Gaufi-Newton method

The least-squares problem has been solved by a generalized Gaufi -Newton method [26,53]. The algorithm of the inverse problem of kinetic parameter identification is available as a code called PARFIT. Nowak and Deuflhard [27] have developed a software package PARKIN for the identification of kinetic parameters. 

The numerical solution of the projection step can be computed iteratively by a Gaufi-Newton method, see also Ch. 7.2.2. There, a nonlinear constrained least squares problem is solved iteratively. The nonlinear functions are linearized about the current iterate and the problem is reduced to a sequence of linear constrained least squares problems. 

The constrained nonlinear least squares system 7.2.1 can be solved iteratively by a Gaufl-Newton method (Step 4). This will be discussed in the next section. Gaufi-Newton methods as well as other modern optimization methods require derivatives of the objective function and constraints with respect to the unknowns, i.e. the computation of so-called sensitivity matrices. Their evaluation will be discussed in Sec. 7.2.3. 

Iterative Solution of Constrained Nonlinear Least Squares Problems by Gaufi-Newton Methods... 

The Gaufi-Newton method generates a sequence of iterates by... 

By comparing (7.2.4) with (7.2.5) the convergence properties of the Gaufi-Newton method can be related to those of Newton s method. In (7.2.4) the second deriva-tives missing. As a consequence of Assumption 7.1.1... 

This can also be seen when considering Theorem 3.4.1 for the case of Gaufi-Newton methods ... 

The other possibility would be to solve fc s, Sj,0) = 0 for Sj as function of sj and thus Sj is no longer a degree of freedom for the Gaufi Newton method. The former type of procedure is often called infeasible path method as for the iterates the constraints are not enforced. This is in contrast to the latter type which belongs to feasible path methods. Feasible path methods are known to converge in general slower than infeasible path methods. 

When applying a Gaufi-Newton method these constraints of the optimization problem have to be linearized in every iteration step. This leads to... 

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