Fixed-Point Bundle Method for Variational Inequalities and Game Equilibria

This repository contains the experiment code and results for the fixed-point bundle method for solving variational inequalities (VIs) and game equilibrium problems.

• vi_numpy.py: implementation for the paper titled “A path-following framework on fiber bundle for variational inequalities” (available at arXiv:2606.00778 and Optimization Online).
• game_numpy.py: implementation for the paper titled “The fixed-point bundle method over product-of-simplex domains arising from game equilibria” (available at arXiv:2609.16158 and Optimization Online).
• main.py: experiment entry points.
• data/: experiment results.

main.py implements two operator models:

• the neural network representation nn_F models general real-analytic VI operators;
• a payoff tensor representation payoff_F models the VI operator for general finite normal-form games.

• test_vi calls vi_numpy.py to solve a VI over a simplex domain with a neural network operator.
• test_game calls game_numpy.py to solve a VI over a product-of-simplex domain with either a neural network operator or a payoff tensor operator.
• group_test runs test_vi or test_game in parallel using multiprocessing, where the number of worker processes is specified by process_num. It is intended for solving multiple VIs or games simultaneously.

The solvers are based on a predictor-corrector framework. vi_numpy.py and game_numpy.py support two corrector options, which are selectable inside the corrector_comp functions. See the papers for a detailed discussion of the differences between the two correctors.

The current implementation uses a naive navigation of the fixed-point bundle, with the sole goal of finding at least one solution for each input VI instance.

• The current singularity-avoidance mechanism uses the simplest stateless or randomized strategy to produce a step along the fiber. For complex problem instances, adaptively determining the singularity-avoidance step based on the algorithm state could further improve performance.
• Some solutions cannot be reached by the current navigation. If convergence to a specific solution is required, the decrease_only option in predictor_corrector can help reach any solution; however, there is no convergence guarantee if decrease_only=True at every step.

If you use this code, please cite the corresponding papers:

@misc{sun2026pathfollowingframeworkfiberbundle,
      title={A path-following framework on fiber bundle for variational inequalities}, 
      author={Hongbo Sun},
      year={2026},
      eprint={2606.00778},
      archivePrefix={arXiv},
      primaryClass={math.OC},
      url={https://arxiv.org/abs/2606.00778}, 
}

@misc{sun2026fixedpointbundlemethodproductofsimplex,
      title={The fixed-point bundle method over product-of-simplex domains arising from game equilibria}, 
      author={Hongbo Sun},
      year={2026},
      eprint={2609.16158},
      archivePrefix={arXiv},
      primaryClass={math.OC},
      url={https://arxiv.org/abs/2609.16158}, 
}