Optimisation is a general term that refers to finding stationary points of a function. Many problems in computational chemistry can be formulated as optimisation of a multidimensional function, NEB optimisation being one of the many. The goal of an optimisation problem can be formulated as follows: find a combination of parameters (independent variables) that optimise a given quantity (the objective function), possibly with some restrictions on the allowed parameter ranges [131]. To know exactly what we are looking for optimality conditions need to be defined. The condition

where is the set of all possible values of the control variable ,defines the global optimum of the function .For an unconstrained problem is infinitely large, and finding the corresponding global minimum may be a difficult task. Similarly, a point is a strong local minimum of if

where is a set of feasible points contained in the neighbourhood of . For a weak local minimum only an inequality must be satisfied in Equation 2.7.

More easily identified optimality conditions could be used instead of Equation 2.6 and Equation 2.7 if is a function with continuous first and second derivatives, namely, the stationary point and the strong local minimum conditions. A stationary point is a point where the gradient vanishes:

(4.8) |

and a strong local minimum is a stationary point where the Hessian matrix

(4.9) |

is positive-definite:

(4.10) |

Formally, optimisation of an NEB qualifies as a nonlinear unconstrained continuous multivariable optimisation problem. There are many algorithms available for solving problems of this type that differ in their computational requirements, convergence and other properties. However, NEB optimisation is augmented with an additional difficulty: due to the projections involved the objective function that is being minimised is unknown. Nevertheless, several optimisation techniques were found to be stable in working with NEB. Most of these are based on steepest-descent methods and, consequently, are not very efficient. The L-BFGS and QVV minimisers discussed in this section are based on the deterministic approach for finding local minima.

The basic structure of any local minimiser can be summarised as follows:

The descent direction is defined as the one along which the directional derivative is negative:(4.11) |

Negativity of the left hand side guarantees that a lower function value will be found along provided that the step is sufficiently small. According to the way algorithms choose the search direction they are classified as non-derivative, gradient and second-derivative. Steepest-descent is an example of a gradient-based method, while BFGS belongs to the class of quasi-Newton methods (or quasi-second-derivative methods). The steepest-descent search direction is defined as the direction antiparallel to the gradient at the current point

(4.12) |

whereas second-derivative-based methods determine the search direction using the Newton-Raphson equation [31]:

The step length can be determined using either `line search' or `trust radius' approaches. Line search is essentially a one-dimensional minimisation of the function , which is performed at every step [131]. In the trust-radius-based approach the step length is adjusted dynamically depending on how well the minimisation algorithm predicts the change in the object function value [9]. There is no clear evidence for the superiority of one method over the other [131]; however, in combination with the limited-memory version of BFGS algorithm the trust ratio approach was found to be more efficient for treating problems with discontinuities such as problems involving periodic boundary conditions with cutoffs [9].