A decentralized Second-Order Method for Dynamic Optimization

This paper considers decentralized dynamic optimization problems where nodes of a network try to minimize a sequence of time-varying objective functions in a real-time scheme. At each time slot, nodes have access to different summands of an instantaneous global objective function and they are allowed to exchange information only with their neighbors. This paper develops the application of the Exact Second-Order Method (ESOM) to solve the dynamic optimization problem in a decentralized manner. The proposed dynamic ESOM algorithm operates by primal descending and dual ascending on a quadratic approximation of an augmented Lagrangian of the instantaneous consensus optimization problem. The convergence analysis of dynamic ESOM indicates that a Lyapunov function of the sequence of primal and dual errors converges linearly to an error bound when the local functions are strongly convex and have Lipschitz continuous gradients. Numerical results demonstrate the claim that the sequence of iterates generated by the proposed method is able to track the sequence of optimal arguments.