274 Optimizing Optimization
For I 1,..., N (^) Iterations.
For S 1,..., N (^) Steps.
Generate a neighbor w n N()
wc
in accordance with Algorithm 2 and
compute (^) ΔUU(()wwc) n.
7. If Δ τ (^) I then redefine w c w n.
- Record UU() (wR )
∗ wc
and wwR∗ c. - The optimal solution, w , is given by ww.max( R)Restarts
R R
∗ N
1Algorithm 2: The definition of a neighbor
- Randomly select an asset with positive weight, i , and one other asset, j i.
- Draw a uniform random number, U Uniform (0, w max ), where w max is
the maximum increment allowed. - If wUic 0 and wUj
c 1
then wwUinci and wwUj
n
j
c ,
Else if wUic 0 and wUjc 1 then wi
n 0
and wwwjncj ic,
Else if wUic 0 and wUj
c 1
then wwwi
n
i
c
j
c 1
and wj
n1,
Else if wUic 0 and wUcj 1 then repeat Steps 2 and 3.Algorithm 3: The threshold sequence
Set an N (^) Iterations -dimensional vector of percentage rejections that will be
used in each stage of the iteration process. For instance, the vector
ξ (0.75, 0.5, 0.25, 0) will reject 25% of the most distant neighbors in
the first iteration, 50% in the second, etc.
Randomly choose w c.
For S 1, ... , N (^) Steps.
Compute w n N()wc in accordance with Algorithm 3 and compute
ΔS|)UU((.)|
wwc n
Compute the empirical distribution, F (^) Δ of Δ (^) S , S 1, ... , N (^) Steps.
Compute the threshold sequence τξIIF^1 () for I 1, ... , N (^) Iterations.
The optimality of z*
The purpose of this section is to investigate the optimality of z *. To this end,
we start by uniformly generating parameters from an unbounded Johnson den-
sity, S U , with supports given by ε [ 0.1, 0.1], λ [0.1, 0.5], γ [0, 0.2],
and η [1, 2]. The random parameters are then used to simulate a vector of
length N { 100, 250, 500 } from the density in accordance with Equation
(11.4). Finally, the quantile estimator, described in Section 11.3.2, is applied
to the simulated series and the parameter estimates are compared to their true