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

Computing optimal mean/downside risk frontiers: the role of ellipticity 181

where λ is a (2  1) vector of Lagrangians. The first-order conditions are:

∂

∂

∂

∂

∂

∂

∂

∂

φ

μ

μ φ

σ

σ

p

p

p

p

xx

 2 E

2

λ 0

(8.2)

and

Ex′ Ψp 0

(8.3)

Here ,

∂

∂

μ

μ

p

x



(^) (8.4)
and
∂
∂
σp
x
x
2
 2 Σ
(8.5)
This implies that:
φμ φ^12 ^2 ΣxEλ
(8.6)
or
φμφ^1
1
2
ΣΣ 2 xE^1 λ
(8.7)
or
λ()(EE E′′ΣΣΨ^11 φμφ 1 ^122 p)
(8.8)
Thus , the general “ solution ” x satisfies:
φμφ 1 ΣΣΣΣΨ^1  222 xEEEE^111 ()(′′φ μφ 1 1  2 p)
(8.9)
We now show that:
xEEEΣΣΨ^111 ()′ p
(8.10)