A Class Of The LQP-based Primal-Dual Algorithm For Solving Variational Inequalities

Since the 1960 s,variational inequality has been an important research content in applied mathematics.Variational inequality is not only a unified expression of optimality conditions for optimization problems,but also an effective modeling approach.In recent years,variational inequalities have been widely applied to equilibrium problems,operational research problems and urban traffic network modeling.Although there is a mature framework for solving variational inequality optimization algorithms,many problems in practical applications require non-negative variables,and it is not easy for existing efficient algorithms to solve their subproblems under non-negative constraints.In order to break through the limitations of existing methods,this paper puts forward a more suitable method for solving variational inequalities with non-negative constraints under the background that more and more practical problems require non-negative variables.On the basis of customized proximal point algorithm,the author transforms the subproblems with non-negative constraints into unconstrained problems by replacing the quadratic regular term with the logarithmic quadratic proximal term,and obtains three algorithms.In the new algorithm,the predictor-correction method is used to solve the subproblems,which reduces the computational difficulty.The convergence analysis of the new algorithm is carried out under the framework of variational inequality,the global convergence and sublinear convergence rate of the new algorithm are obtained.In numerical experiments,this paper chooses nonlinear complementary problem applicable to non-negative variables and compressed sensing problem applicable to general variables respectively to test the new algorithm.When solving the problem applicable to the general variable,the author first replaces the general variable with the difference of two non-negative variables and then applies the new algorithm to solve the problem.Experimental results show that the new algorithm has universal applicability,and the number of iteration steps and calculation time are better than the classical algorithm.The advantages of the new algorithm are more obvious when the problem scale is large and the accuracy is high.