Construct A Quasi-normal Cone For Nonconvex Sets And Its Application In Solving Multi-objective Optimization

Combined homotopy interior point method(denoted as CHIP method) can not only solve the convex optimization problem, but also can solve the problem of nonconvex optimization satisfying some conditions such as Normal cone condition, Weak normal cone condition, Quasi normal cone condition and false cone condition. The modified CHIP method is proposed to expand the use of combined homotopy interior point method, can solve the broader nonconvex optimization problem. In this paper, we mainly study the nonconvex optimization problem and multi-objective optimization problem using combined homotopy interior point method. When the feasible region satisfies the Quasi normal cone condition, combine homotopy interior point method is used to solve the nonconvex problem, we need construct a positive independent map. However, positive independent map structure there is no uniform method, can only research and construction a class of nonconvex region. Based on the existing theoretical research, study the quasi normal cone structure method of a class of nonconvex regions—N. Construct a Combined Homotopy Interior Point method to solve the KKT point of nonconvex optimization according to this quasi normal set. We prove that method has global convergence.Combined homotopy interior point method also can solve the multi-objective optimization problem include direct methods and indirect methods. Nowadays, there are a lot of research results about indirect method but little about direct method. We presented a direct method for solving multi-objective optimization problems. In the process of the homotopy path, ? is not fixed. To a certain extent, provide more choices for decision makers.Through the research of this paper, we further generalize the using range of the combined homotopy interior point method.This paper structure as follows: In the first chapter, the significance of the research and the development of the homotopy interior point method are introduced. In the second chapter, it introduces the basic knowledge and notation and the basic idea of the homotopy algorithm and the algorithm for the prediction and correction of the path following algorithm. In the third chapter, we give the normal independent mapping and the quasi normal cone structure method for the non convex regions in the single objective case.Construct a Combined Homotopy Interior Point method to solve the KKT point of Nonvex optimization according to this quasi-normal set. The algorithm is feasible and effective for solving the non convex optimization problem by numerical examples. In the fourth chapter,we give a new method to solve the multi-objective optimization problem, and give a combination of the KKT point’s homotopy equation and numerical examples.