Theory And Efficient Algorithm For Differential Linear Complementarity System

Complementarity problem is a class of very important optimization problem and has been used in a wide field such as Engineering, Economics and Transportation equilibrium, etc. Theories and algorithms of linear complementa-rity problem have achieved fruitful results. Recently, in the fields of mechanical, kinetic and circuit control, there appear a class of linear complementarity problem with differential equation, called as differential linear complementarity system, but the related research results are very rare and there are still many difficulties on this research. In this thesis, firstly we analyze the solution function of DLCS. Then, based on the Euler implicit time-stepping scheme we propose its improved version for solving DLCS, and analyze its convergence for both Z-matrix and positive semidefinite matrix DLCS. Finally, we give some numerical results.In Chapter1, we introduce the models and concept of DLCS and matrix classification, especially the basic steps of first-order Euler time-stepping scheme approximating the solution of DLCS. We also analyze the relationship and structure between the least element subproblem and quadratic subproblem. For the first-order Euler implicit time-stepping scheme, we study the solution function of the least-element subproblem for DLCS with Z-matrix and obtain its derivative formula in chapter2. About the positive semidefinite matrix DLCS, we analyze the property of the solution to quadratic subproblem while the stability of first-order Euler time-stepping scheme is maintained.In chapter3, considering first-order convergence of solution function in Euler time-stepping scheme, we combine trapezoidal approximation with improved scheme to induce an improved Euler implicit time-stepping scheme without violating its stability. Because the subproblem of positive semi-definite matrix DLCS is a quadratic subproblem which can’t be efficiently solved at present, so we analyze its least norm solution in theoretical. For the least element subproblem with complementarity constraint on the Z-matrix DLCS, we derive the equivalence of the complementary constraint and linear constraint according to the properties of Z-matrix, thus the subproblem is actually equivalent to a linear programming. In chapter4, we report some numerical experiment for the improved Euler time-stepping scheme and compare it with its original version. The numerical results show that the improved Euler scheme works better.