Feedback Stabilization Methods for the Solution of Nonlinear Programming Problems

• Published in: Journal of optimization theory and applications Vol. 161; no. 3; pp. 783 - 806
• Main Author: Karafyllis, Iasson
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.06.2014; Springer Nature B.V
• Subjects: Algorithms; Applications of Mathematics; Calculus of Variations and Optimal Control; Optimization; Construction; Control systems; Control theory; Critical point; Dynamical systems; Engineering; Equilibrium; Feedback; Linear programming; Lyapunov functions; Mathematical analysis; Mathematical programming; Mathematics; Mathematics and Statistics; Neural networks; Nonlinear programming; Operations Research/Decision Theory; Optimization; Ordinary differential equations; Studies; Theory of Computation; Lyapunov functions; Nonlinear programming; Feedback stabilization; Nonlinear systems
• ISSN: 0022-3239; 1573-2878
• DOI: 10.1007/s10957-013-0459-5

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.