A Finite-Time Protocol for Distributed Continuous-Time Optimization of Sum of Locally Coupled Strictly Convex Functions

In this paper we study a distributed optimization problem for continuous time multi-agent systems. In our setting, the global objective for the multi-agent system is to minimize the sum of locally coupled strictly convex cost functions. Notably, this class of optimization objectives can be used to encode several important problems such as distributed estimation. For this problem setting, we propose a distributed signed gradient descent algorithm, which relies on local observers to retrieve 2-hop state information that are required to compute the descent direction. Adaptive gains for the local observer are introduced to render the convergence independent from: i) the structure of the network topology and ii) the local gains of the per-agent signed gradient-descent update law. The finite-time convergence of the local observer and of the proposed signed gradient descent method is demonstrated. Numerical simulations involving a distributed weighted least-square (WLS) estimation problem, with the aim of identifying in the context of an advanced water management system for precision-farming the soil thermal properties in a large-scale hazelnut orchard, have been proposed to corroborate the theoretical findings.