Homotopy Method For Solving Two Kinds Of Problems With General Constraints

The theories of the variational inequality problem and fixed point problems are powerful tools for recent mathematical techniques. It is an effective approach for solving many problems in the field of science by the thought and skill of the variational inequality problem and fixed point problems. And it can also solve many practical problems by simulation experiment. So how to solve the variational inequality problem and fixed point problems is an important research. Many optimization algorithms have been development, such as Newton method, fixed-point interation method and some other interation methods. But it is hard to prove the global convergence of those methods, so we use the homotopy method to solve the variational problems and fixed-point problems in this paper. A distinctive advantage of the homotopy method is that the global convergence of the algorithm generated by it is obtained under weaker conditions. The main works are as follows:Firstly, a combined homotopy interior point method is proposed for solving the variational inequality problem with both equality and inequality constraints on an unbounded set. Under the condition that VIP has no solution at infinity, for almost all interior point of feasible set, existence and convergence of a smooth homotopy pathway to a solution of VIP are proved.Secondly, the equality constraints make a great limitation for chosen starting point, in order to enlarge the scope of initial points, the author gives a suitable perturbation on the equality constraint. Under a new and weaker assumption, existence and convergence of a smooth homotopy pathway to a solution of VIP are proved.Thirdly, the fixed point problems and the variational problems are closely related, so the author use the similar thought which to solve the variational problems to solve the fixed point problems. It means that the method enlarges the choice scope of initial points by a perturbation on equality constraint. Under a new and weaker condition for the defining mapping, a constructive proof of the existence of fixed points and the global convergence are obtained.