Distributed Unbalanced Optimization Design Over Nonidentical Constraints
Qing Huang, Yuan Fan, Songsong Cheng
Abstract
This paper addresses distributed constrained optimization problems involving strongly convex global objective functions represented as the sum of individual convex objective functions, and the corresponding constrained set is the intersection of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> nonidentical closed convex sets. To solve the problem, we introduce the distributed projected sub-gradient algorithm with a row-stochastic weight matrix over unbalanced digraphs. Moreover, based on the condition that the strong convexity of the global objective function and using a non-increasing step size, we analyze that this algorithm converges to the optimal solution with an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$O(\frac{1}{T})$</tex-math></inline-formula> convergence rate, like the centralized counterpart. Finally, we verify the accuracy of the theoretical analysis by examining simulation results.