On the performance of unidimensional search schemes in quadratic convergent algorithms

The performance of the most recently developed unidimensional search schemes coupled with three quadratically convergent algorithms is evaluated through nonquadratic examples. The quadratic convergent algorithms are the rank-one algorithm (Algorithm I), the projection algorithm (Algorithm II) and the Fletcher-Reeves algorithm (Algorithm III). The new search schemes are the exact arithmetic mean (EAM) search, the exact harmonic mean (EHM) search, the exact geometric mean (EGM) search. These are evaluated against the well-known unidimensional searches viz., exact root-mean-square (ERMS) search, the golden section (GOLD) search, the exact quadratic interpolation (EQUA) search, the exact cubic interpolation (ECUB) search and the approximate arithmetic mean (AAM) search, the approximate harmonic mean (AHM) search, the approximate geometric (AGM) search against the approximate root mean square (ARMS) search, the approximate quadratic interpolation (AQUA) search and the approximate cubic interpolation (ACUB) search. The performance is analyzed in terms of the number of function evaluations and CPU-time for convergence. From the extensive numerical experiments conducted the following conclusions are drawn: (a) In general, the relaxation decreases the number of function evaluations and CPU-time. (b) The exact search schemes EAM, EHM and EGM perform better than other exact search schemes. Among the three EAM, EHM and EGM search schemes also, the performance of EGM has been found to be better than EAM and EHM. (c) Among the approximate search schemes, AAM performs much better than the other search schemes considered. (d) The combination of the Algorithm I and AAM and that of Algorithm Ii and AAM is very efficient. (e) The Algorithm III has been found to be not quite efficient in dealing with functions with that flat bottoms.