A new accelerated algorithm with a linesearch technique for convex bilevel optimization problems with applications

We considered a convex bilevel optimization problem when the outer level problem was to find a minimizer of a strongly convex function over the set of solutions of the inner level problem which was in the form of minimization of the sum of a convex differentiable function and a nonsmooth convex function. In this work, we proposed a novel accelerated algorithm by employing both linesearch and inertial techniques for solving a convex bilevel optimization problem. Then, we proved the strong convergence of the sequence generated by our proposed algorithm to an optimal solution of the convex bilevel optimization problems without the continuity assumption on the gradient of the objective function. Moreover, we presented the convergence behavior of the proposed method by some numerical experiments addressing image restoration problems and data classification problems with least squares constraints. Finally, the performances of the restorative image and the data classification of the proposed method were compared with other existing algorithms in the literature. According to the experiment, our proposed algorithm had a better convergence behavior than the others in the literature.