Global Dynamical Solvers for Nonlinear Programming Problems

TL;DR

This paper introduces a family of globally defined dynamical systems for nonlinear programming that guarantees convergence to critical points without convexity assumptions, using an extension of Control Lyapunov Function methodology.

Contribution

It constructs explicit dynamical systems for nonlinear programming with guaranteed convergence and stability properties, independent of convexity, based on an extended Control Lyapunov Function approach.

Findings

Solutions converge to critical points from any initial condition

Strict local minima are locally asymptotically stable

Feasible set is positively invariant

Abstract

We construct a family of globally defined dynamical systems for a nonlinear programming problem, such that: (a) the equilibrium points are the unknown (and sought) critical points of the problem, (b) for every initial condition, the solution of the corresponding initial value problem converges to the set of critical points, (c) every strict local minimum is locally asymptotically stable, (d) the feasible set is a positively invariant set, and (e) the dynamical system is given explicitly and does not involve the unknown critical points of the problem. No convexity assumption is employed. The construction of the family of dynamical systems is based on an extension of the Control Lyapunov Function methodology, which employs extensions of LaSalle's theorem and are of independent interest. Examples illustrate the obtained results.