Looking for the global minimum of the objective function to determine the pseudoinverse matrix through the search method at equalizing free geodesic networks

Abstract

When evaluating the accuracy of equalized free geodetic networks based on the search method of nonlinear programming, a degenerate matrix of unknowns’ normal equations coefficients is formed. It must be pseudoinverse in order to calculate the inverse weight one of the equalized parameters. Performing a mathematical procedure for calculating a pseudoinverse matrix based on the mentioned technology of nonlinear programming at equalizing free geodetic constructions, a situation may arise when the minimization step is incorrectly selected or the approximate values in the desired array of numbers are set roughly. It can lead to finding a local, not a global minimum, as well as to cycling the algorithm. The authors provide the conclusion of the formula and substantiate the conditions on which the elements of the required matrix change, so that the value of the objective function either comes as close as possible to the global minimum, or indicates that it is impossible to achieve it. The correctness of the proposed conditions and the derived formula are confirmed by a test example