Evolution and Optimum Seeking

Multidimensional Strategies 47
Numbering Iteration Direction Variable Comparison Step Remarks
index index values point lengths
k i x 1 x 2 s 1 s 2
(0) 0 0 0 9 - 2 2 starting point
(1) 0 1 2 9 (0) success
(2) 0 2 2 11 (1) failure
(3) 0 2 2 7 (1) success
(4) 1 0 4 5 - 2 ;2 extrapolation
(5) 1 1 6 5 (4),(3) success, success
(6) 1 2 6 3 (5) failure
(7) 1 2 6 7 (5) failure
(8) 2 0 10 3 -,(5) 2 ;2 extrapolation,
success
(9) 2 1 12 3 (8) failure
(10) 2 1 8 3 (8) failure
(11) 2 2 10 1 (8) failure
(12) 2 2 10 5 (8) failure
(13) 3 0 14 1 - 2 ;2 extrapolation
(14) 3 1 16 1 (13) failure
(15) 3 1 12 1 (13),(8) success, failure
(16) 3 2 12 ;1 (15) failure
(17) 3 2 12 3 (15) failure
(8) 4 0 10 3 - 2 ;2 return
(18) 4 1 12 3 (8) failure
(19) 4 1 8 3 (8) failure
(20) 4 2 10 1 (8) failure
(21) 4 2 10 5 (8) failure
(8) 5 0 10 3 - 1 ;1 step lengths
halved
(22) 5 1 11 3 (8) failure
(23) 5 1 9 3 (8) success
(24) 5 2 9 2 (23),(8) success, success
(25) 6 0 8 1 - ;1 ;1 extrapolation
A proof of convergence of the direct search ofHooke and Jeeves has been derived by
Cea (1971) it is valid under the condition that the objective function F (x) is strictly
convex and continuously di erentiable. The computational operations are very simple
and even in unforeseen circumstances cannot lead to invalid arithmetical manipulations
such as, for example, division by zero. A further advantage of the strategy is its small
storage requirement. It is of order O(n). The selected pattern accelerates the search
in valleys, provided they are not sharply bent. The extrapolation steps follow, in an
approximate way, the gradient trajectory. However, the limitation of the trial steps to
coordinate directions can also lead to a premature termination of the search here, as in
the coordinate strategy.
Further variations on the method, which have notachieved such popularity, are due
to, among others, Wood (1960, 1962, 1965 see also Weisman and Wood, 1966 Weisman,