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This paper develops two accelerated Bregman Operator Splitting (BOS) algorithms with backtracking for solving regularized large-scale linear inverse problems, where the regularization term may not be smooth. The first algorithm improves the rate of convergence for BOSVS [

*leader*and that of the

*follower*, respectively. Then, using the notion of Stackelberg's optimization (which is a hierarchical optimization framework), we provide a new result on the existence of optimal control strategies for such an optimization problem, where the

*follower*(which corresponds to the second criterion) is required to respond optimally, in the sense of

*best-response correspondence*to the strategy of the

*leader*(which is associated to the controllability-type problem) so as to achieve the overall objectives. Finally, we remark on the implication of our result in assessing the influence of the target set on the strategy of the

*follower*with respect to the direction of

*leader-follower*(and vice-versa) information flow.

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