Descent structure of local methods

After a brief discussion on the structure of descent methods, we highlight general concepts in the three categories of local methods for unconstrained nonlinear functions nonderivative, first derivative (or gradient), and second derivative methods. 

Independent of the theoretical approach that is used in describing the interatomic interactions, the forces, Eq. (5), can be used in identifying the structures of local total-energy minima. The simplest way is the steepest-descent method where new structures are obtained from old ones by using... 

The basic structure of an iterative local optimization algorithm is one of greedy descent . It is based on one of the following two algorithmic frameworks line-search or trust-region methods. Both are found throughout the literature and in software packages and are essential components of effective... 

Of course, the surface is not quadratic and the Hessian is not constant from step to step. However, near a critical point, the Newton-Raphson method (Eq. (2)) will converge rapidly. The main difficulty is that the convergence of the Newton-Raphson method is local. Thus the method will converge to the nearest critical point to the starting point. Consequently, one must start the optimization with a Hessian that contains no or one negative eigenvalue according to whether a minimum or a saddle point structure is required. For a minimum, the steepest descent direction... 

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