Theoretical Study On Circular Cone And Second-order Cone Complementarity Problems

In recent decades,the research on linear complementarity problems has formed a relatively complete theory system,which includes the theories,algorithms and applications.Especially,in the finite-dimensional Euclidean spacenR,those researches are in a very relatively mature stage,and too many good conclusions have been achieved.Moreover,compared with the classical linear complementarity problem,the cone linear complementarity problem has a more wide range of applications,more extensive application background and stronger application value.Hence,the study on the cone linear complementarity problem is significant.In particular,it is closed concerned by many experts at home and abroad to study the symmetric cone linear complementarity problem by Euclidean Jordan algebra.It is well known that Euclidean Jordan algebra is a very powerful theoretical tool to describe the algebraic features of the cone,the elements of the cone and the projection of the elements onto the cone.Therefore,in this paper,we first introduce some background materials and relevant conclusions,including Euclidean Jordan algebra,second-order cone,and the spectral decomposition of element in the finite-dimensional Euclidean spacenR and the infinite-dimensional Hilbert space,respectively.In addition,the concept and results on circular cone in the spacenR are also introduced,which will be extensively used in subsequent analysis.Then,we study the structure of the solution set of the monotone circular cone linear complementarity problem in the spacenR by the relationship between circular cone and second-order cone.Moreover,by introducing the concept and related properties of complementary function on the circular cone complementarity problem,we give two types of complementary functions that is theoretical basis for designing the algorithms to solve circular cone complementarity problems.Finally,combining with Jordan product and the relevant knowledge about functional analysis,we give the conditions on feasibility and solvability of second-order cone linear complementarity problem in the infinite-dimensional Hilbert space,and consider the structure characteristics of the solution set of monotone second-order cone linear complementarity problem which contributes to future research on cone linear complementarity problems.