On the Computation of the Efficient Frontier of the Portfolio Selection Problem

Algorithm 3

The KTEF-C algorithm.

Inputs 𝐕, 𝐞, 𝐥, 𝐮

 uses H, R, KTEF

  Set 𝑃=0, 𝐽={[𝑟_min𝑖,𝑟_max𝑖]}^𝑃_𝑖=1= an empty sequence,

  𝑆=∅

  Set 𝑟⁻=𝐞𝐇(𝐕, 𝐞, 𝐥, 𝐮, 0), 𝑟⁺=R(𝐕, 𝐞, 𝐥, 𝐮)

 (1) Set 𝑘=1, 𝑎=𝑟⁻.

While 𝑘≤𝑃 and 𝑎=𝑟_min𝑘 do

  𝑘=𝑘+1, 𝑎=𝑟_max𝑘

 If 𝑎<𝑟⁺ then

  If 𝑘>𝑃 then 𝑏=𝑟⁺ else 𝑏=𝑟_min𝑘

  𝑟=(𝑎+𝑏)/2

  else stop

 (2) Set 𝐱=H(𝐕, 𝐞, 𝐥, 𝐮, 𝑟)

  Calculate 𝐿 and 𝑈 from 𝐱 according to (3.6)

  Call KTEF(𝐕, 𝐞, 𝐥, 𝐮, 𝐿, 𝑈)

    on error (KTEF has stoped in a degenerate case)

  set 𝑟=(𝑎+𝑟)/2 go to (2)

  Set 𝑆=𝑆∪{ktef(𝐕,𝐞,𝐥,𝐮,𝐿,𝑈)}

  Set 𝑃=𝑃+1

  Set 𝐽=𝐽∪{[𝑟_min,𝑟_max]} (𝑟_min, 𝑟_max are part of the

  output of KTEF). The new interval should be inserted

  in the right place to preserve the increasing order of the

  sequence 𝐽={[𝑟_min𝑖,𝑟_max𝑖]}^𝑁_𝑖=1

 goto (1)

Output 𝑆