Study Of Computiontial Methods For Two Kinds Of Nonlinear Problems

The nonlinear complementarity problem (NCP) and the secondorder cone program (SOCP) problem are two kinds of the important optimization problems. They arise widely in science and engineering fields, thus how to solve them is significant in both theory and application.The complementarity problems are closely related to many fields including the nonlinear programming, the min-max problem, game theory, fixed point theory, variational inequlity and so on, and have wide applications in mechanics, economic science, transprotation science fields. Thus they have drawn great attention, and great achievement in both theory and algorithm has been made. Recently, there has been an increasing interest in solving NCP by using so-called smoothing Newton method. In Chapter 2, we consider NCP(F) with P₀-mapping. First, we convert equivalently NCP(F) into a smoothing system of equations by defining a new smoothing function, and present a smoothing Newton method for it. Second, we prove that the sequence produced by iteration is bounded and its every accumula tion point is a solution of NCP(F) when the solution set of NCP(F) is nonempty and bounded. Third, when NCP(F) has a unique solution, the convergence rate is proved to be locally superlinear and quadratic under the mild conditions. Compared with the existing methods, the proposed method has no requirments of the boundedness of search step and the strict complementarity, and its convergence rate can be controlled by designing Newton equation and line search. Finally, illustructive examples demonstrate the feasibility and efficiency of the proposed method.SOCP problem is also one of the importantly convex optimization problems. Not only does it arise widely in engineering fields, but also many optimization problems can be converted to it, thus how to solve it is a focus. As we know, many algorithms for SOCP problem are developed, but they almost all are the traditional iterative methods, and might not obtain the solution in real-time due to stringent requirment on computational time. Differing from the traditional algorithms, the neural network has many advantages on computation and real-time applications due to its inherent massive parallelism and electric circuit implementation. Since Hopfield proposed artificial neural network and it is applied successfully to optimization problems, the theory, methodology and application of neural network have been widely investigated, and many significant results have been achieved. In Chapter 3, we consider SOCP problem. By using two smooth functions, the second-order cone constraints are converted into smoothing convex ones. Thus, SOCP problem can be converted approximately into two convex optimization problems, and two neural networks for the resulting problems are presented by using projection theory. The proposed neural networks are shown to be stable in the sense of Lyapunov under some mild conditions and converge to an optional solution of the original problem with any accuracy. Finally, illustructive examples demonstrate the feasibility and efficiency of the two neural networks.