Fixed Point Problem Homotopy Interior Point Algorithms And Complexity

This paper studies the problem solving and effective fixed point algorithm, Brouwer fixed point theorem is a very well-known fundamental theorem, which is an important tool in many areas, such as mathematical programming, economics, engineering and management. For the problem of fixed points can be divided into theory and algorithms, the main problems in the former existence of solutions, which mainly studies how to construct effective algorithms and convergence analysis, the effective algorithm to do some research.The rise of interior point algorithms began in 1984, made by the Karmarkar, Karmarkar algorithm is a calculation of the theoretical and actual performance is better than the simplex method has polynomial complexity of Xing's linear programming algorithm, while allowing linear programming nonlinear programming problem with two independent systems have been linked. Kamrarkar algorithm is starting from the initial interior point, from within the region gradually become feasible optimal solution, the Karmarkar algorithm is also known as interior point method. Smoothing Newton algorithm for solving mathematical programming, variational inequality problems in the classical algorithm. Solving Variational Smoothing Newton algorithm is the variational inequality problem into solving Robinson equation, and then using smooth approximation of non-smooth equations, which can then be solved using Newton-type method to design algorithms.In this paper, drawing on existing theoretical results of interior-point algorithm, smoothing algorithm and Newton homotopy method are given in the-under the conditions ofγ-cone neighborhood Homotopy Interior Point algorithm used to solve fixed point problems, and discussed the algorithm the convergence of nature, given the algorithm complexity analysis, prove that the algorithm has polynomial complexity, that is, the total complexity of the algorithm is O(n3.5L). Finally, using this algorithm to solve the problem is not fixed-point numerical examples. Through this research, the group to further promote the use of Homotopy methods. And use MATLAB programming the numerical experiment, numerical results show that the proposed algorithm is feasible.