Theoretical Study On Lorentz Cone Linear Complementarity Problems On Hilbert Space

In finite-dimensional Euclidean space, the cone linear complementarityproblem is a hot research topic at home and abroad. In particular, the use ofEuclidean Jordan algebra to study the cone linear complementarity problemattract the close attention of many domestic and foreign scholars. However, sofar, the use of Jordan algebra to study the cone linear complementarity prob-lem is in the preliminary stage in infinite-dimensional Hilbert space. Therefore,this thesis gives further theoretical study on this topic. Details are as follows:Firstly, we introduce the basic material about Jordan product and theform of the Lorentz cone in the infinite-dimensional Hilbert space, and thenstudy the properties about them. Furthermore, we give several concepts forthe linear transformation on Hilbert space, and discuss some interconnectionsamong them.Secondly, we introduce the concepts for the column-sufficiency and therow-sufficiency of the linear transformation. Moreover, we establish somenecessary and sufficient conditions for the row-sufficiency and the column-sufficiency of the linear transformation. That is, an equivalent conditionfor the row-sufficiency of the linear transformation is that the KKT pointof the quadratic programming is a solution of the Lorentz cone linear com-plementarity problem. The necessary and sufficient condition for the column-sufficiency of the linear transformation is that the solution set (if nonempty)of the Lorentz cone complementarity problem is convex. These conclusions ininfinite-dimensional Hilbert space are the generalization of the correspondingconclusions in the finite-dimensional Euclidean space.Thirdly, we establish the relationship between the P-properties of thebounded linear transformation and the solution of the Lorentz cone linearcomplementarity problem in infinite-dimensional Hilbert space. Moreover, weconclude some necessary conditions , several sufficient conditions and an equiv- alent condition of the bounded linear transformation having the GUS-property.These results in infinite-dimensional Hilbert space are also the generalizationof the setting of the finite-dimensional Euclidean space.Finally, corresponding to the properties of the w-solution on the cone lin-ear complementarity problem in finite-dimensional Euclidean space, we studythe related properties of the w-solution on the Lorentz cone linear comple-mentarity problem in infinite-dimensional Hilbert space. We establish sev-eral necessary and/or sufficient conditions for the linear transformation hav-ing w-uniqueness property. Furthermore, we discuss the w-unique and w-Pproperties for the Lyapunov-like transformation. These conclusions in infinite-dimensional Hilbert space coincide with the corresponding conclusions in thesetting of the finite-dimensional Euclidean space.