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The efficiency of stochastic methods in solving difficult non-convex problem has been shown
on many practical examples. Notably, we have addressed the problems of robust structured
control and fault diagnosis of industrial systems. These topics lead indeed to non-convex
constrained optimization problems which are known to be difficult to deal with using
conventional methods. Since stochastic methods does not require strong assumptions such as
linearity, differentiability, convexity etc. they can be used to find out, in a straightforward
manner, if not the optimal solution but at least a suboptimal one, which is very useful for the
practitioner.
APPENDIX
A1. List of test functions F1 to F9
2
2
Easom's function (F1) (2 variables): J ( q)
= −cos(
q1)
cos( q2)
exp( −((
q1
− π ) + ( q2
− π ) ))
search domain: -100 ≤ qi ≤ 100, i = 1, 2; global minimum: qop t= (π, π), J(qop) = -1.
Goldstein-Price's function (F2) (2 variables); search domain: -2 ≤ qi ≤ 2, i = 1, 2; global
minimum: qopt = (-1, 0), J(qop) = 3. For a complete definition of this function see Socha & Dorigo,
(2008).
Hartmann's function (F3) (3 variables); search domain: 0 ≤ q i≤ 1, i = 1, 3; global minimum:
qopt = (0.1146, 0.5556, 0.8525), J(qop) = -3.86278. For a complete definition of this function see
Socha & Dorigo, (2008).
Rastrigin's function (F4) (5 variables)
5
∑ i=
1
2
J ( q)
= 50 + ( q −10
cos( 2π
q ))
i
search domain: -3 ≤ qi ≤ 3, i = 1, 5; global minimum: qop t= 0, J(qop) = 0.
i
Zakharov's function (F5) (5 variables)
( ) ( ) 4
5
2
2 5
5
J ( q)
= ∑ q + 0.
5 0.
5
= 1 ∑ +
i i iq
i=
1 i ∑ iq
i=
1 i
search domain: -6 ≤ qi ≤ 12, i = 1, 5; global minimum: qop t= 0, J(qop) = 0.
Michalevicz's function (F6) (10 variables)
∑ = ∑ = ⎟ 2
10 ⎛ 5 ⎛ iq ⎞⎞
⎜
i
J ( q)
= −
⎜
sin( q ) sin ⎜
⎟
i 1 i 1 i
⎝
⎝ π ⎠⎠
search domain: 0 ≤ qi ≤ π, i = 1, 10; global minimum: J(qop) = 9.66015.
Levy's function (F7) (30 variables)
qi
−1
zi
= 1+ , i = 1,
L,
30
4
29
= 1 ∑ i=
1
20
2
2
2
2 2
J ( q)
sin ( π z ) + [( zi
−1)
( 1+
10sin
( πzi
+ 1))]
+ ( z30
−1)
( 1+
sin ( 2πz30
+ 1))
search domain: -10 ≤ qi ≤ 10, i = 1, 30; global minimum: qop t= (1,1,…,1), J(qop) = 0.