Residual Feedback for Transformed-State Equilibrium Seeking via General Variational Inequalities

Abstract

Motivated by networked systems in which equilibrium conditions apply to a regulated state rather than directly to the decision variable, we study transformed-state general variational inequalities. Using a projection-residual reformulation, we develop residual feedback methods that operate in the decision space without inverting the state mapping H. For a one-step method, we establish an 𝒪(T^−1/2) best-iterate residual rate under solution-restricted cocoercivity and linear convergence under solution-restricted strong monotonicity and residual Lipschitz continuity. We also obtain an 𝒪(T^−1/2) best-iterate residual rate for a predictor-corrector method applied to the induced GVI residual under monotonicity and Lipschitz continuity. The assumptions are imposed directly on the computable residual and are illustrated by affine network and rank-deficient examples. Finally, we extend the residual framework to generalized quasi-variational inequalities with decision-dependent feasible sets.

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BibTeX RIS

BibTeX

@misc{smith2026residual,
  title = {{Residual Feedback for Transformed-State Equilibrium Seeking via General Variational Inequalities}},
  author = {Griffin Smith and Afrooz Jalilzadeh},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.36286},
  eprint = {2609.36286},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.36286},
}

RIS

TY  - UNPB
TI  - Residual Feedback for Transformed-State Equilibrium Seeking via General Variational Inequalities
AU  - Griffin Smith
AU  - Afrooz Jalilzadeh
PY  - 2026
DA  - 2026/09/28
PB  - arXiv
DO  - 10.48550/arxiv.2609.36286
UR  - https://arxiv.org/abs/2609.36286
AB  - Motivated by networked systems in which equilibrium conditions apply to a regulated state rather than directly to the decision variable, we study transformed-state general variational inequalities. Using a projection-residual reformulation, we develop residual feedback methods that operate in the decision space without inverting the state mapping H. For a one-step method, we establish an 𝒪(T^−1/2) best-iterate residual rate under solution-restricted cocoercivity and linear convergence under solution-restricted strong monotonicity and residual Lipschitz continuity. We also obtain an 𝒪(T^−1/2) best-iterate residual rate for a predictor-corrector method applied to the induced GVI residual under monotonicity and Lipschitz continuity. The assumptions are imposed directly on the computable residual and are illustrated by affine network and rank-deficient examples. Finally, we extend the residual framework to generalized quasi-variational inequalities with decision-dependent feasible sets.
ER  -

Abstract

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