Weak To Be Non-convex Optimization In The Normal Cone Condition Combined Homotopy Algorithm,

Combined homotopy interior point method is used to solve not only the convex problem, but also the non-convex problem which satisfies some conditions. The modified homotopy interior point method expands its using range. However, when it is to use to solve the problem we to need construct other mapping, and we need to judge that the non-convex field satisfies the conditions or not. The judgement is very important for us to use this method to solve the programming problem. So, how to judge the positive independent mapping is necessary. In the paper, first of all, the positive independent mapping and positive independent vector is studied systemly. And we give three sufficient and necessary conditions and some sufficient conditions about the positive independent mapping and vector to study their characteristics and give some specific examples to explain the constructive method about the positive independent mapping.For convex programming problem, CHIP can get the optimum solution of the problem when the problem has the optimum solution. Moreover, for the non-convex programming problem under the external normal cone condition and quasi-cone condition, the CHIP also have convergence. Nowadays, the papers which have been published can solve many non-convex programming problem under different conditions. For instance, we can use it to solve the problem which has equality constraint and inequality constraint, the feasible region which like star, and can solve multiple objective programming and so on. But it can solve more complex problem now.In this paper, we define the weak qusi-normal cone condition and study the hom-otopy interior method under the quasi-normal cone condition to solve constrained non-convex programming program.We proof the existence and global convergence of the homotopy path about the non-linear homotopy function. And numerical examples are presented to show the effectiveness of the algorithm.