On Portfolio Optimization Using Fuzzy Decisions.
ON PORTFOLIO OPTIMIZATION USING FUZZY DECISIONS *)
Supian SUDRADJAT and Vasile PREDA
Departement of Mathematics, Faculty of Mathematics and Natural Sciences Padjadjaran University,
Bandung, Indonesia Jl. Raya Bandung Sumedang km.21 Bandung 40600 Indonesia, email :
adjat03@yahoo.com
Faculty of Mathematics and Computer Science Bucharest University Romania 14 Academiei Str.,
Sector 1, Bucharest 010014 email : preda@fmi.unibuc.ro
Abstract.
We consider stochastic optimization problems involving stochastic dominance constraints. We develop portfolio optimization
model involving stochastic dominance constrains using fuzzy decisions and we concentrate on fuzzy linear programming
problems with only fuzzy technological coefficients and application/implementation of modified subgradient method to fuzy
linear programming problems.
AMS subject classifications. Primary, 90C15, 90C29, 90C46, 90C48, 90C70;
Secondary, 46N10, 60E15, 91B06
1. Introduction
The problem of optimizing a portfolio of finitely many assets is a classical problem in theoretical and
computational finance. Since the seminal work of Markowitz [11, 2] it is generally agreed that portfolio
performance should be measured in two distinct dimensions: the mean describing the expected return,
and the risk which measures the uncertainty of the return. As one of theoretical approach to the
portfolio selection problem is that of stochastic dominance (see [18, 7]).
The rest of the paper is organized in the following manner. Section 1 provides some newly obtained
results on stochastic dominance, motivation for posing the portfolio selection problem in the fuzzy
decisions theory framework. Section 2 describes the formulation of the portfolio selection problems.
Section 3, consider an overview of portfolio problem fuzzy technological coefficient. Section 4, give a
solution of defuzzificated problems.
2. Portfolio problem
Let R ,...,
R n be random returns of assets 1 ,..., n . We assume that E [ R j ] 0 we look for a compromise between the mean and the risk.
Here,
The second approach is to select a certain utility function u : R ® R and to formulate
the following optimization problem
max E u (R ( x ) ) .
(2.3)
x Î X
[
]
It is usually required that the function u(∙) is concave and nondecreasing, thus representing preferences
of a riskaverse decision maker.
In this paper the portfolio optimization, we shall consider stochastic dominance relations between
random returns in (2.1) if to avoid placing the decision vector, x, in a subscript expression, we shall
simply write
F ( h ; x ) = F R ( x ) ( h ) and F2 ( h ; x ) = F 2 R ( x ) ( h ) .
We say that portfolio x dominates portfolio y under the FSD rules, if
F ( R ( x ); h ) £ F ( R ( y ); h ) for all h Î R ,
where at least one strict inequality holds. Similarly, we say that x dominates y under the SSD rules
( R ( x ) f SSD R ( y )) , if
F2 ( R ( x ); h ) £ F 2 ( R ( y ); h ) for all h Î R ,
with at least one inequality strict. Recall that the individual returns R j have finite expected values and
thus the function F 2 ( R ( x ); × h ) is well defined.
Stochastic dominance relations are of crucial importance for decisions theory. It is known that
R ( x ) f FSD R ( y ) if and only if
(2.4)
E[ u ( R ( x ))] ³ E [ u ( R ( y ))]
for any nondecreasing function u(∙) for which these expected values are finite. Furthermore,
R ( x ) f SSD R ( y ) if and only if (2.4) holds true for every nondecreasing and concave u(∙) for which
these expected values are finite [4, 5, 8].
A portfolio x is called SSDefficient (or FSDefficient) in a set of portfolios X if there is no y Î X
such that ( R ( y ) f SSD R ( x )) (or ( R ( y ) f FSD R ( x )) ).
We shall focus our attention on the SSD relation, because of its consistency with riskaverse
preferences: if ( R ( y ) f SSD R ( x )) , then portfolio x is preferred to y by all riskaverse decision
makers.
