Convergence of Primal-Dual Gradient Dynamics-Based Algorithm for Non-Strictly Convex Objective Functions

Due to its scalability in large-scale systems and potential for distributed implementation, the primal-dual gradient dynamics (PDGD)-based algorithm has been widely used in various fields where optimization problems have strictly or strongly convex objective functions. By contrast, optimization problems involving convex but non-strictly convex objective functions are more ubiquitous in applications. However, the algorithm’s convergence is only established for solving affine equality constrained optimization problems that have non-strictly convex objective functions, which may limit the algorithm’s further application. In view of this issue, this paper studies the PDGD-based algorithm’s convergence for solving optimization problems that have non-strictly convex objective functions and affine inequality constraints. Firstly, the optimality analysis is conducted based on the Karush-Kuhn-Tucker conditions. Secondly, the asymptotic convergence of the algorithm is established by using the invariance principle of the discontinuous Carathéodory system. Finally, numerical simulations are employed to verify the effectiveness of the algorithm.