Aggregate Smoothing Algorithm Of Absolute Value Equation

Absolute value equation under certain conditions can be equivalent conversion into linear complementarily problem, can also be equivalent transformation into a bilinear programming problem. Absolute value equation in energy, environmental protection,national defense and many other fields have a wide range of applications, as an important branch of mathematics problems. The absolute value equation problem originates from the study of interval linear equations and linear complementary problem, as a kind of nonlinear equations, the solving process is also a problem.In this paper, under the condition of satisfy regularity of interval matrix, using the homotopy method, using different forms of smoothing of the absolute value function,construct two different forms of homotopy equation, and proves that the smooth solution curve exists, the point is the absolute value equation solution.Paper first introduces the progress of the absolute value equation problem, the research status of homotopy method and the results obtained, and the main content of the study and conclusion in this paper. Absolute value equation is given in the preliminary knowledge and the related concepts of homotopy method, including the equivalent form of absolute value equation theorem, the homotopy method, the basic idea of constructing homotopy equation and so on. The absolute value function is expressed as the great function of form, to discuss the nature of the great function of condensation of the smoothing function. Different smoothing function is established based on the absolute value function is two homotopy equation, the interval matrix satisfy regularity under equivalent conditions, using the theorem and the implicit function theorem, prove the homotopy path exists, classification by using one dimensional manifold theorem, prove the limit point is the absolute value equation solution. Finally, summarized the full text of the main outcomes for further work in the future are put forward.