Homotopy Method For Solving Generalized Nash Equilibrium Problem

The theory of generalized Nash equilibrium is the core concept in the theory of economic countermeasures.It is mainly used to describe the reality of market competition and has a great influence on sociology and economics.At present,the research methods of generalized Nash equilibrium include: transforming the problem into quasi variational inequality for discussion,penalty function method,studying with nikaido isoda function,traditional homotopy method and so on,but these methods are either difficult to solve,or the global convergence can only be achieved under strong assumptions.In order to overcome the disadvantages,this paper solves the generalized Nash equilibrium problem with equality and inequality constraints based on homotopy method.By constructing a new homotopy mapping,the global convergence of the algorithm is proved under unbounded conditions.The computational efficiency of numerical examples is obviously improved with this algorithm.The main contents of this paper are as follows:(1)Appropriate perturbations are added to existing homotopy equality constraints to construct homotopy maps,and a new homotopy interior point method is proposed to solve the generalized Nash equilibrium problem.First,a set of unbounded assumptions are given,on which the existence of the internal path and the global convergence of the algorithm are obtained;second,the selection range of the initial point only needs to satisfy the inequality constraint,which enlarges the selection range of the initial point.Numerical examples show the effectiveness of the method.(2)Based on content above,a new homotopy equation is constructed by introducing two quadratic continuous differentiable maps,which weakens the assumptions,generalizes the solution to the generalized Nash equilibrium problem to a more general case,and expands the selection of initial points Range.umerical examples prove the effectiveness of the new homotopy method and have higher computational efficiency.(3)On the basis of content above,a combined homotopy method with moving boundary is proposed to solve the generalized Nash equilibrium problem.It removes the restriction that the initial point must be an inner point,thus extends the selection range of the initial point to the outside of the constraint region.Numerical examples show the effectiveness of the method.