A GENERAL CONVERGENCE THEORY FOR QUASI-NEWTON METHODS FOR CONSTRAINED OPTIMIZATION

[摘要] In this thesis we study the local convergence of quasi-Newton methods for nonlinear optimization problems with nonlinear equality constraints. A general theory for analyzing the local convergence of the sequence {x(,k)} generated by the diagonalized quasi-Newton method is developed. Conditions on the multiplier update that allow one to determine whether the convergence is q-linear in the x variable alone or in the pair (x,(lamda)) where (lamda) is the correspondent multiplier are specified. Two characterizations of q-superlinear convergence of the sequence {x(,k)} are given. The satisfaction of linearized constraints seems necessary to obtain q-superlinear convergence in the x variable. The use of the DFP or the BFGS secant updates requires the Hessian at the solution to be positive definite. The second order sufficiency conditions insure the positive definiteness only in a subspace of R(;;n). Conditions are given so we can safely update with either update. A new class of algorithms is proposed which generate a sequence {x(,k)} converging 2-step q-superlinearly. We propose an algorithm that converges q-superlinearly if the Hessian is positive definite in R(;;n) and it converges 2-step q-superlinearly if the Hessian is positive definite only in a subspace.

[发布机构] Rice University