Push-Sum Based Algorithm for Constrained Convex Optimization Problem and Its Potential Application in Smart Grid

Qian Xu, Zao Fu, Bo Zou, Hongzhe Liu, and Lei Wang

Dear Editor,

This letter aims to investigate the optimization problem where the decision variable is contained in a closed convex set. By combing the gradient decent-like method with the push-sum algorithm framework,we design the distributed iterative formulas under the condition that the considered graphs sequence is time-varying and unbalanced.Under some standard assumptions on the problem setting and the decaying step-sizes, we analyze the convergence property of the generated variables sequence under the proposed iterative formulas.

The distributed optimization has been widely and deeply researched during the recent decades due to its great potential application value in various practical fields. Accordingly, a great deal of excellent results with both the continuous-time [1]–[5] and discrete-time algorithms [6]–[16] developed have been obtained. Here, we only care about the results with discrete-time algorithms developed.

Fig. 1. Unbalanced graphs sequence with four agents.

Fig. 2. Behaviors of the variables xi(k) under iterative rules (2).