Optimizing Optimization: The Next Generation of Optimization Applications and Theory (Quantitative Finance)

Heuristic portfolio optimization: Bayesian updating with the Johnson family of distributions 273

λ

p 

p

m

p

n

11 24

⎛^21

⎝

⎜⎜

⎜

⎞

⎠

⎟⎟

⎟

⎛

⎝

⎜⎜

⎜

⎞

⎠

⎟⎟

⎟

⎛

⎝

⎜⎜

⎜

⎞

⎠

⎟⎟

⎟⎟

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

//2

2

1

0

p

mn



⎛

⎝

⎜⎜

⎜⎜

⎞

⎠

⎟⎟

⎟⎟⎟

()λ

ε

λ











xx

p

p

n

p

m

p

mn

zz

22

21

2

⎛

⎝

⎜⎜

⎜

⎞

⎠

⎟⎟

⎟

⎛

⎝

⎜⎜

⎜⎜

⎞

⎠

⎟⎟

⎟⎟⎟

Parameters estimates of the S L distribution

ηγη=

⎛

⎝

⎜⎜

⎜⎜

⎞

⎠

⎟⎟

⎟⎟

⎛

⎝

⎜⎜

⎜⎜

⎞

⎠

⎟⎟

⎟⎟

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

2

,

1

1/2

z

m

p

m

p

p

m

p

ln

 ln



⎦⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

⎛

⎝

⎜⎜

⎜⎜

⎞

⎠

⎟⎟

⎟⎟

⎛

⎝

⎜⎜

⎜⎜

⎞

⎠

⎟⎟

⎟⎟

,1,

2

1

21

λε









xx

p

m

p

m

p

zz

Parameters estimates of the S N distribution

ηγη λε



 

2

,

2

,1,0

z

m

xxzz

Threshold acceptance pseudocode

The utility maximization algorithm comprises three constituent blocks: (1) the
optimization routine; (2) the definition of a neighbor; and (3) choice of thresh-
old sequence. The pseudocode for each of them is described below.

Algorithm 1: The optimization routine

1. 
   

   Compute the threshold sequence, τ  { τ (^) I } , in accordance with
   Algorithm 3.

   
   


2. 
   

   For R  1,..., N (^) Restarts.

   
   


3. 
   

   Randomly generate a current solution w c by drawing random weights
   from a beta distribution, Beta( α, β ), such that || w c || 1  1. The α and β
   control the sparsity of the weight vector. We set α  1.5 and β  ( N  1)
   α , where N is the number of assets in the universe.