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

TL;DR

This paper introduces a control barrier function-based method to design safe gradient flows for constrained nonlinear optimization, ensuring feasibility and stability of solutions in continuous time.

Contribution

It develops a novel safe gradient flow approach that guarantees feasible set invariance and stability using control barrier functions, combining primal-dual dynamics.

Findings

The safe gradient flow ensures feasible solutions at all times.

Local minimizers are stable under certain constraint qualifications.

Compared to other methods, it offers advantages in stability and feasibility.

Abstract

This paper considers the problem of designing a continuous-time dynamical system that solves a constrained nonlinear optimization problem 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 it returns a feasible solution regardless of when it is terminated. Our 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 resulting closed-loop system, termed safe gradient flow, 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 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.