non-monotone inclusions of the form by exploiting the monotonicity of operator pairs and warped resolvents. It develops (i) constructive approaches to obtain monotone pairs in linear/quadratic settings and (ii) a Generalized Inertial Proximal Point Algorithm (GIPPA) with inertial terms that converges weakly, strongly, or linearly under mild assumptions; and it provides numerical experiments illustrating improved performance over GPPA. The results extend monotone operator theory to non-monotone inclusions and suggest robust operator-splitting schemes based on pair monotonicity for large-scale optimization. The approach unifies kernel design and inertial proximal methods, enabling scalable solutions to nonconvex or nonmonotone variational problems with practical impact.

Abstract

Monotonicity of pairs of operators is an extension of monotonicity of operators, which plays an important role in solving non-monotone inclusions. One of challenging problems in this new tool is how to design the associated mappings to obtain the monotone pairs. In this paper, we solve this problem and propose a Generalized Inertial Proximal Point Algorithm (GIPPA) using warped resolvents under the monotonicity of pairs. The weak, strong and linear convergence of the algorithm under some mild assumptions are established. We also provide numerical examples illustrating the implementability and effectiveness of the proposed method.
Paper Structure (10 sections, 12 theorems, 37 equations, 3 figures)

This paper contains 10 sections, 12 theorems, 37 equations, 3 figures.

Table of Contents

  1. Keywords:
  2. Mathematics Subject Classification (MSC 2020):
  3. Introduction
  4. Mathematical Background and Notations
  5. Constructing Associated Mappings to Obtain Monotone Pairs
  6. Global Monotonicity
  7. Local Strong Monotonicity and Quasi-Newton's Algorithm
  8. Generalized Inertial Proximal Point Algorithm (GIPPA) and Convergence Analysis
  9. Numerical Examples
  10. Conclusion and Perspectives

Key Result

Lemma 2.5

For all $x,y\in\mathcal{H}$ and all $\beta\in\mathbb{R}$, the following identity holds:

Figures (3)

  • Figure 1: The rate of convergence of $\|x_n-x^*\|_2$ when $v=v_1$.
  • Figure 2: The iterate error $\|x_n-x^*\|_2$ when $v=v_2$
  • Figure 3: The rate of convergence of $\|x_n-x^*\|_2$.

Theorems & Definitions (21)

  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Lemma 2.8: Opial's Lemma Opial
  • Theorem 3.1
  • Remark 3.2
  • ...and 11 more