Evolution and Optimum Seeking

180 Comparison of Direct Search Strategies for Parameter Optimization
with concentric, oblique ellipses, or ellipsoids as the contour lines or surfaces. The condition
number of the matrix of coe cients increases quadratically with the number of
parameters (see Appendix A, Sect. A.1). In general, the time required to calculate one
value of the objective function increases as O(n 2 ) for a quadratic problem, because, for a
full form matrix, n
2 (n + 1) distinct second order terms aij xi xj must be evaluated. The
objective function of Problem 1.2 has been formulated with the intention of reducing the
computation time per function call to O(n), without it being such a particular quadratic
problem that one of the strategies could nd it especially advantageous. The strategy
comparison for this problem could thereby be made for much larger numbers of variables
for the prescribed maximum computation time (Tmax = 8 hours). The storage requirement
for the full matrix A would also have been an obstacle to numerical tests with many
parameters.
To enable comparison of the experimental and theoretical results, the required number
of iterations, line searches, orthogonalizations, objective function calls, and the computation
time were measured in going from the initial values
to an approximation
x (k)
i
x (0)
i
; x i
= x i + (;1)i
p n for i = 1(1)n
1
10 x(0)
i ; x i for i =1(1)n
The interval of uncertainty of the variables thus had to be reduced by at least 90%.
The distance covered is e ectively independent ofthenumber of variables. The above
conditions were tested after each iteration, and as soon as they were satis ed the search
was terminated. The convergence criteria of the strategies themselves were not suppressed,
but they could not generally take e ect as they were much stricter. If they did actually
operate it could be regarded as a failure of the method being applied.
The results of the rst test are given in Tables 6.3 and 6.4. The number of function
calls and the number of iterations or other characteristic processes involved are displayed
in Figures 6.2 to 6.13 as a function of the number of parameters n on a log-log scale. As
the data show, the computation time and e ort of a strategy increase sharply with n.
The large range in the number of variables compared to other investigations allows the
trends to be seen clearly. To facilitate an overall view, the computation times of all the
strategies are plotted as a function of the number of variables in Figures 6.14 and 6.15.