Passivity-based generalization of primal–dual dynamics for non-strictly convex cost functions

In this paper, we revisit primal–dual dynamics for convex optimization and present a generalization of the dynamics based on the concept of passivity. We hypothesize that supplying a stable zero to one of the integrators in the dynamics allows one to eliminate the assumption of strict convexity on the cost function. Then, we prove this hypothesis based on the passivity paradigm together with the invariance principle for Carathéodory systems. Additionally, we show that the presented algorithm is also a generalization of existing augmented Lagrangian-based primal–dual dynamics and discuss the benefit of the presented generalization in terms of noise reduction and convergence speed. Finally, the presented algorithm is demonstrated through simulation of lighting control in a building.