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

Algorithm 1

The KTEF procedure.

Inputs 𝐕, 𝐞, 𝐥, 𝐮, 𝐿, 𝑈.

 Step 1 Set 𝑁^′=𝐿∪𝑈, 𝑁={1,…𝑛}∼𝑁^′, extract the vectors 𝐞_𝑁, 𝐞_𝑁′

    and the submatrices 𝐕₀, 𝐖 and 𝐙 (see (A.1) in the Appendix),

    and the vector 𝐛 of active bounds.

 Step 2 If #(𝑁)≤1 the case is degenerate (STOP).

 Step 3 Calculate the inverse matrix 𝐕₀⁻¹.

 Step 4 Calculate 𝐴, 𝐵, 𝐶, 𝐷, 𝐸, 𝐹.

 Step 5 Calculate 𝜆₀, 𝜆₁, 𝜇₀, 𝜇₁ according to (A.14).

 Step 6 Calculate 𝐠_𝑁,𝐡_𝑁 according to (A.11), as well as

    𝐠=(𝐠_𝑁,𝐛), 𝐡=(𝐡_𝑁,𝟎).

 Step 7 Calculate 𝝀_𝐿0, 𝝀_𝐿1, 𝝁_𝑈0, 𝝁_𝑈1 according to (A.15).

 Step 8 Define a set 𝐿𝐵 of lower bounds for 𝑟 containing:

    (i) (−𝐸𝐶+𝐴+𝐴𝐹)/𝐶,

    (ii) (𝐥_𝑖−𝐠_𝑖)/𝐡_𝑖 for 𝑖∈𝑁 provided that 𝐡_𝑖>0,

    (iii)(𝐮_𝑖−𝐠_𝑖)/𝐡_𝑖 for 𝑖∈𝑁 provided that 𝐡_𝑖<0,

    (iv) −𝝀_0𝑖/𝝀_1𝑖 for 𝑖∈𝐿 provided that 𝝀_1𝑖<0, (where 𝝀₀=(𝝀_𝐿0,𝟎),

        𝝀₁=(𝝀_𝐿1,0)),

    (v) −𝝁_0𝑖/𝝁_1𝑖 for 𝑖∈𝑈 provided that 𝝁_1𝑖<0 (where 𝝁₀=(𝝁_𝑈0,𝟎),

        𝝁₁=(𝝁_𝑈1,𝟎),

 Step 9 Define 𝑟_min=max𝐿𝐵.

 Step 10 Define a set 𝑈𝐵 of upper bounds for 𝑟 containing:

    (i) (𝐥_𝑖−𝐠_𝑖)/𝐡_𝑖 for 𝑖∈𝑁 provided that 𝐡_𝑖<0,

    (ii) (𝐮_𝑖−𝐠_𝑖)/𝐡_𝑖 for 𝑖∈𝑁 provided that 𝐡_𝑖>0,

    (iii) −𝝀_0𝑖/𝝀_1𝑖 for 𝑖∈𝐿 provided that 𝝀_1𝑖<0,

    (iv) −𝝁_0𝑖/𝝁_1𝑖 for 𝑖∈𝑈 provided that 𝝁_1𝑖>0,

 Step 11 Define 𝑟_max=min𝑈𝐵.

 Step 12 If 𝑟_min≥𝑟_max the case is degenerate (STOP).

 Step 13 Calculate 𝑎, 𝑏, 𝑐 according to (A.19).

Outputs 𝑟_min, 𝑟_max, 𝐠, 𝐡, 𝜆₀, 𝜆₁, 𝝀_𝐿0, 𝝀_𝐿1, 𝝁_𝑈0, 𝝁_𝑈1, 𝑎, 𝑏, 𝑐.