Distributed Continuous-Time Optimization: Nonuniform Gradient Gains, Finite-Time Convergence, and Convex Constraint Set

• Published in: IEEE transactions on automatic control Vol. 62; no. 5; pp. 2239 - 2253
• Main Authors: Peng Lin, Wei Ren, Farrell, Jay A.
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.05.2017; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Asymptotic properties; Consensus; Constraints; Continuous time systems; Convergence; Convex functions; convex set constraint; Distributed algorithms; distributed optimization; finite-time convergence; Heuristic algorithms; Linear programming; Multi-agent systems; Multiagent systems; nonuniform gradient gains; Optimization; Optimization algorithms
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2016.2604324

In this paper, a distributed optimization problem with general differentiable convex objective functions is studied for continuous-time multi-agent systems with single-integrator dynamics. The objective is for multiple agents to cooperatively optimize a team objective function formed by a sum of local objective functions with only local interaction and information while explicitly taking into account nonuniform gradient gains, finite-time convergence, and a common convex constraint set. First, a distributed nonsmooth algorithm is introduced for a special class of convex objective functions that have a quadratic-like form. It is shown that all agents reach a consensus in finite time while minimizing the team objective function asymptotically. Second, a distributed algorithm is presented for general differentiable convex objective functions, in which the interaction gains of each agent can be self-adjusted based on local states. A corresponding condition is then given to guarantee that all agents reach a consensus in finite time while minimizing the team objective function asymptotically. Third, a distributed optimization algorithm with state-dependent gradient gains is given for general differentiable convex objective functions. It is shown that the distributed continuous-time optimization problem can be solved even though the gradient gains are not identical. Fourth, a distributed tracking algorithm combined with a distributed estimation algorithm is given for general differentiable convex objective functions. It is shown that all agents reach a consensus while minimizing the team objective function in finite time. Fifth, as an extension of the previous results, a distributed constrained optimization algorithm with nonuniform gradient gains and a distributed constrained finite-time optimization algorithm are given. It is shown that both algorithms can be used to solve a distributed continuous-time optimization problem with a common convex constraint set. Numerical examples are included to illustrate the obtained theoretical results.