64
4.5 Decomposition
It follows that the dual functional can be expressed as a weighted sum of 2
+ T
functions 4.30–4.32. In order to analyze their properties and to develop a numerical method we need to
find a proper representation of the utility function
u
. We represent the function
u
by its slopes between break points. Let us denote the values of
u
at its break points by
k l
k l
y u
u =
, υ
, 1
, ,
1 =
= k
m l
. We introduce the slope variables
k l
k l
y u
−
= β
, υ
, 1
, ,
1 =
= k
m l
The vectors ,....
,..., ,
,...,
2 2
1 1
1 1
m m
β β
β β
β = is nonnegative, because u is
nondecreasing. As
u
is concave,
k l
k l
1 +
≥ β
β ,
υ ,
1 ,
1 ,
1 =
− =
k m
l . We can represent
the values of u at break points as follows
∑
−
− −
=
l l
k l
k l
k l
k l
y y
u
1
β ,
1 ,
1 −
= m l
. The function 4.24 takes on the form
⎥ ⎦
⎤ ⎢
⎣ ⎡
− −
− =
− =
∑
− ≤
≤ ≤
≤
l l
k l
k t
k l
k l
k l
m l
k l
k l
k l
m l
k t
k t
y y
y y
u u
D max
] [
max ,
1 1
1
θ β
θ θ
. In this way we have expressed
,
k t
k t
u D
θ as a functions of the slope vector
m
R ∈
β and of
+
∈R
k t
θ . We denote
⎥ ⎦
⎤ ⎢
⎣ ⎡
− −
− =
∑
− ≤
≤
l l
k l
k t
k l
k l
k l
m l
k t
k t
y y
y B
max ,
1 1
θ β
θ β
. 4.34
Observe that
k t
B is the maximum of finitely many linear functions in its domain. The domain is a convex polyhedron defined by
k l
k t
β θ ≤
≤ .
65 Consequently,
k t
B is a convex polyhedral function. Therefore its subgradient at a point
,
k t
θ β
of the domain can be calculated as the gradient of the linear function at which the maximum in 4.34 is attained. Let
l be the index of this linear function. Denoting by
l
δ the l th unit vector in
m
R we obtain that:
⎟⎟⎠ ⎞
⎜⎜⎝ ⎛
− −
−
∑
−
,
1
l l
k l
k l
k l
l
y y
y
δ is the subgradient of
,
k t
k t
B θ
β .
Similarly, function 3.32 can be represented as a function
k T
B
1 +
of the slope vectors
β :
∑ ∑
= −
+
− =
m l
l l
k l
k l
k l
k l
k T
y y
B
1 1
1
β π
β .
It is linear in β and its gradient has the form
1 1
k l
k l
l l
k l
n l
l
y y
− =
−
∑ ∑
π δ
. Finally, denoting by
j the index at which the maximum in 4.30 is attained, we see that the vector with coordinates
k t
j k
t
r p
, υ
, 1
, ,
1 =
= k
T t
, 4.35
is a subgradient of
k
D .
Summing up, with our representation of the utility function by its slopes, the dual function is a sum of T + 2 convex polyhedral functions with known domains.
Moreover, their subgradients are readily available. Therefore the dual problem can be solved by nonsmooth optimization methods see Dentcheva and
Ruszczynski [41], Beale [8] and Bonnans and Shapiro [18]. We have developed a specialized version of the regularized decomposition method described in
Dentcheva and Ruszczy´nski [41] and Ogryezak and Ruszczy’nski [127]. This approach is particularly suitable, because the dual function is a sum of very many
polyhedral functions.
66 After the dual problem is solved, we obtain not only the optimal dual solution
ˆ ,
ˆ θ
β , but also a collection of active cutting planes for each component of the dual function.
Let us denote by
k
j the collection of active cuts for
k
D . Each cutting plane for
k
D provides a subgradient 4.35 at the optimal dual solution. A convex
combination of these subgradients provides the subgradient of
k
D that enters the
optimality conditions for the dual problem. The coefficients of this convex combination are also identified by the regularized decomposition method. Let
k
g denote this subgradient and let
, J
j v
k j
∈
the corresponding coefficients. Then
∑ ∑
∈ =
=
J j
1
k j
k jt
k t
T t
k t
k
v r
p g
δ ,
where
≥
k j
v
,
∑
∈
=
J j
1
k j
v
. For each t the subgradient of
k t
B with respect to
k t
θ entering the optimality conditions is
} 34
. 4
in maximizer
is :
{ ˆ
l y
conv v
k l
k t
∈
. Therefore
ˆ
1
= −
∑
= T
k t
k t
k
v p
g
l
, Using these relations we can verify that
vˆ
are the vector of optimal portfolio returns in scenarios
T t
, 1
= . Thus the optimal portfolio has the weights
⎩ ⎨
⎧ ∉
= ∈
= .
, ˆ
, ,
ˆ J
j x
J j
v x
j j
j
67
68
C
HAPTER 5
A FUZZY APPROACH TO PORTFOLIO OPTIMIZATION
5.1 Introduction