Abstract
This paper deals with a modified version of Newton's method for unconstrained minimization in which a search direction of the form s = - ag + ß p is obtained by fitting a quadratic model to the objective function in the plane of the Newton direction p and the steepest descent direction -g. This composite search direction can be used as a fall-back option when the standard Newton step proves ineffective - e.g., when the Hessian matrix is not positive definite and p represents a move towards a saddle point or maximum.
The method proposed for determining a and ß resembles a two variable trust-region approach. Two prototype algorithms are discussed: one uses s in a trust-region framework and the other performs a line search. In both methods the computational overheads are comparable with those for a standard Newton method because only one linear system needs to be solved per iteration.
Keywords: Unconstrained minimization, Newton method, Two-dimensional Line Search.
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