The Combined Homotopy Method for Solving a Class of Programming Problems with a Bounded Feasible Set

In this paper, we study the following nonlinear nonconvex programming problem: { min f(x) : s.t.g
_i
(x) ≤ 0. i ∈ M, M = {1, 2,...,m}. Under the condition that the feasible set is bounded and connected, but it has a point that the boundary is not regular at this point, we propose the combined homotopy method to solve this problem by constructing a new constraint function and a combined homotopy equation. The convergence of the method is proved and the existence of a smooth homotopy path from any interior point to a solution of the problem is established. Our method is very different from previous homotopy method. Numerical examples show that this method is feasible and effective.