r - The Hong Kong Polytechnic University

The dual is
min
*
w, b, ξξ ,
s.t
1
min
* αα , 2
n
1
T
*
ww+ C∑( ξ + ξ
i i
)
2
i=
1
T
w Φ + b− y ≤ ε + ξ
y
( x )
i i i
( x )
−w
Φ −b
≤ ε + ξ
T
*
i i i
*
, ≥ 0, i = 1, 2, L,
i i
ξ ξ
l
l
*
T
* * *
( α−α ) Q( α− α ) + ε∑( α + α
i i ) + ∑yi( α −α
i i )
l
∑( )
i= 1 i=
1
α − α = ≤α α ≤ C i = L l
* *
s. t 0, 0 , , 1, 2, ,
i i i i
i=
1
l
T
Where Q is an l× l positive semi-definite matrix, Q = K( x , x ) ≡ Φ( x ) Φ( x ), and K ( x , x ) is the
ij i j i j
i j
kernel.
The approximate function is:
l
*
f ( x) = ∑( − α + α ) K( x , x ) + b
(4)
i i i
i=
1
SVM-based reliability analysis method with UDM
The limit state function can be calculated by SVM as
g( x) = z = f ( x )
(5)
Where X= {x 1 ,x 2 ,…,x n } is the vector of random variables.
In FORM and SORM, the first-order and second-order derivatives of the structural response need to be
computed. So the polynomial function, the Gaussian radial basis function and the sigmoid function can be used
as the kernel functions in SVM-based reliability analysis.
Polynomial function
A polynomial mapping is a popular method for non-linear modeling:
T
K ( xc , ) = ( xc + 1)
q
(6)
i
i
Where q ≥ 2 , and c
i
are the support vectors.
Gaussian Radial Basis Function
'
q
q−1
∂z
⎛ ⎛ ⎞ ⎞
⎛ ⎞
= − + ⎜ + ⎟ = +
∂x
j i= 1
⎝ ⎝ j= 1 ⎠ ⎠
i= 1 ⎝ j=
1 ⎠
2
q−1
m
l
m
l
∂ z ∂ ( j) * ⎛ ( j)
⎞ ( j) ( k) * ⎛ ( j)
⎞
= q ci ( − α + α ) 1 ( 1) ( )
1
i i ⎜ x c
j i
+ ⎟ = qq− ∑c c − α + α xc
i i i i ⎜∑
+
j i ⎟
∂x ∂x ∂x
j k k i= 1 ⎝ j=
1 ⎠
i= 1 ⎝ j=
1 ⎠
m l m l
( ) ( j) ( j) ( )
( j)
∑⎜ α α
1
i i ∑x c q x
j i ⎟ ∑c − α + α c
i i i ⎜∑ j i
1 ⎟
(7)
∑ ∑ (8)
Gaussian radial basis function can be chosen as the kernel function.
2
⎛ x−
c ⎞
i
K ( xc , ) = exp −
i ⎜
2 ⎟
(9)
⎝ 2σ
⎠
∂ z =− −
∂x
j
1 ci
m
*
( j ) x −
2 ∑( − α + α )( x c ) exp
i i j i ⎜−
σ = 2
i 1
⎛
⎝
2
2
2
m
m
∂ z ∂ ⎛ 1 ⎛
* ( j)
− ⎞⎞
1 ⎛∂x
i
* j 1 ( j) ( k)
⎞ ⎛ − ⎞
i
= −
2 ( − α + α )( x ) exp
2
2 ( ) 2 ( )( ) exp
i i j
− ci
− =− − α + α − x
i i
j ci xk
ci
⎜− 2
xj xk x ⎜ ∑
x c
k
σ ⎜
i= 1 2σ
⎟
⎟ ∑ ⎜ − − ⎟
x c
∂ ∂ ∂ ⎝ ⎠ σ i=
1 ⎝∂xk
σ ⎠
⎜ 2σ
⎟(11)
⎝ ⎠ ⎝ ⎠
2
σ
2
⎞
⎟
⎠
q−2
(2)
(3)
(10)
-401-