55 max
x f
4.6 Subject
to
k i
k it
k jt
n j
j
y s
r x
≥ +
∑
=
1
,
m i
, 1
=
,
T t
, 1
=
,
υ ,
1 =
k
, 4.7
,
2 1
k i
k k
it T
t k
t
y Y
F s
p ≤
∑
=
,
m i
, 1
=
,
υ ,
1 =
k
, 4.8
≥
k it
s m
i ,
1 =
,
T t
, 1
=
,
υ ,
1 =
k
, 4.9
X ∈
x
. 4.10
Proof. If
n
x R
∈
is a feasible point of 4.2–4.4, then we can set
+ =
⎟⎟⎠ ⎞
⎜⎜⎝ ⎛
− =
∑
k jt
n j
j k
i k
it
r x
y s
1
,
m i
, 1
=
,
T t
, 1
=
,
υ ,
1 =
k
. The pair
, s
x
is feasible for 4.7–4.10. On the other hand, for any pair
, s
x
, which is feasible for 4.7–4.10, we get
⎟⎟⎠ ⎞
⎜⎜⎝ ⎛
− ≥
∑
= k
jt n
j j
k i
k it
r x
y s
1
,
m i
, 1
=
,
T t
, 1
=
,
υ ,
1 =
k
. Taking the expected value of both sides and using 3.21 we obtain
, ,
;
2 2
k i
k k
i k
y Y
F y
x R
F ≤
m i
, 1
=
,
υ ,
1 =
k
. Proposition 4.1 implies that
x
is feasible for problem 4.2–4.4. □
4.3 Optimality and duality
From now on we shall assume that the probability distributions of the returns and of the reference outcome
k
Y
are discrete with finitely many realizations. We also assume that the realizations of
k
Y
are ordered:
k m
k
y y
...
1
,
υ ,
1 =
k
. The probabilities of the realizations are denoted by
m i
k i
, 1
, =
π
. We define the set
U
of functions
R R
→ :
u
satisfying the following conditions:
⋅ u
is concave and nondecreasing;
⋅ u
is piecewise linear with break points
k i
y
,
m i
, 1
=
,
υ ,
1 =
k
;
= t
u
for all
k m
y t
≥
,
υ ,
1 =
k
. It is evident that
U
is a convex cone.
56
Let us define the Lagrangian of 4.2–4.4,
R R
→ ×
υ
U
n
L :
, as follows
∑
=
− +
=
υ υ
1
,
k k
k k
k
Y u
E x
R Eu
x f
u x
L
. 4.11 where
,...,
1
υ υ
u u
u =
. It is well defined, because for every
U ∈
k
u
and every
n
x R
∈
the expected value
] [
x R
u
k k
E
exists and is finite.
Theorem 4.4 If
xˆ
is an optimal solution of 4.2–4.4 then there exists a function
υ υ
U ∈
ˆ u
, such that
ˆ ,
max ˆ
, ˆ
υ υ
u x
L u
x L
x
X
∈
=
4.12 and
] ˆ
[ ]
ˆ ˆ
[
k k
k k
Y u
x R
u E
E =
,
υ ,
1 =
k
, 4.13 where
ˆ ,...,
ˆ ˆ
1
υ υ
u u
u =
. Conversely, if for some function
υ υ
U ∈
ˆ u
an optimal solution
xˆ
of 4.12 satisfies 4.3 and 4.13, then
xˆ
is an optimal solution of 4.11–4.13.
Proof. By Proposition 4.2 problem 4.2–4.4 is equivalent to problem 4.6–4.10. We
associate Lagrange multipliers
m
R ∈
μ
with constraints 4.8 and we formulate the Lagrangian
Λ
:
R R
R R
→ ×
×
m mT
n υ
as follows:
⎟ ⎠
⎞ ⎜
⎝ ⎛
− +
= Λ
∑ ∑∑
= = =
T t
k it
k t
k i
k k
m i
k i
s p
y Y
F x
f s
x
1 2
1 1
, ,
,
υ υ
υ
μ μ
where
,...,
1
υ υ
s s
s =
,
,...,
1
υ υ
μ μ
μ =
and
,...,
1 k
m k
k
μ μ
μ =
. Let us define the set
⎭ ⎬
⎫ ⎩
⎨ ⎧
= =
= ≥
+ ×
∈ =
∑
= +
υ
υ υ
, 1
, ,
1 ,
, 1
, :
,
1
k T
t m
i y
s r
x R
s x
Z
k i
n j
k it
k jt
j mT
X
. Since
Z
is a convex closed polyhedral set, the constraints 4.8 are linear, and the objective function is concave, if the point
ˆ ,
ˆ
υ
s x
is an optimal solution of problem
57
4.2–4.4, then the following Karush-Kuhn-Tucker optimality conditions hold true. There exists a vector of multipliers
ˆ ≥
μ
such that:
ˆ ,
, max
ˆ ,
ˆ ,
ˆ
,
υ υ
υ υ
μ μ
υ
s x
s x
Z s
x
Λ =
Λ
∈
4.14 and
υ μ
, 1
, ,
1 ,
, ˆ
1 2
= =
= ⎟
⎠ ⎞
⎜ ⎝
⎛ −
∑
=
k m
i s
p y
Y F
T t
k it
k t
k i
k k
i
. 4.15 We can transform the Lagrangian
Λ
as follows:
∑∑∑ ∑∑
= = =
= =
− +
= Λ
υ υ
υ υ
μ μ
μ
1 1
1 2
1 1
, ,
,
k k
it k
i m
i T
t k
i k
i k
k m
i k
i
s p
y Y
F x
f s
x
∑ ∑∑
∑∑
= = =
= =
− +
=
m i
k it
k i
k T
t k
t k
i k
k m
i k
i
s p
y Y
F x
f
1 1
1 2
1 1
,
μ μ
υ υ
. For any fixed
x
the maximization with respect to
υ
s
such that
Z s
x ∈
,
υ
yields
+ =
⎟⎟⎠ ⎞
⎜⎜⎝ ⎛
− =
∑
n j
k jt
j k
i k
it
r x
y s
1
, ,
1 ,
, 1
, ,
1 ,
] [
υ
= =
= −
=
+
k T
t m
i x
R y
t k
k i
where
t k
x R
] [
is the t-th realization of the portfolio return. Define the functions
R R
→ :
k i
u
,
m i
, 1
=
,
υ ,
1 =
k
by
+
− −
= η
η
k i
k i
y u
, and let
1
η μ
η
μ
∑
=
=
m i
k i
k i
k
u u
k
,
,...,
1 k
m k
k
μ μ
μ =
. Let us observe that
υ μ
U ∈
k
k
u
. We can rewrite the result of maximization of the Lagrangian
Λ
with respect to
s
as follows:
∑ ∑∑
∑ ∑
= = =
= =
=
+ +
= Λ
m i
t k
k i
k i
k T
t k
t k
i k
k m
i k
i s
x R
u p
y Y
F x
f s
x
1 1
1 2
1 1
1
] [
, ,
, max
μ μ
μ
υ υ
υ υ
υ
∑∑ ∑∑
= = = =
+ +
=
υ μ
υ
μ
1 1
2 1
1
] [
,
k t
k k
T t
k t
k i
k k
m i
k i
x R
u p
y Y
F x
f
k
.
