  A dynamic method to solve the Non-linear Programming (NLP) problem with
Equality Constraints (ECs) and Inequality Constraints (IECs) is proposed.
Inspired by the Lyapunov continuous-time dynamics stability theory in the
control field, the optimal solution is analogized to the stable equilibrium
point of a finite-dimensional dynamic system and it is solved in an asymptotic
manner. Under the premise that the Karush-Kuhn-Tucker (KKT) optimality
condition exists, the Dynamic Optimization Equation (DOE), which has the same
dimension to that of the optimization parameter vector, is established and its
solution will converge to the optimal solution of the NLP globally with a
theoretical guarantee. Using the matrix pseudo-inverse, the DOE is valid even
without the linearly independent regularity requirement on the nonlinear
constraints. In addition, the analytic expressions of the Lagrange multipliers
and KKT multipliers, which adjoin the ECs and the IECs respectively during the
entire optimization process, are also derived. Via the proposed method, the NLP
may be transformed to the Initial-value Problem (IVP) to be solved, with mature
Ordinary Differential Equation (ODE) integration methods. Illustrative examples
are solved and it is shown that the dynamic method developed may produce the
right numerical solutions with high efficiency.

Optimization and Control