Two differential equation systems for inequality constrained optimization

This paper presents two differential systems, involving first and second order derivatives of problem functions, respectively, for solving inequality constrained optimization problems. Local minimizers to the optimization problems are proved to be asymptotically stable equilibrium points of the two differential systems. First, the Euler discrete schemes with constant stepsizes for the two differential systems are presented and their convergence theorems are demonstrated. Second, an Euler discrete scheme of the second differential system with an Armijo line search rule, is proposed and proved to have the locally quadratic convergence rate. The numerical results based on solving the second system of differential equations show that the Runge–Kutta method for the second system has good stability and the Euler discrete scheme the Armijo line search rule has fast convergence.