| We examine the nonlinear equality constrained problem (NEP) in a projected or partitioned form to reveal the basic structure of the Hessian matrix. The concepts of quasi-Newton methods are then introduced by considering unconstrained optimization problems. Drawing on this background, we present quasi-Newton methods for NEP and explore properties of the approximate Hessian matrix which insure good performance of Sequential Quadratic Programming (SQP) type algorithms: positivity, quasi-Newton relation, symmetry, and the least-change property. Current quasi-Newton methods and the strategies they use to provide these properties are then examined. Although it is particularly desirable to maintain positivity of the null-space projection of the Hessian matrix, we show that existing methods fail to do this in a simple and reliable manner. Consequently, our goal in this thesis is to find new updates which incorporate these desirable properties of the Hessian matrix.; Using a variational approach, we derive several new quasi-Newton updates for NEP. These updates are derived using the partitioned forms of the Hessian matrix, allowing us to incorporate the desired properties for each partition. Two of the update formulas, the Lower Half Update (LHU) and the Partitioned BFGS Update (PBU), are shown to be particularly successful. The LHU and PBU formulas are of the least-change type and satisfy a standard quasi-Newton equation, retain symmetry of the full Hessian, and keep the null-space projection of the Hessian positive definite. In special cases, these new updates simplify to projected forms of the Broyden, Powell Symmetric Broyden, and Broyden-Fletcher-Goldfarb-Shanno formulas. These new updates are particularly attractive since they require only one gradient evaluation per iteration. They may be used at nearly every iteration, even when we are far from the solution, without requiring special line search conditions or step size restrictions.; Next, we discuss implementation of the LHU and PBU formulas. In particular, we examine the cubic equation which must be solved at every iteration, its accurate solution, and the selection of the cubic root.; We prove R-superlinear convergence and present extensive computational results including comparisons with existing quasi-Newton methods for the nonlinear equality problem. |
