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FIXP-completeness for exact variational inequality problems

Establish FIXP-completeness for computational formulations of the exact versions of variational inequality problems, such as generalized quasi-variational inequalities, quasi-variational inequalities, and variational inequalities, under appropriate formulations that parallel the approximate versions analyzed in the paper.

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Background

The paper proves PPAD-completeness for approximate computational formulations of generalized quasi-variational inequalities (GQVI), quasi-variational inequalities (QVI), and variational inequalities (VI), using separation-oracle representations of convex sets/correspondences and linear arithmetic circuits for functions.

In the conclusion, the authors point to exact (rather than approximate) versions of these problems and posit that they should admit FIXP-completeness results. FIXP captures the complexity of finding fixed points of algebraic circuits; establishing FIXP-completeness for exact variational inequality formulations would extend the work from approximate (PPAD) to exact (FIXP) settings, aligning with related developments cited in the literature.

References

We have not considered exact versions of variational inequality problems, but conjecture that it should be possible to prove FIXP-completeness for appropriate formulations (see ).

— The Computational Complexity of Variational Inequalities and Applications in Game Theory (2411.04392 - Kapron et al., 7 Nov 2024) in Conclusion and Future Work (Section 6)