2.2. The Hooke-Jeeves Method

The Hooke-Jeeves (HJ) local search algorithm method was proposed by Hooke and Jeeves in 1961 [45]. Because it is a pattern search method, it can be used when the objective function is irregular.

The point of departure for the Hooke-Jeeves method is to define the following parameters:

d—the orthogonal n base for linearly independent orthogonal vectors,

τ—the initial length of the searching step dependent on the area of searching and the distribution of the objective function,

γ—the ratio of decreasing the searching step,

τ_end—the minimal length of the step which is the criterion of the end of the searching process,

x0—the starting point of the procedure.

Each iteration in the HJ method consists of two moves:

the exploratory move, in which the distribution of value of the objective function is tested within a small selected area of the base point, utilising trial steps along all directions of the orthogonal base d;

the pattern move involves moving in a strictly determined manner to the next base point in which another exploratory move is considered, but only on condition that at least one of the trial steps taken was successful.

A step is successful if it leads to the decrease in the value of the objective function. If none of the steps were successful, you return to the previous base point and the search cycle starts again with a decreased length of step τ.

The algorithm ends its work as soon as the ratio of the step τ achieves the assumed final value τ_end. The HJ algorithm is shown in detail in Figure 3. f(x) stands for objective function.

Flowchart of the Hooke-Jeeves method.

Unlike the SA method, the HJ method is totally deterministic. The same solution will always be obtained for a given starting point and the same search parameters.

The HJ method is simple and relatively fast-converging, which, in combination with no need to calculate the gradient of the objective function, makes it attractive if the objective function has no analytical form and is obtained based on the empirical data. The practical applications of the HJ method include different fields of widely understood engineering [46,47,48,49,50]. The method is not very popular in geodesy and measurement data processing. Work [51] is the exception in this area.