Control Barrier Function Based Design of Gradient Flows for Constrained Nonlinear Programming

Abstract

This paper considers the problem of designing a continuous time dynamical system to solve constrained nonlinear optimization problems such that the feasible set is forward invariant and asymptotically stable. The invariance of the feasible set makes the dynamics anytime, when viewed as an algorithm, meaning that it is guaranteed to return a feasible solution regardless of when it is terminated. The system is obtained by augmenting the gradient flow of the objective function with inputs, then designing a feedback controller to keep the state evolution within the feasible set using techniques from the theory of control barrier functions. The equilibria of the system correspond exactly to critical points of the optimization problem. Since the state of the system corresponds to the primal optimizer, and the steady-state input at equilibria corresponds to the dual optimizer, the method can be interpreted as a primal-dual approach. The resulting closed-loop system is locally Lipschitz continuous, so classical solutions to the system exist. We characterize conditions under which local minimizers are Lyapunov stable, drawing connections between various constraint qualification conditions and the stability of the local minimizer. The algorithm is compared to other continuous time methods for optimization.

Sampled-Data Online Feedback Equilibrium Seeking: Stability and Tracking

“…Multiple iterations (mostly inspired by iterative algorithms in nonlinear optimization) serve this purpose, including • (projected) gradient iterations [6], [31] • primal-dual saddle-point dynamics [6], [9], [32] • safe gradient flows [33] • regularized primal-dual iterations [10] • quasi-Newton flows [34] • sequential convex programming [22] • and others. We refer to [22], for a general framework encompassing most of the algorithms listed above.…”