Solving Non-Monotone Inclusions Using Monotonicity of Pairs of Operators

TL;DR

This work addresses solving non-monotone inclusions $0 \in F(x)$ in a Hilbert space by exploiting the monotonicity of pairs $(F,v)$. It introduces Generalized Proximal Point Algorithms (GPPA) built on warped resolvents $J_{\gamma F}^v$ and transformed resolvents $T_{F}^v$, establishing weak, strong, and linear convergence under mild assumptions. A key contribution is showing that $T_{\gamma F}^v$ is single-valued and firmly nonexpansive when $(F,v)$ is monotone, enabling fixed-point iterations that converge to zeros of $F$; the paper also provides robust convergence results for GPPA1/GPPA2 and presents practical quadratic programming applications. The transformed resolvent offers advantages when $v^{-1}$ is not well-behaved, and the results extend proximal-point-type methods to structured non-monotone problems, with implications for large-scale nonconvex optimization.

Abstract

In this paper, under the monotonicity of pairs of operators, we propose some Generalized Proximal Point Algorithms to solve non-monotone inclusions using warped resolvents and transformed resolvents. The weak, strong, and linear convergence of the proposed algorithms are established under very mild conditions.

Theorems & Definitions (24)

• Example 2.1
• Example 2.2
• Example 2.3
• Example 2.4
• Lemma 2.5: Opial's Lemma
• Definition 3.1
• Proposition 3.2
• Remark 3.3
• Proposition 3.4
• Remark 3.5