Minimization by Random Search - Department of Mathematics
Minimization by Random Search - Department of Mathematics
Minimization by Random Search - Department of Mathematics
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MINIMIZATION BY RANDOM SEARCH TECHNIQUES<br />
The remaining table gives the average number <strong>of</strong> function evaluations IL, the<br />
standard deviation a and the maximum number <strong>of</strong> function evaluations recorded in 20<br />
runs, when solving a number <strong>of</strong> classical problems in global optimization. Algorithm 3<br />
is used with variants a or b as indicated. Computations were halted when the known<br />
minimum was attained (with a tolerance <strong>of</strong> .001 in the norm <strong>of</strong> x). These results<br />
compare favorably to any other reported in the literature, cf. [15]. Note however that<br />
in [15] the number <strong>of</strong> function evaluations recorded include those required to "verify"<br />
that the optimal solution has been reached.<br />
TABLE 4.<br />
Algorithm 3, xo randomly generated, 20 runs.<br />
Problem SQRN 5 SQRN 7 SQRN 10 Hartm. 3 Hartm. 6 6 Hump C<br />
Variant 3a 3a 3a 3a 3a 3b<br />
Mean numb.<br />
fct. eval. u 187 273 246 149 158 135<br />
Stand. dev. o 86 157 198 78 14 32<br />
Max, numb.<br />
fct. eval. 405 644 936 345 185 Not avail.<br />
m = 5,7, 10<br />
SQRNm = - 7= [(x - a )T. (x - a') + ci]- -, S = [0, 10]4 C R.<br />
I 1 2 3 4 5 6 7 8 9 10<br />
0.4 1.0 8.0 6.0 3.0 2.0 5.0 8.0 6.0 7.0<br />
0.4 1.0 8.0 6.0 7.0 9.0 5.0 1.0 2.0 3.6<br />
al 0.4 1.0 8.0 6.0 3.0 2.0 3.0 8.0 6.0 7.0<br />
0.4 1.0 8.0 6.0 7.0 9.0 3.0 1.0 2.0 3.6<br />
c1 0.1 0.2 0.2 0.4 0.4 0.6 0.3 0.7 0.5 0.5<br />
d= 3,6.<br />
Hartm d= -<br />
d=3<br />
3.0 0.1<br />
[aij]= 10.0 10.0<br />
30.0 35.0<br />
c = [1, 1.2, 3, 3.2]<br />
d=6<br />
[a ]=<br />
10.0 0.05<br />
3.0 10.0<br />
17.0 17.0<br />
3.5 0.1<br />
1.7 8.0<br />
8.0 14.0<br />
c = [1, 1.2,3,3.2]<br />
6 Hump C. function<br />
ciexp(- lai(x -pij)2), S = [0, ]d C R d.<br />
3.0 0.1 -<br />
10.0 10.0<br />
0.3689 0.4699 0.1091<br />
[Pij] = 0.117 0.4387 0.8732<br />
30.0 35.0 0.2673 0.747 0.5547<br />
3.0<br />
3.5<br />
1.7<br />
10.0<br />
17.0<br />
8.0<br />
17.0<br />
8.0<br />
0.05<br />
10.0<br />
0.1<br />
14.0<br />
[Pj] =<br />
= [ - 3,3] x [- 11.51.5] c R2<br />
0.1312<br />
0.1696<br />
0.5569<br />
0.0124<br />
0.8283<br />
0.5886<br />
0.2329<br />
0.4135<br />
0.8307<br />
0.3736<br />
0.1004<br />
0.9991<br />
f(x) = 4x2- 2.1x + x? + X2 4x2 + 4x4.<br />
I 1 3 1 XJX2 ~2 2-<br />
0.2348<br />
0.1451<br />
0.3522<br />
0.2883<br />
0.3047<br />
0.6650<br />
0.03815<br />
0.5743<br />
0.8828<br />
0.4047<br />
0.8828<br />
0.8732<br />
0.5743<br />
0.1091<br />
0.0381<br />
Addendum. In "On Accelerations <strong>of</strong> the Convergence in <strong>Random</strong> <strong>Search</strong> Meth-<br />
ods," to appear in Operation Research Verfahren, K. Marti obtains related convergence<br />
results for the conceptual algorithm as well as for more structured algorithmic<br />
procedures.<br />
29
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