A homotopy method for solving a class of nonlinear programming problems

In this paper, we study the following nonlinear nonconvex programming problem: {min f(x), s.t.g_i(x) ≤ 0, ∈ M, M = {1, 2,⋯, m}. Under the condition that the feasible set is bounded and connected, and the feasible set does not satisfy the pseudo-normal cone conditions, we propose the combined homotopy method to solve this problem by constructing new constraint functions 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. Numerical examples show that this method is feasible and effective.