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Control barrier function-based design of gradient flows for constrained nonlinear programming
A. Allibhoy, J. Cortés
IEEE Transactions on Automatic Control 69 (6) (2024), 3499-3514

Abstract

This paper considers the problem of designing a continuous-time dynamical system that solves constrained nonlinear optimization problems and makes the feasible set forward invariant and asymptotically stable. The invariance of the feasible set makes the dynamics anytime, when viewed as an algorithm, meaning that it returns a feasible solution regardless of when it is terminated. This makes the proposed dynamics an attractive option for interconnection with other dynamical processes. Our novel design approach augments the gradient flow of the objective function with inputs defined by the constraint functions, treats the feasible set as a safe set, and synthesizes a safe feedback controller using techniques from the theory of control barrier functions. The equilibria of the dynamics correspond exactly to critical points of the optimization problem. The resulting closed-loop system, termed safe gradient flow, is locally Lipschitz and can be viewed as a primal-dual flow, where the state corresponds to the primal variables and the inputs correspond to the dual ones. We provide a detailed suite of conditions based on constraint qualification under which (both isolated and nonisolated) local minimizers are asymptotically stable with respect to the feasible set and the whole state space. Comparisons with other continuous-time methods for optimization in a simple example illustrate the advantages of the safe gradient flow.

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