BFGS Method And Its Applications In Solving Constrained Optimization Problems

This thesis is concerned with the BFGS method for unconstrained optimization and its applications to sequential quadratic programming (SQP) method, reduced Hessian SQP method and sequential quadratically constrained quadratic programming (SQCQP) method for constrained optimization.We first introduce the previous works on the thesis in Chapter 1. In Chapter 2, we study the convergence properties of BFGS method for solving nonconvex unconstrained problems. It is well-known that the BFGS method is one of the most famous quasi-Newton algorithms for solving unconstrained optimization problems because of its favorable numerical experiment and fast convergence property. However, The BFGS method with exact line search or Wolfe-type inexact line search or Armijo inexact line search need not converge for solving nonconvex minimization problem. In this chapter, we introduce a perturbation strategy in BFGS method to develop a perturbed BFGS method. We prove that under mild condition, the perturbed BFGS method with Wolfe line search is convergent globally and superlinearly even if the objective function is nonconvex. Moreover, the proposed method maintains the affine invariance of the BFGS method and enjoys more favorable numerical experiment results than the BFGS method and the modified BFGS methods.The BFGS update technique is usually used by other methods for solving some related prblems such as nonlinear equations, constrained optimization problems, stochastic programming and semi-infinite programming. In next three chapters, we mainly analyze SQP method, reduced Hessian SQP method, SQCQP metod for solving constrained optimization problems by employing BFGS update technique.In Chapter 3, we study the convergenc properties of SQP methods under mild conditions. The global convergence of the existed SQP methods usually requires that the quasi-Newton matrices are uniformly positive definite and bounded. However, it is unknown whether there exists the qusi-Newton method satisfying the above conditions. By employing BFGS update technique, we propose a perturbed SQP method for solving general constrained problems. We prove that the proposed method converges globally under mild condition. In particular, the global convergence does not require the uniformly positive definiteness and the boundedness of the quasi-Newton matrices, while existed SQP methos usually require these conditions. Besides, we propose some strategies for global convergence in SQP methods. Numerical results show that the proposed strategies are practically efficient.SQP methods are usually used to solve small and medium-scale problems. In Chapter 4, we study reduced Hessian SQP methods for sloving large-scale optimization problems. The existed reduced Hessian methods slove only equality constrained problems and their global convergence require the linear independence and the uniformly positive definiteness of the reduced Hessian of Lagrangian function. The reason using the requirement of linear independence is that the existed quasi-Newton update formulas generate only quasi-Newton matrices with fixed order. On the other hand, the update formulas play important roles in analyzing the global convergence of reduced Hessian SQP methods. We propose a new quasi-Newton update formula which can generate quasi-Newton matrices with variable order, and then propose a reduced Hessian method for solving general equality constrained problems (including degnerate problems) and prove that the proposed method globally converges without above two requirements. Moreover, by using the proposed update formula, we extends the reduced Hessian method to solve constrained problems with inequality constraints.In Chapter 5, we study the sequential quadratically constrained quadratic programming method for solving inequality constrained problems. It is well-known that traditional SQP methods may occur Maratos effect. Several authors have considered SQCQP-type methods which use information up to second-order for both the constraints and objective function. SQCQP methods are free of the Maratos effect and thus enjoy fast convergence property. However, the existed SQCQP methods solve only some special problems such as convex programming, or problems with convex feasible set or some problems with strong conditions. By employing the perturbed strategy or BFGS update technique we propose two SQCQP methods for general inequality constrained optimization and prove that the proposed SQCQP methods are globally convergent under mild condition and are of superlinear rate of convergence at least.In Chapter 6, we report some numerical results for the proposed methods. The numerical results show that the proposed methods are practically efficient and thus support the proposed methods...