A Mixed Extragradient Algorithm with Double Acceleration for Non-Lipschitz Bilevel Split Variational Inequality Problems

Résumé

In this paper, we study the problem of approximating solutions of the bilevel split variational inequality problem (BSVIP) in real Hilbert spaces. The operator associated with the upper-level problem is assumed to be strongly monotone and uniformly continuous, while the operators in the lower-level problem are quasimonotone and uniformly continuous. To address this problem, we propose a new algorithm obtained by combining the modified subgradient extragradient method with a modified Tseng extragradient method. In contrast to existing modified subgradient extragradient approaches for solving the BSVIP, the proposed method avoids the computation of projections onto two auxiliary half-spaces containing the feasibility sets, thereby reducing the computational complexity of each iteration. Moreover, the step-size rules employed in the algorithm are self-adaptive and do not require prior knowledge of the norm of the bounded linear operator or the Lipschitz constants of the involved mappings. Under mild assumptions on the control parameters, we establish strong convergence of the proposed scheme. The algorithm further incorporates two inertial extrapolation terms, which help accelerate the convergence process. Numerical experiments are provided to illustrate the effectiveness and practical advantages of the proposed method when compared with several existing algorithms. The obtained results extend, unify, and improve a number of previously known results in the literature.

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