An Adaptive Primal-Dual Subgradient Algorithm for Online Distributed Constrained Optimization

Deming Yuan, Daniel W. C. Ho, Guoping Jiang

Abstract: In this paper, we consider the problem of solving distributed constrained optimization over a multiagent network that consists of multiple interacting nodes in online setting, where the objective functions of nodes are time-varying and the constraint set is characterized by an inequality. Through introducing a regularized convex-concave function, we present a consensus-based adaptive primal-dual subgradient algorithm that removes the need for knowing the total number of iterations T in advance. We show that the proposed algorithm attains an O(T <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2+c</sup> ) [where c ∈ (0, 1/2)] regret bound and an O(T <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1-c/2</sup> ) bound on the violation of constraints; in addition, we show an improvement to an O(T <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sup> ) regret bound when the objective functions are strongly convex. The proposed algorithm allows a novel tradeoffs between the regret and the violation of constraints. Finally, a numerical example is provided to illustrate the effectiveness of the algorithm.