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Abstract: This paper presents two differential systems, involving first and second order derivatives of problem functions, respectively, for solving equality-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, we construct algorithms in which directions are computed by these two systems and the stepsizes are generated by Armijo line search to solve the original equality-constrained optimization problem. The constructed algorithms and the Runge–Kutta method are employed to solve the Euler discrete schemes and the differential equation systems, respectively. We prove that the discrete scheme based on the differential equation system with the second order information has the locally quadratic convergence rate under the local Lipschitz condition. The numerical results given here show that Runge–Kutta method has better stability and higher precision and the numerical method based on the differential equation system with the second information is faster than the other one. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]