The starting point for our model is the assumption that a reference random return Y having a finite
expected value is available. It may be an index or our current portfolio. Our intention is to have the
return of the new portfolio, R (x ) , preferable over Y. Therefore, we introduce the following
optimization problem [4]:
(2.10)
max f ( x )
Subject to R ( x ) f ( 2 ) Y ,
(2.11)
x Î X
(2.9)
2
*)
International Congress Industry and Appplied Mathemaic (ICIAM07) ElvetiaZurich
Switzerland 1620 July 2007
Here f : X ® R is a concave continuous functional. In particular, we may use
f ( x ) = E[ R ( x )]
and this will still lead to nontrivial solutions, due to the presence of the dominance constraint
Proposition 2.1 [4] Assume that Y has a discrete distribution with realizations y i , i = 1 , m . Then
relation (2.10) is equivalent to
(2.12)
E [( y i - R ( x )) + ] £ E [( y i - Y ) + ], "i = 1 , m .
Let us assume now that the returns have a discrete joint distribution with realizations r jt , t = 1, T ,
j = 1, n , attained with probabilities p t , t = 1, T . The formulation of the stochastic dominance
relation (2.10) resp. (2.12) simplifies even further. Introducing variables s it representing shortfall of
R(x) below yi in realization t, i = 1, m and t = 1, T , we obtain the following result.
Proposition 2.2 The problem (2.9)(2.11) is equivalent to the problem:
max f ( x )
(2.13)
(2.14)
Subject to -
n
år
j =1
jt
x j - s it £ - y i , i = 1, m , t = 1, T ,
T
å p s £ F ( Y ; y ) , i = 1, m ,
(2.15)
t = 1
t it
2
i
s it ³ 0 , i = 1, m , t = 1, T
(2.16)
n
å x £ 1
(2.17)
j =1
j
n
- å x j £ -1
(2.18)
j =1
x j ³ 0 ,
(2.19)
j = 1, n ,
and problema (2.13)–(2.19) can be written as
n
max j ( X ) = max å c j X j
(2.20)
j = 1
(2.21)
Subject to
n + mT
åa
j =1
(2.22)
ij
X j £ b i , i = 1, mT + m + 2 ,
X j ³ 0 , j = 1 , n + mT ,
Where X = ( x 1 ,..., x n , s 11 ,..., s 1 T , s 21 ,..., s 2 T ,..., s m 1 ,..., s mT )
ì- r ij , j = 1 , n , i = Km +1 , ( K +1 ) m , K = 0 , ( T -1 )
ï
a ij = í -1 , i = Km +1 , ( K +1 ) m , K = 0 , ( T -1 ) , and j = n +1 , n +T ( i -1 ) +1
ï0
, otherwise
î
ì1 , j = 1 , n , i = mT + 1
a ij = í
î0 , otherwise
3
*)
International Congress Industry and Appplied Mathemaic (ICIAM07) ElvetiaZurich
Switzerland 1620 July 2007
ì- 1 , j = 1 , n , i = mT + 2
a ij = í
î 0 , otherwise
ìp
, j = n +T ( K -1 ) +1 , n +TK , K =1 , m , i = mT + 3 , mT + m + 2
a ij = í j - n -T ( K -1 )
, otherwise
î 0
In the next section we extended this result to fuzzy decisions theory.
3. Portfolio problems with fuzzy technological coefficients
In this section presents an approach to portfolio selection using fuzzy decisions theory.
We consider a linear programming problem (2.20) – (2.22) with fuzzy technological coefficients [13].
n
max j ( X ) = max å c j X j
(3.1)
j = 1
(3.2)
Subject to
n + mT
å a~ X
j =1
ij
j
£ b i , i = 1, mT + m + 2 ,
X j ³ 0 , j = 1 , n + mT .
(3.3)
~ is a fuzzy number for any i and j.
Assumption 3.1. a
ij
In this case we consider the following membership functions:
(i) 1. For i = Km + 1, ( K + 1 ) m , K = 0 , ( T - 1 ) and j = 1, n
ì 1
if t < - r ji ,
ï
m a ( t ) = í( -r ij + d ij - t ) / d ij if - r ij £ t < -r ij + d ij ,
ï 0
if t ³ - r ij + d ij .
