Stability of primal–dual gradient dynamics and applications to network optimization

Convergence of Caratheodory solutions for primal-dual dynamics in constrained concave optimization

“…One way of tackling this problem, see e.g., [8], is to interpret the dynamics as a state-dependent switched system, formulate the latter as a hybrid automaton as defined in [15], and then use the invariance principle for hybrid automata to characterize its asymptotic convergence properties. However, this route is not valid in general because one of the key assumptions required by the invariance principle for hybrid automata is not satisfied by the primal-dual dynamics.…”

“…The first contribution of this paper is an example that illustrates this point. Our second contribution is an alternative proof strategy that arrives at the same convergence results of [8]. We consider an inequality constrained concave optimization problem described by continuously differentiable functions with locally Lipschitz gradients.…”

“…In the case when one deals with inequality constraints, the primal-dual dynamics requires a projection in the dual variables which is not necessary in the case of equality constraints. Such dynamics, first studied in the classic works [1,12], have found numerous applications in recent times, particularly in network resource allocation problems [8,5,9], network optimization [10,18,20], and in distributed stabilization and optimization of power networks [16,24,17,23]. These network optimization problems have aggregate objective functions and constraints that can be expressed locally which together * A journal version of this paper has been submitted to Systems and Control Letters.…”

“…Various instances of this argument are also combined to provide a global convergence result. Instead, [8] takes an indirect approach and characterizes convergence using classical notions such as invariant sets and LaSalle functions. This work models the primal-dual dynamics as a hybrid automaton, as defined in [15], and employs a LaSalle Invariance Principle for hybrid automata to show the asymptotic convergence of the solutions.…”

Averaging approach to distributed convex optimization for continuous-time multi-agent systems

“…Most of the available algorithms, such as the widely used distributed algorithms based on subgradient [14] and projected subgradient [15], are developed in discrete-time mainly due to the overwhelming ability of digital computers to execute the algorithms discretely. Recently, more and more distributed convex optimization algorithms are explored in continuous-time since continuous-time set up is favored for utilizing more techniques (the elegant Lyapunov argument in [4] for example) to prove the algorithm convergence, and is beneficial for adopting differential geometry viewpoint which is extremely powerful when the optimization is constrained (see for example [21]). …”