A Homotopy Method Without The Normal Cone Condition For Solving Inequality Constrained Nonlinear Programming

Homotopy method is one of powerful numerical methods for solving nonlinear program-ming. For almost all initial points of the strictly feasible set, the global convergence of homotopy method can be generally established, under conditions of the nonemptiness and boundness of the interior of the feasible set, the Positive Linear Independent Constraint Qualification and the normal cone condition. Among the three kinds of convergence conditions, the Positive Linear Independent Constraint Qualification is equivalent to the Mangasarian-Fromovitz Constraint Qualification, therefore, the former two are basic for solving nonlinear programming, and the normal cone condition is the generalization of the convexity condition. In this paper, a brief introduction of the background of the nonlinear programming and some numerical methods are introduced. And the equivalence between the Positive Linear Independent Constraint Qualifi-cation and the Mangasarian-Fromovitz Constraint Qualification is proven. Secondly, the four typical normal cone conditions, the corresponding homotopy function and the global conver-gence of the homotopy methods are listed. Thirdly, the relationship of the four typical normal cone conditions is proposed for the first time, by proving or listing conterexamples. Finally, a new homotopy method is constructed through constructing the auxiliary mapping satisfying some certain conditions—the norm preserving map. Its global convergence is proved under only the conditions of the nonemptiness and boundness of the interior of the feasible set and the Positive Linear Independent Constraint Qualification, but without the normal cone condition in any form, hence, its global convergence is relatively weak. Moreover, a general construction of the norm preserving map for the general inequality constrained nonlinear programming is given.