î
ij
2. For i = Km + 1, ( K + 1 ) m , K = 0 , ( T - 1 ) and j=n+T(iKm1)+K+1
ì1
if t < -1
ï
m a ( t ) = í( -1 + d ij - t ) / d ij if - 1 £ t < -1 + d ij ,
ï0
if t ³ -1 + d ij ,
î
ij
(ii) For i = mT + 3, mT + m + 2 , K = 1, m and j = n + T ( K - 1 ) + 1 , n + TK
ì
1
if t < p
ï
m a ( t ) = í( p j - n -T ( K -1 ) + d ij - t ) / d ij if p
ï
0
if t ³ p
î
,
£ t < p
+ d ij ,
j - n -T ( K -1 )
ij
j - n -T ( K -1 )
j - n -T ( K -1 )
+ d ij ,
j - n -T ( K -1 )
where t Î R and d ij > 0 for all i = 1, mT + m + 2 , K = 0, ( T - 1 ) and j = 1 , n + mT .
4
*)
International Congress Industry and Appplied Mathemaic (ICIAM07) ElvetiaZurich
Switzerland 1620 July 2007
For defuzzification of this problem, we first fuzzify the objective function. This is done by
calculating the lower and upper bound of the optimal values first. The bounds of the optimal
values z l and z u are obtained by solving the standard linear programming problems
(3. 4)
(3. 5)
z1 = max j ( X )
Subject to
n + mT
åa
j =1
ij
X j £ b i , i = 1, mT + m + 2
(3.6)
X j ³ 0 , j = 1 , n + mT
and
(3.7)
z 2 = max j ( X )
(3.8)
Subject to
n + mT
å aˆ X
j =1
(3.9)
ij
j
£ b i , i = 1, mT + m + 2
X j ³ 0 , j = 1 , n + mT
where
ì- r ij + d ij , j =1 , n , i = Km +1 , ( K +1 ) m and K = 0 , ( T -1 )
ï
ˆij = í -1 + d ij , i = Km +1 , ( Km +1 ) m , j = n +1 , n + T ( i -1 ) +1 , and K = 0 , ( T -1 )
a
ï d
, otherwise
î ij
ìï1 + d ij , j = 1 , n , , i = mT + 1
ˆij = í
a
, otherwise
ïî d ij
ìï - 1 + d ij , j = 1 , n , i = mT + 2
ˆ ij = í
a
, otherwise
ïîd ij
The objective function takes values between z 1 and z 2 while technological coefficients vary
between a ij and aij + d ij . Let zl = min( z 1 , z 2 ) and z u = max( z 1 , z 2 ) . Then z l and z u are
called the lower and upper bounds of the optimal values, respectively.
Assumption 3.2. The linear crisp problem (3.4)(3.6) and (3.7)(3.9) have finite optimal values. In this
n
case the fuzzy set of optimal values, G, which is subset of R , is defined as [10]
n
ì
0
if
c j X j < z l
ï
å
=
j
1
ï
n
ï n
(3.10) m G ( X ) = í ( å c j X j - z l ) /( z u - z l ) if z l £ å c j X j £ z u
j =1
ï j =1
n
ï
if å c j X j ³ z u
ï1
j =1
î
n + mT
The fuzzy set of the ith constraint, C i , which is a subset of R
, is defined by:
(i) 1. For i = Km + 1, ( K + 1 ) m and K = 0, ( T - 1 )
5
*)
International Congress Industry and Appplied Mathemaic (ICIAM07) ElvetiaZurich
Switzerland 1620 July 2007
(3.11)
n
ì
, b i < -år ji X j
ï0
j = 1
ï
n
n
n
n
ï
m C ( X ) = í ( b i + år ij X j ) / åd ij X j , - år ij X j £ b i < å( -r ij + d ij ) X j
j =1
j =1
j =1
j =1
ï
n
ï
, b i ³ å( -r ji + d ij ) X j
ï1
j =1
î
i
2. For i = Km + 1, ( K + 1 ) m and K = 0, ( T - 1 )
n ( i , K )
ì
b
X j
0
,
<
ï
å
i
j
n
=
+
