Research On Uncertain Linear Complementarity Problems

Complementarity is not only a hot issue in mathematical programming research,but also has a wide range of applications in operations research.In the classic linear complementarity problem model,it is usually assumed that the data is accurate.However,there are inevitably uncertain factors in real life,so the stochastic linear complementarity problem appears,its research is mainly based on probability theory,but when faced large earthquakes,devastating military experiments,and unprecedented infectious diseases without large numbers of samples,the problems cannot be solved by knowledge of probability theory.In 2004,Professor Liu Baoding proposed the uncertainty theory in the absence of a large number of samples.Based on the uncertainty theory,this thesis research the uncertainty linear complementarity problem.The main research contents are as follows:(1)First,based on the uncertainty theory and linear complementarity problem,uncertain variables are introduced into the linear complementarity problem,and the uncertain linear complementarity problem is proposed.The uncertain linear complementary problem is transformed into an uncertain equation system with the same solution set by NCP function,and a model for uncertain expected residual minimization is established,the existence of the model solution is proved by level set.Secondly,in order to find the solution of the uncertain linear complementarity problem,the expectation of the uncertain variable is discretize by the Stieltjes integral method,and the uncertain problem is transformed into an approximate solution of the deterministic problem.The existence and convergence of the approximate solution are studied,and it is proved that the approximate solution converges to the solution of the original problem.Finally,the related properties of uncertain linear complementarity problems with fixed coefficient matrices are studied.(2)The uncertain variables are further extended to the second-order cone linear complementarity problem,and the uncertain second-order cone linear complementarity problem is proposed.The uncertain second-order cone linear complementary problem is transformed into an uncertain equation system with the same solution set by NCP function,and a model for uncertain expected residual minimization is established.Combined with Jordan algebra and spectral decomposition theorem,the boundedness of the level set is studied,and the existence of the model solution is proved.Secondly,in order to find the solution of the uncertain linear complementarity problem,the expectation of uncertain variables is discretize by the integration by parts method,and the uncertain problem is transformed into an approximate solution of the deterministic problem.The existence and convergence of approximate solutions are studied,we prove that every accumulation point of minimizers of discrete approximation problems is a solution of the uncertain expected residual minimization problem.Finally,some properties of global optimal solutions are studied.