A Projection Anderson Acceleration Algorithm for Non-quasi-monotone Variational Inequalities with Hölder Continuous Terms

This paper proposes a Projection Anderson Acceleration (PAA) algorithm for variational inequality (VI) problems, where the underlying function is expressed as a finite sum of Hölder continuous functions and satisfies the $\sigma$-star strongly monotone condition that includes the star strong monotonicity with $\sigma =1$ as a special case. In the absence of quasi-monotonicity and Lipschitz continuity, we establish the evaluation complexity of the PAA algorithm for finding an approximate solution. Specifically, for the star strongly monotone VI problems, the PAA algorithm needs at most $O(\log ({\epsilon }^{-1}){\epsilon }^{2(\underset{‾}{\alpha }-1)})$ evaluations of the function and projection for finding an $\epsilon$-approximate solution, where $\underset{‾}{\alpha }\in (0,1]$ is the smallest exponent of H{ö}lder continuity of the component functions. We illustrate our theoretical results by numerical examples of non-quasi-monotone complementarity problems and non-monotone semilinear elliptic equations with non-Lipschitz terms.