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Iterative Splitting Methods for Stochastic Dynamic SVIs

Saeed Hashemi Sababe, Ehsan Lotfali Ghasab

TL;DR

This work develops iterative, split-variational-inclusion algorithms for dynamic, stochastic, and multi-agent SVIs in Banach spaces, relaxing traditional monotonicity requirements. By leveraging resolvent-based updates and self-adaptive schemes, the authors establish weak and, under stronger assumptions, strong convergence results for three related problems: dynamic SVI, stochastic SVI, and coupled SVI. Theoretical guarantees are complemented by numerical experiments in resource allocation under uncertainty, time-varying operators, and multi-agent equilibrium scenarios, illustrating robustness to noise and time variation. The proposed framework broadens the applicability of SVIs to non-static, uncertain, and decentralized settings with practical convergence and error-analysis insights.

Abstract

This paper extends split variational inclusion problems to dynamic, stochastic, and multi-agent systems in Banach spaces. We propose novel iterative algorithms to handle stochastic noise, time-varying operators, and coupled variational inclusions. Leveraging advanced splitting techniques and self-adaptive rules, we establish weak convergence under minimal assumptions on operator monotonicity. Numerical experiments demonstrate the efficacy of our algorithms, particularly in resource allocation and optimization under uncertainty.

Iterative Splitting Methods for Stochastic Dynamic SVIs

TL;DR

This work develops iterative, split-variational-inclusion algorithms for dynamic, stochastic, and multi-agent SVIs in Banach spaces, relaxing traditional monotonicity requirements. By leveraging resolvent-based updates and self-adaptive schemes, the authors establish weak and, under stronger assumptions, strong convergence results for three related problems: dynamic SVI, stochastic SVI, and coupled SVI. Theoretical guarantees are complemented by numerical experiments in resource allocation under uncertainty, time-varying operators, and multi-agent equilibrium scenarios, illustrating robustness to noise and time variation. The proposed framework broadens the applicability of SVIs to non-static, uncertain, and decentralized settings with practical convergence and error-analysis insights.

Abstract

This paper extends split variational inclusion problems to dynamic, stochastic, and multi-agent systems in Banach spaces. We propose novel iterative algorithms to handle stochastic noise, time-varying operators, and coupled variational inclusions. Leveraging advanced splitting techniques and self-adaptive rules, we establish weak convergence under minimal assumptions on operator monotonicity. Numerical experiments demonstrate the efficacy of our algorithms, particularly in resource allocation and optimization under uncertainty.
Paper Structure (11 sections, 14 theorems, 91 equations, 3 figures, 3 algorithms)

This paper contains 11 sections, 14 theorems, 91 equations, 3 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proposed Algorithms
  4. Weak Convergence Analysis
  5. Strong Convergence Analysis
  6. Error Analysis
  7. Numerical Experiments
  8. Results:
  9. Results:
  10. Results:
  11. Conclusion

Key Result

Lemma 2.7

For a maximal monotone operator $\mathcal{T}: \mathcal{B} \to \mathcal{B}^*$ and $\gamma > 0$, the resolvent operator $J_\mathcal{T}^\gamma$ satisfies:

Figures (3)

  • Figure 1: Convergence of the dynamic SVI algorithm.
  • Figure 2: Convergence of the stochastic SVI algorithm.
  • Figure 3: Convergence of the multi-agent coupled SVI algorithm.

Theorems & Definitions (29)

  • Definition 2.1: Monotone Operator 4
  • Definition 2.2: Maximal Monotone Operator 4
  • Definition 2.3: Resolvent Operator 4
  • Definition 2.4: Stochastic Operator 2
  • Definition 2.5: Dynamic Operator 3
  • Definition 2.6: Coupled Variational Inclusion 5
  • Lemma 2.7: Resolvent Properties 4
  • Theorem 2.8: Existence of Solutions for Coupled SVIs 7
  • Theorem 2.9: Convergence of Iterative Methods for Dynamic SVIs 3
  • Theorem 2.10: Stochastic Convergence 2
  • ...and 19 more