Continuous Benchmark Functions - Self-Adaptive Heuristics for Evolutionary Computation - page 140

Schwefel's Problem 2.40

Minimize

5

f ( x )= − x i (A.14)

i =1

constraints

⎧
⎨

x j ≥ 0 , for j =1 ,..., 5

g j ( x )= (A.15)

i =1 (9 + i ) x i + 50000

5

⎩

− ≥ 0 , for j =6

minimum x ∗ = (5000 , 0 , 0 , 0 , 0) T with f ( x ∗ )=

− 5000, g 2 to g 6 active.

Schwefel's Problem 2.41

Minimize

5

f ( x )= − ( ix i ) (A.16)

i =1

constraints like problem 2.40.
minimum x ∗ = 0 , 0 , 0 , 0 , 50000

14 T

with f ( x ∗ )=

250000
14

− ,

g j active for j =1 , 2 , 3 , 4 , 6.

g01

Minimize

4 4 13

( x i ) f ( x )=5 ( x i ) − 5 − ( x i ) (A.17)

i =1 i =1 i =5

constraints:

g 1 ( x )=2 x 1 +2 x 2 + x 10 + x 11 − 10 ≤ 0

g 2 ( x )=2 x 1 +2 x 3 + x 10 + x 12 − 10 ≤ 0

g 3 ( x )=2 x 2 +2 x 3 + x 11 + x 12 − 10 ≤ 0

g 4 ( x )= − 8 x 1 + x 10 ≤ 0

g 5 ( x )= − 8 x 2 + x 11 ≤ 0 (A.18)

g 6 ( x )= − 8 x 3 + x 12 ≤ 0

g 7 ( x )= − 2 x 4 − x 5 + x 10 ≤ 0

g 8 ( x )= − 2 x 6 − x 7 + x 11 ≤ 0

g 9 ( x )= − 2 x 8 − x 9 + x 12 ≤ 0

feasible intervals 0 ≤ x j ≤ 1for j =1 ,..., 9, 0 ≤ x j ≤ 100 for j =10 , 11 , 12

1, minimum x ∗ =(1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 3 , 3 , 3 , 1) T with f ( x ∗ )=

0 ≤ x 13 ≤ − 15,

feasible starting point x (0) =(0 . 5 ,..., 0 . 5) T f ( x (0) )=0 . 5.

with