Transform Nonlinear Programming Problem to Initial-Value Problem to Solve, Part 1: Within the Feasible Region

A dynamic method to solve the general Nonlinear Programming (NLP) problem, inspired by the Lyapunov continuous-time dynamics stability theory in the control field, is proposed. The optimal solution is regarded as the stable equilibrium point of a finite-dimensional dynamic system and 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 within the feasible region and its solution will converge to the optimal solution of the NLP globally. The expressions of the Lagrange multipliers and the KKT multipliers during the entire optimization process are also derived. Using the matrix pseudo-inverse, the DOE is valid even without the linearly independent regularity requirement on the nonlinear constraints. Via the control based method, it is shown that the NLP may be transformed to the Initial-value Problem (IVP) to be solved efficiently, with mature ordinary differential equation integration methods.