Distributed Nonsmooth Nonconvex Optimization: A Multiagent Approach With Finite-Time Consensus

Distributed nonsmooth nonconvex optimization is prevalent in practical applications. However, the inherent nonsmoothness and nonconvexity of such problems pose significant challenges to the development of efficient optimization approaches. This article proposes a multiagent system with finite-time consensus for solving this problem. A smooth approximation technique is leveraged to handle the nonsmoothness in the problem, and a state-dependent gain function is incorporated into the proposed approach to handle the nonconvexity in constraints. The states of the multiagent system remain within their local feasible regions and reach consensus in a finite time. In addition, the states are proven to be convergent to the critical-point set of the problem under consideration. Furthermore, the states are proven to be convergent to a globally optimal solution, under the nonsmooth Polyak–Łojasiewicz condition or other generalized convexity conditions. The simulation results are elaborated to substantiate the effectiveness and viability of the proposed approach.