Feedback Stabilization Methods for the Solution of Nonlinear Programming Problems

In this work, we show that, given a nonlinear programming problem, it is possible to construct a family of dynamical systems, defined on the feasible set of the given problem, so that: (a) the equilibrium points are the unknown critical points of the problem, which are asymptotically stable, (b) each dynamical system admits the objective function of the problem as a Lyapunov function, and (c) explicit formulas are available without involving the unknown critical points of the problem. The construction of the family of dynamical systems is based on the Control Lyapunov Function methodology, which is used in mathematical control theory for the construction of stabilizing feedback. The knowledge of a dynamical system with the previously mentioned properties allows the construction of algorithms, which guarantee the global convergence to the set of the critical points.