Global Dynamical Solvers for Nonlinear Programming Problems
This provides a theoretical framework for solving general nonlinear programming problems via dynamical systems, but the practical impact is unclear as no empirical results or comparisons are given.
The authors construct a family of globally defined dynamical systems for nonlinear programming problems that guarantee convergence to critical points from any initial condition, with strict local minima being locally asymptotically stable, without requiring convexity. The method is demonstrated with examples.
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.