A Line Search Filter Two Piece Update Of Reduced Hessian Method For Nonlinear Constrained Optimization

Optimization,which is an energetic subject,makes research on how to find the optimal solutions among many feasible plans.Nonlinear programming(NLP)is one of the most important branches of optimization,and has been widely applied in many fields such as finance,trade,management and military research.Recently,Fletcher and Leyffer proposed filter method,replacing the traditional merit function method,as a tool to guarantee global convergence in algorithms for nonlinear programming(NLP). The underlying concept of filter methods is that trial points are accepted if they improve the objective function or improve the constraint violation instead of a combination of those two measures defined by a merit function.The motivation given by Fletcher and Leyffer for the development of the filter method is to avoid the necessity of determining a suitable value of the penalty parameter in the merit function.In addition,the filter approach offers another important advantage regarding robustness.If the trial step size becomes too small to guarantee sufficient progress toward a solution of the problem,the proposed filter method reverts to a feasibility restoration phase,whose goal is to deliver a new acceptable iterate by decreasing the constraint violation,or to converge to a local minimizer of infeasibility if this is not possible.Reduced Hessian Algorithm,which is developed from Sequential Quadratic Programming method,now has become one of the very popular and most effective methods for solving nonlinear equality constrained optimization problems.Recent research has focused on methods that solve such problem by recurring an approximation to second derivative information projected on to the tangent space of the constraints.The main motivation of Reduced Hessian Algorithm is that only the projected matrix enters into the optimality conditions for the nonlinear problem and this reduced the size of matrix which need to be computed in each iteration.Gurwitz[3]proposed the two-piece update of a projected Hessian matrix algorithm for nonlinear programming.The main idea of her method is to perform a two piece-update of a one-sided projected Hessian.This method maintains one piece of the projected Hessian as a positive definite matrix,which is updated by DFP or BFGS method,and uses Broyden's method for updating the other piece.But Gurwitz didn't mention the global convergence,which will be discussed in this paper.Inspired by the ideas of Fletcher & Leyffer and Gurwitz,we proposed a line search filter two piece update of reduced Hessian method for solving nonlinear constrained programming.In our algorithm,the search direction of the line search method is obtained by two piece update of reduced Hessian method,which reduces the size of the matrix that need to be computed in each iteration.Filter line search method is used to ensure the global convergence.Under mild assumptions it is shown that every limit point of the sequence of iterates generated by the algorithm is feasible,and that there exists at least one limit point that is a stationary point for the problem under consideration.In addition,we also show that the proposed method does not suffer from Maratos effect by using second order correction step.Consequently,fast local convergence to second order sufficient local solutions is achieved.The numerical experiments are reported to show the effectiveness of the proposed algorithm.