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Asymptotic convergence of constrained primal-dual dynamics
Ashish Cherukuri, Enrique Mallada, Jorge Cortes
TL;DR
Constrained primal-dual dynamics has discontinuities that complicate classical convergence analysis, and a hybrid-automaton invariance approach is not valid in general. The paper instead uses Caratheodory solutions, projected dynamical-system results, and an invariance principle for discontinuous systems to prove global asymptotic stability of primal-dual optimizers and convergence of every solution.
Problem
The paper addresses how to rigorously establish convergence for constrained primal-dual dynamics when its discontinuous right-hand side makes standard stability tools and hybrid-automaton invariance analysis unavailable in general.
Method
The authors formulate solutions in the Caratheodory sense, use projected dynamical systems to establish existence, uniqueness, and initial-condition continuity, and apply a discontinuous-system invariance principle.
Results
The primal-dual optimizers are globally asymptotically stable, and each solution converges to an optimizer.
Takeaways & Limitations
Classical stability tools provide a conceptually simple and versatile framework for analyzing the asymptotic convergence of these discontinuous dynamics.
Takeaways & Limitations
A hybrid-automaton proof strategy is not valid in general because the corresponding solutions lack continuity in the hybrid sense.
Abstract
from arXiv · showhide
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.
1. Introduction
The paper gives a rigorous stability-based convergence analysis for constrained primal-dual dynamics despite its discontinuous right-hand side. It replaces an invalid hybrid-automaton invariance route with Caratheodory-system tools and Lyapunov-oriented analysis.
- Motivation: Primal-dual dynamics is a continuous-time algorithm for finding primal and dual solutions of inequality-constrained convex or concave optimization problems.It has been applied to wireless network resource allocation and power-network stabilization and optimization.
- Motivation: The paper seeks a rigorous convergence analysis using classical notions from stability analysis.The authors emphasize the conceptual simplicity and versatility of Lyapunov-like methods.
- Motivation: Because the dynamics has a discontinuous right-hand side, standard Lyapunov and LaSalle results are not directly applicable.A hybrid-automaton approach is also invalid in general because the corresponding automaton need not be continuous, a key invariance-principle assumption.
- Approach: The paper characterizes solution behavior and uses the invariance principle for discontinuous Caratheodory systems to establish convergence.It analyzes existence, uniqueness, and continuity with respect to initial conditions before proving the asymptotic result.
- Results: Primal-dual optimizers are globally asymptotically stable, and every solution of the dynamics converges to an optimizer.These are the paper’s principal convergence conclusions.
2. Preliminaries
The preliminaries define Caratheodory solutions, invariance, omega-limit sets, and Lie derivatives for discontinuous systems, then introduce projected dynamical systems and their solution properties.
- Discontinuous dynamical systems: A Caratheodory solution is absolutely continuous and satisfies the differential equation almost everywhere on its time interval.This solution notion is used for discontinuous dynamical systems.
- Discontinuous dynamical systems: An invariant set contains every solution that starts within it for the entire subsequent evolution.Omega-limit sets capture limiting behavior of solutions defined on [0, ∞).
- Discontinuous dynamical systems: The invariance principle requires compactness and invariance, unique solutions, invariant omega-limit sets, and a nonpositive Lie derivative.Under these conditions, solutions converge to the largest invariant subset where the Lie derivative is zero.
- Projected dynamical systems: Projection onto a closed convex set is single-valued and 1-Lipschitz.The point projection minimizes Euclidean distance to the set.
- Projected dynamical systems: Projected dynamical systems restrict a vector field at the boundary so solutions remain in the constraint set.They coincide with the original vector field in the interior and are generally discontinuous.
- Projected dynamical systems: A Lipschitz vector field on a closed convex polyhedron yields globally defined unique projected-system solutions that depend continuously on initial conditions.Convergence of trajectories from converging initial conditions is uniform on every compact time interval.
3. Problem statement
The paper formulates constrained concave optimization through primal-dual dynamics and explains why hybrid-automaton invariance tools are unsuitable for its discontinuous behavior. It instead motivates Caratheodory solutions and classical stability analysis for studying convergence.
- Optimization problem: The problem considers strictly concave f and convex g with locally Lipschitz gradients, under Slater’s condition and zero duality gap.The primal-dual optimizers are represented by saddle points of the Lagrangian and satisfy the KKT conditions.
- Primal-dual dynamics: Caratheodory solutions are used for the discontinuous primal-dual dynamics, making equilibria equivalent to points satisfying the KKT conditions.This solution notion is adopted to connect the dynamics directly to optimization optimality conditions.
- Hybrid-automaton formulation: The associated hybrid automaton divides the state space into two modes according to whether the projection in the dual dynamics is active.Its domains, transitions, guards, and state-preserving reset map encode switching between projected and unprojected vector fields.
- Failure of hybrid continuity: Arbitrarily close initial conditions can produce executions whose first mode transitions occur at times that are not arbitrarily close, so the hybrid automaton is not continuous.One execution never switches modes, while a nearby execution switches from mode 2 to mode 1 and back to mode 2.
- Classical stability approach: Despite hybrid-automaton discontinuity, solutions of the primal-dual dynamics remain close when initialized sufficiently close, motivating continuity with respect to the initial condition in the continuous state.This property, together with existence and uniqueness, supports the paper’s classical stability-based convergence analysis.
4. Convergence analysis of primal-dual dynamics
The paper proves asymptotic convergence of constrained primal-dual dynamics by treating it as a projected dynamical system and applying an invariance principle for discontinuous Caratheodory systems. Solutions exist uniquely, depend continuously on initial conditions, and converge to primal-dual optimizers.
- Convergence strategy: The proof applies the invariance principle for discontinuous Caratheodory systems to establish asymptotic convergence.The required hypotheses include solution properties and invariance of omega-limit sets.
- Well-posedness: The primal-dual dynamics can be represented as a projected dynamical system, enabling existence, uniqueness, and continuity results.Monotonicity bounds solutions in suitable compact sublevel sets, where the vector field can be handled with Lipschitz arguments.
- Well-posedness: A unique global solution exists from every initial point and remains within a Lyapunov sublevel set.Solutions starting from convergent initial-condition sequences converge uniformly on every compact time interval.
- Main result: The set of primal-dual solutions is globally asymptotically stable, and every trajectory converges to a single optimizer.The singleton omega-limit conclusion follows from invariance and Lyapunov stability of primal-dual optimizers.
5. Conclusions
The paper establishes asymptotic convergence of Caratheodory solutions using classical stability tools and explains why a hybrid-automaton proof strategy is not generally valid. The approach also motivates future robustness analysis and extensions to other optimization problems.
- Conclusions: The paper establishes asymptotic convergence of Caratheodory solutions to a primal-dual optimizer using classical stability theory.The technical approach combines projected dynamical systems with an invariance principle for discontinuous Caratheodory systems.
- Conclusions: A counterexample shows that interpreting the dynamics as a hybrid automaton is not generally valid because hybrid-sense solution continuity can fail.This limitation rules out that proof strategy in general.
- Future directions: The approach opens the possibility of characterizing robustness against unmodeled dynamics, disturbances, and noise.The paper also identifies semidefinite and quadratically constrained quadratic programs as planned extension targets.