Accelerated bregman operator splitting with backtracking
Yunmei Chen Xianqi Li Yuyuan Ouyang Eduardo Pasiliao
Inverse Problems & Imaging 2017, 11(6): 1047-1070 doi: 10.3934/ipi.2017048

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 [5] in terms of the smooth component in the objective function by incorporating Nesterov's multi-step acceleration scheme under the assumption that the feasible set is bounded. The second algorithm is capable of dealing with the case where the feasible set is unbounded. Moreover, it allows more aggressive stepsize than that in the first scheme by properly selecting the penalty parameter and jointly updating the acceleration parameter and stepsize. Both algorithms exhibit better practical performance than BOSVS and AADMM [21], while preserve the same accelerated rate of convergence as that for AADMM. The numerical results on total-variation based image reconstruction problems indicate the effectiveness of the proposed algorithms.

keywords: Bregman operator splitting accelerated ADMM convex optimization Barzilai-Borwein stepsize backtracking total variation image reconstruction
On the hierarchical optimal control of a chain of distributed systems
Getachew K. Befekadu Eduardo L. Pasiliao
Journal of Dynamics & Games 2015, 2(2): 187-199 doi: 10.3934/jdg.2015.2.187
We consider a chain of distributed systems governed by a degenerate parabolic equation, which satisfies a weak Hörmander type condition, with a control distributed over an open subdomain. In particular, we consider two objectives that we would like to accomplish. The first one being of a controllability type that consists of guaranteeing the terminal state to reach a target set starting from an initial condition; while the second one is keeping the state trajectory of the overall system close to a given reference trajectory over a finite time interval. We introduce the following framework. First, we partition the control subdomain into two disjoint open subdomains that are compatible with the strategy subspaces of the 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.
keywords: distributed systems Stackelberg's optimization. hierarchical systems Degenerate parabolic equations

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