The Research On Positive Definite Property Of B_κ In Sequential Quadratic Programming

In this paper, we are concerned with the sequence quadratic programming (SQP) methods for solving constrained optimization problems. The basic idea of an (SQP) methods is to approximate the constrained problem by a sequence of quadratic programming (QP) problems. The objective function of the QP problem is a quadratic function which is an approximation of the Lagrangian function of the constrained problem and the constraints of the QP problem are linear approximation of the constraints of the constrained problem.In an SQP method, it is important to keep the Hessian of the objective function in the QP problem to be positive definite. When the Hessian is positive definite, the QP subproblem is a strictly convex quadratic programming. It has a unique solution and can be easily solved by existing algorithms. Moreover, the solution of the QP problem provides a descent direction for most merit functions. Therefore, it is desirable to globalize the method.There have developed some technique to ensure the positive definiteness of the Hessian of the quadratic objective function. However, some of them are restrictive. For some of these methods, the convergence is not known.In this paper, based on a modified BFGS update formula proposed by Li and Fukushima, we proposed a modified BFGS method for solving equality and inequality constrained optimization problem. An attractive property of this method is that the Hessian of the quadratic objective function is always positive definite. In addition, by using an exact penalty function, we globalize the proposed method. Under suitable conditions, we obtain the global convergence of the globalized method. Numerical experiments show that the proposed method is effective.We also apply the SQP method with MBFGS update to solve an equivalent constrained optimization problem of the nonlinear complementarity problem. Under certain conditions, We prove the global convergence of the method.