Second Order Derivatives Based Differential Approaches To Nonconvex Constrained Optimization

The aim of the dissertation is to study second order derivatives based differential equation approaches to nonconvex constrained optimization problems. There are three reasons for selecting such a subject: one is that this may provide a theoretical support to some artificial neural network methods as there are many artificial neural networks for optimization problems characterized by systems of differential equations; and the second one is that effective numerical algorithms for differential equations can be applied to solving constrained optimization problems; the last one is that second order derivatives based differential equation methods usually have fast convergence rates. The dissertation focuses on the study of differential equation systems via a specific space transformation, modified Evutushenko-Zhadan systems, and nonlinear Lagrangian based differential equation systems. The main results can be summarized as follows:1. Chapter 2, with the help of a space transformation, constructs a first order derivatives based and a second order derivatives based differential equation systems for inequality constrained optimization problems. The two systems are proved to have the properties: KKT points of an inequality constrained optimization problem are their asymptotically stable equilibrium points and the whole solution trajectories are in the feasible region of the problem if they start from initial feasible points. Moreover, we demonstrate the convergence theorems for the discrete schemes of the two differential systems, including the locally quadratic convergence rate of the discrete algorithm for second order derivatives based differential equation system. Finally, we give numerical examples based on these two discrete methods and the numerical results show that the second order derivatives based differential equation algorithm is faster than the other one.2. Chapter 3 has two parts. The first part presents tow modified Evtushenko-Zhadan systems involving first order derivatives and the second order derivatives of problem functions, respectively, for equality constrained optimization problems. It is proved that KKT points of an equality constrained optimization problem are their asymptotically stable equilibrium points. The Euler discrete schemes for the both differential...