Asymptotic convergence of constrained primal–dual dynamics

Abstract

This paper studies the asymptotic convergence properties of the primal-dual dynamics designed for solving constrained concave optimization problems using classical notions from stability analysis. We motivate the need for this study by providing an example that rules out the possibility of employing the invariance principle for hybrid automata to study asymptotic convergence. We understand the solutions of the primal-dual dynamics in the Caratheodory sense and characterize their existence, uniqueness, and continuity with respect to the initial condition. We use the invariance principle for discontinuous Caratheodory systems to establish that the primal-dual optimizers are globally asymptotically stable under the primal-dual dynamics and that each solution of the dynamics converges to an optimizer.

Port-Hamiltonian Formulation of the Gradient Method Applied to Smart Grids

“…Hence on the largest invariant set S, λ must satisfy A T (λ−λ) = 0 which implies λ =λ. By applying a Caratheodory variant of LaSalle's invariance principle we conclude that (x, λ, µ) → Ω 1 as t → ∞ (Cherukuri et al (2015)). …”

Global exponential stability of primal-dual gradient flow dynamics based on the proximal augmented Lagrangian

“…where ∂g is the subgradient of g. The following lemma establishes the relation betweenw and the optimality conditions (7); see [10] for details.…”

“…This Lyapunov function allows us to derive a worst-case lower bound on the exponential decay rate. In contrast to [1], [7], [8], [15], our gradient flow dynamics are projection-free and there are no nonsmooth terms in the Lyapunov function.…”

“…For a projected variant of the primaldual dynamics that could account for inequality constraints, a Krasovskii-based Lyapunov function was combined with LaSalle's invariance principle to show the global asymptotic stability in [1]. The invariance principle was also specialized to discontinuous Carathéodory systems and a quadratic Lyapunov function was used to show the global asymptotic stability of projected primal-dual gradient flow dynamics under globally-strict or locally-strong convexity-concavity assumptions [7], [8]. Additional information about the utility Financial support from the National Science Foundation under awards ECCS-1708906 and ECCS-1809833 is gratefully acknowledged.…”

Distributed Optimization Decomposition for Joint Economic Dispatch and Frequency Regulation

“…The proof of Proposition 2 follows from [14] and uses the machinery developed in [30] to handle projections (10b) -(10c). By substituting the line flows v s (t) = BC θ s (t) into (8) and eliminating θ s (t), we can show that the entire system (8) - (10) is a primal-dual algorithm of F R (see [14,Theorem 5]).…”