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

Abstract

In this paper, we revisit primal-dual dynamics for convex optimization and present a generalization of the dynamics based on the concept of passivity. It is then proved 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 based on the passivity paradigm together with the invariance principle for Carathéodory systems. We then show that the present algorithm is also a generalization of existing augmented Lagrangian-based primal-dual dynamics, and discuss the benefit of the present generalization in terms of noise reduction and convergence speed.

Smooth Dynamics for Distributed Constrained Optimization With Heterogeneous Delays

“…We conclude this section by giving the definition of passivity [5]. Consider a system Σ described by a state model with state x ∈ R m , input u ∈ R n and output y ∈ R n .…”

“…When f i lacks strict convexity, an extra modification is needed. In this subsection, we add the phase lead compensator into the dynamics (4), which serves to provide stable zeros and avoid possible oscillations for the algorithm under general convexity [5].…”

“…Note that we only apply the phase lead compensator to (4a), and (8) is a simplified version of the algorithm in [5]. Then, the overall distributed algorithm becomeṡ…”

“…, ν N ) and ν * = ν(x * , ξ * , λ * , µ * ) = 0. First, we focus on subsystem (9a)-(9c) and obtain the following lemma whose proof can be found in [5,Lemma 7].…”

“…However, these methods are either restricted to affine constraints or not in a distributed structure. Recently, a phase lead compensation technique is adopted as a generalized method to ensure convergence [5].…”