58
4.16 Furthermore, we can obtain a similar expression for the sum involving
k
Y
:
+ =
= = = =
∑ ∑∑
∑∑
− =
,
1 1
1 2
1 1
m l
k l
k i
k l
k m
i k
i k
i k
k m
i k
i
y y
y Y
F
π μ
μ
υ υ
+ =
= =
∑ ∑∑
− =
1 1
1 m
i k
l k
i k
i k
m l
k l
y y
μ π
υ
.
1 1
k l
k m
i m
l k
l
y u
k
μ
π
∑∑
= =
− =
Substituting into 4.16, we obtain
[ ]
∑
=
− +
= Λ
υ μ
μ υ
υ
μ
υ
1
] [
] [
, ,
max
k k
k k
k s
Y u
x R
u x
f s
x
k k
E E
,
υ μ
u x
L =
, 4.17
Setting
k k
k
u u
μ
= ˆ
,
υ ,
1 =
k
we conclude that the conditions 4.14 imply 4.12, as required. Furthermore, adding the complementarity conditions 4.15 over
m i
, 1
=
, and using the same transformation we get 4.13.
To prove the converse, let us observe that for every
υ υ
U ∈
ˆ u
we can define
, ˆ
ˆ ˆ
k i
k k
i k
k i
y u
y u
+ −
− =
μ m
i ,
1 =
,
υ ,
1 =
k
. with
ˆ
k
u
−
and
ˆ
k
u
+
denoting the left and right derivatives of
k
uˆ
:
t t
u u
u
k k
t k
− −
=
↑ −
η η
η
η
ˆ ˆ
lim ˆ
and
t t
u u
u
k k
t k
− −
=
↓ +
η η
η
η
ˆ ˆ
lim ˆ
. Since
k
uˆ
is concave,
ˆ ≥
k
μ
. Using the elementary functions
+
− −
= η
η
k i
k i
y u
we can represent
k
uˆ
as follows:
∑
=
=
m i
k i
k i
k
u u
1
ˆ ˆ
η μ
η
,
υ ,
1 =
k
. Consequently, correspondence 4.17 holds true for
k
μˆ
, and
ˆ
υ
u
. Therefore, if
xˆ
is the maximizer of 4.12, then the pair
ˆ ,
ˆ
υ
s x
, with
ˆ ,...,
ˆ ˆ
1 υ
υ
s s
s =
,
+ =
⎟⎟⎠ ⎞
⎜⎜⎝ ⎛
− =
∑
n j
k jt
j k
i it
r x
y s
1
ˆ ˆ
υ
,
59
is the maximizer of
ˆ ,
,
υ υ
μ s
x Λ
, over
Z s
x ∈
,
υ
. Our result follows then from standard sufficient conditions for problem 4.6–4.10 see,e.g.,Rockafellar [150,
Theorem. 28.1]. We can also develop duality relations for our problem. With the Lagrangian 4.11 we
can associate the dual function
, max
υ υ
u x
L u
D
x
X
∈
=
. We are allowed to write the maximization operation here, because the set
X
is compact and
,
υ
u x
L
is continuous. The dual problem has the form
⎪⎩ ⎪
⎨ ⎧
∈
∈
. ,
min
υ υ
υ
υ υ
U
U
u u
D
u
4.18 The set
υ
U
is a closed convex cone and
⋅ D
is a convex functional, so 4.18 is a convex optimization problem.
□
Theorem 4.5 Assume that 4.11–4.13 has an optimal solution. Then problem 4.18
has an optimal solution and the optimal values of both problems coincide. Furthermore, the set of optimal solutions of 3.31 is the set of functions
U ∈
uˆ
satisfying 4.11– 4.13 for an optimal solution
xˆ
of 4.11–4.13.
Proof. The theorem is consequence of Theorem 4.4 and general duality relations in
convex non-linear programming see Beale [10, Theorem. 2.165]. Note that all constraints of our problem are linear or convex polyhedral, and therefore we do not need
any constraints qualification conditions here. ■
4.4 Splitting
