The Active Set Smoothing Constraint Homotopy Method For Solving Multi-constrained Nonlinear Programming

Based on the active set strategy,smoothing function and homotopy method,a two-stage algorithm is proposed to solve the nonlinear programming problems containing only inequality constraints.Under certain conditions,the convergence of the method is proved.The preliminary numerical experiments show that effectiveness of the method.This paper is based on the smoothing function defined by the piecewise cubic polynomial function and the existing homotopy methods for solving nonlinear programming problems.By taking the homotopy parameter and the smoothing parameter as the same parameter,An active set smoothing constrained homotopy map is proposed to solve the multi-constrained nonlinear programming problems with only inequality constraints.Under the boundedness of the feasible set,positive linear independent constraint qualification and the normal cone condition for the feasible set,for almost any starting point in the interior of the feasible set,the set of zero points of the homotopy map defines a smooth homotopy path starting from this point and converging to the stable point of the original problem.Because this method adopts the active set strategy,it only needs to calculate the gradient of a small number of constraint functions and the Hessian matrix in the calculation,so it has high computational efficiency.When the smoothing parameter is small,the smoothing constrained homotopy method will have an ill-conditioned problem.In response to this problem,this paper presents an active set strategy based on the first-order optimality condition,which uses an approximate solution calculated by the homotopy method to meet a certain accuracy,based on the first-order optimality condition of the original problem,construct the equivalent non-ill-conditioned smoothing equations of the first-order optimality condition,and solve it by Newton method,the stable point of the original problem can be obtained.Under general conditions,the convergence of the strategy is proved.Preliminary numerical experiments show that this strategy avoids the ill-conditioned problem of the active set smoothing constraint homotopy method when the smoothing parameter is small,and effectively improves its computational efficiency.