mlsanityData to 7 Oct 2026

Generalized monotone operators and their averaged resolvents

arXiv 1902.09827 ↗First posted 26 Feb 2019math.OCcited by 27

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Abstract (arXiv)

The correspondence between the monotonicity of a (possibly) set-valued operator and the firm nonexpansiveness of its resolvent is a key ingredient in the convergence analysis of many optimization algorithms. Firmly nonexpansive operators form a proper subclass of the more general - but still pleasant from an algorithmic perspective - class of averaged operators. In this paper, we introduce the new notion of conically nonexpansive operators which generalize nonexpansive mappings. We characterize averaged operators as being resolvents of comonotone operators under appropriate scaling. As a consequence, we characterize the proximal point mappings associated with hypoconvex functions as cocoercive operators, or equivalently; as displacement mappings of conically nonexpansive operators. Several examples illustrate our analysis and demonstrate tightness of our results.

Key facts. Cited by 27 papers in the map (data through 7 Oct 2026). First posted on arXiv 26 Feb 2019.

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Citing papers per year: 2021: 1; 2022: 5; 2023: 9; 2024: 5; 2025: 3; 2026: 4.

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  1. 2022ProxSkip: Yes! Local Gradient Steps Provably Lead to Communication Acceleration! Finally!cited by 134
  2. 2022Accelerated Single-Call Methods for Constrained Min-Max Optimizationcited by 31
  3. 2023Solving stochastic weak Minty variational inequalities without increasing batch sizecited by 21
  4. 2023Escaping limit cycles: Global convergence for constrained nonconvex-nonconcave minimax problemscited by 21
  5. 2021Halpern-Type Accelerated and Splitting Algorithms For Monotone Inclusionscited by 20
  6. 2022Accelerated Algorithms for Constrained Nonconvex-Nonconcave Min-Max Optimization and Comonotone Inclusioncited by 19
  7. 2023On the Convergence of No-Regret Learning Dynamics in Time-Varying Gamescited by 14
  8. 2023Single-Call Stochastic Extragradient Methods for Structured Non-monotone Variational Inequalities: Improved Analysis under Weaker Conditionscited by 14
  9. 2022Clipped Stochastic Methods for Variational Inequalities with Heavy-Tailed Noisecited by 13
  10. 2022Beyond the Golden Ratio for Variational Inequality Algorithmscited by 12
  11. 2024Stochastic Halpern iteration in normed spaces and applications to reinforcement learningcited by 11
  12. 2023Variance Reduced Halpern Iteration for Finite-Sum Monotone Inclusionscited by 10
15 more
  1. 2024COMAL: A Convergent Meta-Algorithm for Aligning LLMs with General Preferencescited by 8
  2. 2023Sublinear Convergence Rates of Extragradient-Type Methods: A Survey on Classical and Recent Developmentscited by 5
  3. 2025Expected Variational Inequalitiescited by 5
  4. 2025Min-Max Optimisation for Nonconvex-Nonconcave Functions Using a Random Zeroth-Order Extragradient Algorithmcited by 5
  5. 2024Revisiting Inexact Fixed-Point Iterations for Min-Max Problems: Stochasticity and Structured Nonconvexitycited by 4
  6. 2023Stable Nonconvex-Nonconcave Training via Linear Interpolationcited by 3
  7. 2026Solving Stochastic Variational Inequalities without the Bounded Variance Assumptioncited by 3
  8. 2024Variance-Reduced Fast Krasnoselkii-Mann Methods for Finite-Sum Root-Finding Problemscited by 2
  9. 2026Unbiased and Biased Variance-Reduced Forward-Reflected-Backward Splitting Methods for Stochastic Composite Inclusionscited by 2
  10. 2023Convergence of the Chambolle-Pock Algorithm in the Absence of Monotonicitycited by 1
  11. 2025Nonconvex Regularization for Feature Selection in Reinforcement Learningcited by 1
  12. 2023Extragradient-Type Methods with O (1/k) Last-Iterate Convergence Rates for Co-Hypomonotone Inclusionscited by 0
  13. 2024HPPP: Halpern-type Preconditioned Proximal Point Algorithms and Applications to Image Restorationcited by 0
  14. 2026RAMPAGE: RAndomized Mid-Point for debiAsed Gradient Extrapolationcited by 0
  15. 2026Non-Convex Sparse Reinforcement Learning via Non-Monotone Inclusionscited by 0