Brief Notes #9
CALCULATION OF RELIABILITY INDEX, β
Z ~ (0, I)
X ~ (m, Σ)
1. Select initial unit vector, α o
1. Select initial unit vector, α o
2. Find z *o = β o α o
where β o = min{β : βα o ∈ D F }
2. Find x *o = m + δα o ∈ Β and calculate
β o as:
3. Linearize Β around z *o
Let γ be the unit external vector:
o
γo =
grad g(z)
grad g(z)
z *o
⎡ ∂g (z) ⎤
⎢
⎥
∝ ⎢ ∂z i
⎥
⎢ #
⎥
z *o ⎦⎥
⎣⎢
4. Go back to step 2 with α o replaced with α1 = γ o .
Iterate until convergence in β and z * .
−1
β o = δ(α T
o Σ αo )
1
2
3. Linearize Β at x *o . (Calculate γ as in
o
z space).
4. The next search direction α1 is that of the β
point according to the linearized boundary.
From analytical geometry,
α1 ∝ Σ γ o .
5. Go back to Step 2 until convergence.