The Theory Of Generalized-F-Complementarity Problem And Its Algorithm Design

The complementary problem is a kind of mathematical model that mainly solves the complementarity relationship between two sets of variables.It is widely used in optimization theory and arithmetic,mathematical physics equation and economics,etc.This paper has formed a relatively mature theoretical system for the study of complementary problems.The F-complementarity problem is a complementarity problem arising from the perturbation of function F,which has a more general form than the complementary problem and further expands the application range of the complementary problem.Therefore,this paper studies the theory and algorithm of generalized F-complementary problem,which is of great significance to classical complementarity problem and F-complementary problem.In addition,the generalized F-complementarity problem also has important applications in daily production,such as image processing.In this paper,we study the generalized F-complementarity problem from two aspects of theory and algorithm.In theory,this paper first studied the existence and stability of the solution about generalized F-complementarity problem.Then we proved the equivalence of the generalized F-complementarity problem with convex programming problem,the zero point problem of multi-valued function and the variational inequality problem,which it provides the theoretical basis for the later algorithm design.In this paper,we designed a total of three algorithms.The first algorithm transformed the generalized F-complementarity problem into a convex programming problem.We devised an adjacent point algorithm for solving generalized F-complementarity problems and proved its convergence.This algorithm does not require the perturbation function F to be smooth and expand its applicability.The second algorithm designed a fixed point algorithm for a class of monotone generalized F-complementary problems,and we proved its feasibility and convergence.Under the condition of orthogonal projection mapping,this algorithm directly proves the equivalence between the generalized F-complementarity problem and the fixed point problem,which is the most commonly used algorithm for solving the fixed point problem.The last algorithm is a predictive-correction algorithm designed for the generalized F-complementary problem,and we also analyzed its convergence.