On Local Error Bound Analysis For Generalized Nonlinear Complementarity Problems

The generalized nonlinear complementarity problem is a research hotspot in the field ofcomputational science and operational research cross, it is extended directly from the classicalnonlinear complementarity problems. In recent years, many algorithms have been proposed byscholars, including Newton algorithm, L-M algorithm and trust region algorithm, etc. The erroranalysis plays an important role both in numerical computation and mathematical programming,specifically, in the convergence analysis of algorithm. Therefor, to explore the error boundproperties of the generalized nonlinear complementarity problem has important theoretical andpractical value.In this paper, we study the local error bounds of the generalized nonlinear complementarityproblem. we first transform the generalized nonlinear complementarity problem into a nonsmoothsystem of equations by using nonlinear complementary function. Next we analyze the local errorbound of the function, and propose the sufficient condition while it satisfies the local error boundproperties. Then a Levenberg-Marquardt method is proposed, and it has a local superlinear orquadratical convergence under an error bound condition, which is much weaker than the standardnonsingularity condition.In this article, we study the generalized nonlinear complementarity problem of wideapplication background problems. We mainly study the equivalent transformation, local error bound,convergence analysis, etc. The main contents of this article are as follows:Chapter1. Describe the problem’s background, state its research purpose and the researchstatus, introduce the main content of this paper.Chapter2. Introduce some definitions and properties related to the generalized nonlinearcomplementarity problem, such as complementary functions, smooth/nonsmooth functions, localerror bound condition and so on.Chapter3. Introduce some equivalent transform methods, and proposed two methods while theproblem is under the condition m n.Chapter4. Study the local error bound of equations. Under suitable conditions, we propose thesufficient condition while it satisfies the local error bound properties.Chapter5. Propose a modified L-M method to solve the generalized nonlinearcomplementarity problem, then prove it has a local superlinear or quadratical convergence under anerror bound condition.Chapter6. We summarize this paper and talk about the future research work.