A Two-Inertial Forward–Backward Algorithm with Adaptive Line Search for Convex Bilevel Optimization and Applications

This paper proposes a two-inertial forward–backward algorithm with adaptive line search for solving convex bilevel optimization problems in real Hilbert spaces. The lower-level problem is reformulated as a fixed point problem associated with the forward–backward operator, while the upper-level objective is incorporated through a viscosity approximation framework. The proposed method employs adaptive line search to avoid requiring prior knowledge of a global Lipschitz constant and incorporates two-inertial extrapolation terms to improve practical performance. Under mild assumptions, we establish the strong convergence of the generated sequence to the unique viscosity-selected solution. Numerical experiments on image restoration, sparse signal recovery, digital twin optimization, and aerospace topology optimization illustrate the effectiveness of the proposed method and its competitive performance when compared with several recent bilevel optimization algorithms.