Distributed Continuous-Time Algorithms for Time-Varying Constrained Convex Optimization

Abstract

This paper is devoted to the distributed continuous-time optimization problems with time-varying objective functions and time-varying constraints. Different from most studied distributed optimization problems with time-invariant objective functions and constraints, the optimal solutions in this paper are time varying and form a trajectory. First, for the case where there exist only time-varying nonlinear inequality constraints, we present a distributed control algorithm that consists of a sliding-mode consensus part and a Hessian-based optimization part coupled with the log-barrier penalty functions. The algorithm can guarantee the asymptotical tracking of the optimal solution with a zero tracking error. Second, we extend the previous result to the case where there exist not only time-varying nonlinear inequality constraints but also linear equality constraints. An extended algorithm is presented, where quadratic penalty functions are introduced to account for the equality constraints and an adaptive control gain is designed to remove the restriction on knowing the upper bounds on certain information. The asymptotical convergence of the extended algorithm to the vicinity of the optimal solution is studied under suitable assumptions. The effectiveness of the proposed algorithms is illustrated in simulation. In addition, one proposed algorithm is applied to a multi-robot multi-target navigation problem with experimental demonstration on a multi-crazyflie platform to validate the theoretical results.