Analysis Of BFGS-SQP-L Method In Equality Constrained Optimization With Errors

In this paper,for a class of equality constraints optimization problems,in which both objective functions and their gradients have certain errors(in short,EOEP),we give two algorithms,i.e.,BFGS-SQP-L and MBFGS-SQP-L,and discuss the convergence of the two algorithms.The main idea of BFGS-SQP-L algorithm proposed in this paper is to apply extended difference technology to choose the search direction under a SQP framework,and based on the obtained search direction,the inexact L-step length of Lagrange function satisfying Armijo-Wolfe condition is taken.For this purpose,we propose a series of sufficient conditions to ensure the existence of step size satisfying Armijo-Wolfe condition,some beneficial results of the difference technique and good iterative properties.Finally,on the basis of proving some convergence results,we point out that the algorithm is locally convergent under certain conditions.For the same EOEP,MBFGS-SQP-L algorithm also is proposed under a SQP framework,where the penalty function method is used to solve the search direction,and at the same search direction,the inexact Lstep length of Lagrange function satisfying Armijo-Wolfe condition is selected.In addition,it can be deduced that the algorithm is locally convergent under certain conditions,and for the optimization problem involving error functions it is globally convergent.Finally,we compare and analyze the two proposed algorithms,and point out that they are both adaptive BFGS methods that can handle errors,while the purpose of extending the difference technique and penalty function is to make the corresponding algorithms robust.