Fundamentals - Preventive Biomechanics - page 118

U W [ U G [ U B U i ¼ U ð p i ; 1 Þ i ¼ 1 ; 2 ; 3

ð 3 : 371 Þ

3.2-Determination of the coordinates of centroid m of all function values
except for the best in R 2 ( 3.372 ), and generally in R n ( 3.373 ).

m ¼ 1

2 ð p G p B Þþ p B
¼ 1

2 ð p G þ p B Þ
¼ p 1 G ; 1 þ p 1 B ; 1
2

ð 3 : 372 Þ

h i T

p 2 G ; 1 þ p 2 B ; 1
2

In n-dimensions:

X

n

m ¼ 1
n

p i with p i 6¼ p W : ð 3 : 373 Þ

i ¼ 1

3.3-Construction of vertex r by reflection of the triangle Dp B p W p G through the
side p B p G . The potentially new triangle is Dp B p G r.

r ¼ p W þ 2 ð m p W Þ
¼ð p G þ p B Þ p W
¼½ p 1 G ; 1 þ p 1 B ; 1 p 1 W ; 1

ð 3 : 374 Þ

p 2 G ; 1 þ p 2 B ; 1 p 2 W ; 1 T

Objective function evaluation at vertex r.

Case distinction
If U ð r Þ \U ð p B Þ go to step 5 otherwise proceed with step 4.
4. Step: Test if value of U ð r Þ is smaller than the function values at all other
vertices.

Case distinction
(i) If U ð r Þ U ð p i ; 1 Þ for one single vector p i ; 1 with ð i ¼ 1 ; ... ; n Þ go to step

6

(ii) If U ð r Þ ! U ð p i ; 1 Þ for all p i ; 1 with ð i ¼ 1 ; ... ; n Þ go to step 7

(iii) If U ð r Þ U ð p i ; 1 Þ for with ð i ¼ 1 ; ... ; n Þ

more than one vector p i ; 1

replace p W with r and go to step 9
5. Step: Expansion of the simplex Dp B p G r to generate the potentially new tri-
angle Dp B p G e (cf. Fig. 3.27 b ) by determination of the coor dinates of point
e through shifting of r along mr about the distance of p W m.
e ¼ p W þ 3 ð m p W Þ
¼ 3

2 ð p G þ p B Þ 2p W
¼ 2r m

ð 3 : 375 Þ