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

TL;DR

The paper develops a safe gradient flow for constrained nonlinear programming by marrying gradient descent with control barrier function-based safety to keep the feasible set forward invariant. It offers two equivalent viewpoints: (i) a safety-driven controller augmenting the gradient flow, and (ii) a continuous approximation of the projected gradient flow that converges to the tangent projection as a parameter grows, revealing a primal-dual structure via Lagrange multipliers. The authors prove local Lipschitz well-posedness, identify equilibria with KKT points, and establish stability results: isolated local minimizers are asymptotically stable (relative to the feasible set and globally under stronger qualifications) and exponentially stable under LICQ with strict complementarity and second-order sufficiency, while nonisolated minimizers are semistable under analytic assumptions. Global convergence to KKT points is shown under real-analytic data and bounded feasibility, with a lower bound on the design parameter ensuring convergence; the framework is contrasted with barrier, penalty, and projected-dynamics methods, highlighting invariance, smoothness, and applicability to infeasible initial conditions.

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.

Figures (3)

• Figure 1: Intuition behind the design of the safe gradient flow. Grey lines are the level curves of the objective function and the shaded region is $\mathcal{C}$. In (a), the initial condition is $x_0$ and the minimizer is $x^*$, with $-\nabla f(x)$ in black and $-\nabla g(x)$ in gray at both points. In (b), the dashed line is a trajectory of $\dot{x} = -\nabla f(x) - u \nabla g(x)$ starting from $x_0$. The black vectors are $-\nabla f(x)$, the gray vectors are $-u\nabla g(x)$, and the red vectors are $\dot{x}$. Deep in the interior of $\mathcal{C}$, one has $u \approx 0$, as following the gradient of $f$ does not jeopardize feasibility while minimizing it. As the trajectory approaches the boundary, $u$ increases to keep the trajectory in $\mathcal{C}$.
• Figure 2: Projected gradient flow versus continuous approximation. The solution of the projected gradient flow is in black and solutions of $\dot{x} = \mathcal{G}_\alpha(x)$ for varying values of $\alpha$ are in the colors corresponding to the colorbar. All solutions start from the same initial condition, marked by the black dot.
• Figure 3: Comparison of methods minimizing $f(x) = 0.25\lVert x\rVert^2 - 0.5x_1 + 0.25x_2$ subject to $x_2 \geq 0$ and $x_1 \leq x_2$ (see also AH-SB-GH-FD:21 for a comparison of additional methods). The blue-shaded region is the feasible set and the grey curves are level sets of the objective function. The initial condition is denoted by the purple dot, and the global minimizer is denoted by a blue dot. (a) The trajectory converges to the global minimizer, and the trajectory remains inside the feasible set for all time but it is nonsmooth. (b) The trajectory is smooth and remains inside the feasible set but does not converge to the global minimizer. However, by choosing $\mu$ small enough, the trajectory can be made to converge arbitrarily close to the minimizer. (c) The trajectory is smooth, but does not remain inside the feasible set or converge to the global minimizer. However, by choosing $\epsilon$ small enough, the trajectory can be made to converge arbitrarily close to the minimizer. (d) Initialized with $u(0) = 0$, the trajectory does not remain inside the feasible set, but it converges to the global minimum. (e) The trajectory is smooth, converges to the global minimizer, and remains inside the feasible set. However, this method may not be well-defined for nonconvex problems (f) The trajectory is smooth, converges to the global minimizer, and remains inside the feasible set.

Theorems & Definitions (58)

• Lemma II.1: Safe feedback control
• Lemma IV.1: Vector control barrier function for \ref{['eq:augment']}
• proof
• Remark IV.2: Connection with the Literature
• Remark IV.3: Inequality Constraints via Quadratic Slack Variables
• Proposition IV.4: $\mathcal{G}_{\alpha}$ approximates the projected gradient
• proof
• Lemma IV.5: Necessity of optimality conditions
• proof
• Proposition IV.6: Equivalence of two constructions of the safe gradient flow