Abstract
Recently, distributed convex optimization using a multiagent system has received much attention by many researchers. This problem is frequently approached by combing the consensus algorithms in the multiagent literature and the gradient algorithms in the convex optimization literature. Compared with unconstrained distributed optimization, the constrained case is more challenging, and it is usually tackled by the projected gradient method. However, the projected gradient algorithm involves projection nonlinearity and thus is hard to analyze. To avoid gradient projection, in this paper, we present a novel distributed convex optimization algorithm in continuous time by using mirror design. The resulting optimization dynamics is smooth without using gradient projection and is designed in a primal-dual framework, where the primal and dual dynamics are respectively aided by the mirror descent and the mirror ascent. As for the merit of mirror design in our paper, it avoids gradient projection in the optimization dynamics design and removes the difficulty of analyzing projection nonlinearity. Furthermore, the mirror base primal-dual optimization dynamics facilitates more convenience construction of Lyapunov functions in the stability analysis.
Highlights
Optimization is an important field in mathematics, and many engineering applications can be converted into optimization problems [1,2,3,4,5,6,7,8,9,10]
The modified primal dynamics facilitates the use of Bregman divergence and Frenchel coupling in the stability analysis, and the redesigned dual dynamics for the evolution of positive Lagrange multiplier does not include projection and reduces the complexity of convergence analysis
One popular method to deal with constraints is based on the projection, which causes the optimization dynamics to be nonsmooth
Summary
Optimization is an important field in mathematics, and many engineering applications can be converted into optimization problems [1,2,3,4,5,6,7,8,9,10]. We note that the dual dynamics designed in this way is nonsmooth due to the projection operator involved to keep the evolution of the Lagrangian multipliers within the nonnegative orphan, and it is difficult to analyze As another contribution of this paper, we pursue a novel line of designing the dual dynamics via the mirror descent method. The third superiority of our method in comparison with existing results lies in the fact that it avoids using gradient projection in the optimization algorithm design It removes the difficulty of analyzing the resulting nonsmooth optimization dynamics and makes the simulation easier. The frequently used primal-dual algorithm for the optimization problem in the existing literature is modified in our paper via the mirror descent method, giving rise to new primal and dual dynamics.
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