1
ï
n ( i , K )
n ( i , K )
n ( i , K )
n
ï
(3.12) m C ( X ) = í( b i + å X j ) / åd ij X j , - å X j £ b i < å( -1 + d ij ) X j
j =n +1
j =1
j =n +1
j =n +1
ï
n ( i , K )
ï
1
, b i ³ å( -1 + d ij ) X j
ï
j =n +1
î
i
where n(i,K)=n+T(iKm1)+K+1
(ii) For i = mT + 3, mT + m + 2 , and K = 1, m
n +TK
ì
<
b
p j - n -T ( K -1 ) X j ,
0
,
ï
å
i
=
+
-1 )
j
n
T
K
(
ï
n +TK
n +TK
n +TK
n +TK
ï
£
<
p
X
d
X
p
X
b
)
/
,
( p j -n -T ( K -1 ) +d ij ) X j ,
(3.13) m C ( X ) = í( b
å
å
å
å
i
j -n -T ( K -1 ) j
ij j
j -n -T ( K -1 ) j
i
=
+
=
+
=
+
=
+
-1 )
j
n
T
K
j
n
T
K
j
n
T
K
j
n
T
K
(
1
)
(
1
)
(
1
)
(
ï
n +TK
ï
, b i ³ å( p j -n -T ( K -1 ) +d ij ) X j .
ï 1
j =n +T ( K -1 )
î
i
By using the definition of the fuzzy decisions proposed by Bellman and Zadeh [1], we have
m D ( X ) = min( mG ( X ), min ( mC j ( X ))) .
j
i.e.
max ( m D ( X )) = max min( mG ( X ), min ( mC ( X ))) .
x ³ 0
j
x ³ 0
j
Consequently, the problem (3.1)(33) becomes to the following optimization problem
(3.14)
max l
m G ( X ) ³ l
(3.15)
(3.16)
m C ( X ) ³ l , i = 1 , mT + m + 2 ,
(3.17)
X j ³ 0 , 0 £ l £ 1 , j = 1, mT .
i
By using (3.10) and (3.11)(3.13), we obtain the following theorem.
Theorem 3.1 The problem (3.1)(3.3) is reduced to one of the following crisp
problems :
6
*)
International Congress Industry and Appplied Mathemaic (ICIAM07) ElvetiaZurich
Switzerland 1620 July 2007
max l
(3. 18)
(3.19)
n
l ( z 1 - z 2 ) - å c j X j + z 2 £ 0
j = 1
(3.20)
n + mT
å a ˆ ( l ) X - b £ 0 , i = 1, mT + m + 2 ,
j =1
(3.21)
ij
j
i
X j ³ 0 , 0 £ l £ 1 , j = 1, mT .
where
ì- r ij + ld ij , j =1 , n , i = Km +1 , ( K +1 ) m , and K = 0 , ( T -1 ) ,
ï
aˆ ij ( l) = í -1 + ld ij , i = Km +1 , ( K +1 ) m , K = 0 , ( T -1 ) and j = n +1 , T ( i -1 ) +1 ,
ï ld
, otherwise,
î ij
ìï1 + ld ij , j = 1 , n , i = mT + 1 ,
aˆ ij ( l ) = í
, otherwise ,
ïî ld ij
ìï- 1 + ld ij , j = 1 , n , i = mT + 2 ,
aˆ ij ( l ) = í
, otherwise,
ïî ld ij
ìï p
+ ld ij , j = n + T ( K - 1 ) + 1 , n + TK , i = mT + 3 , mT + m + 2
aˆ ij ( l ) = í j - n -T ( K -1 )
, otherwise .
ïîld ij
Notice that, the constraints in problem (3.18)(3.21) containing the cross product term l X
j
are not
convex. Therefore the solution of this problem requires the special approach adopted for solving
general nonconvex optimization problem.
4. Solution of defuzzificated problems
In this section, we present the modified subgradient method [6] and use it for solving the defuzzificated
problems (3.18)(3.21) for nonconvex constrained problems and can be applied for solving a large class
of such problems.
Notice that, the constraints in problem (3.18)(3.21) generally are not convex. These problems may be
solved either by the fuzzy decisive set method, which is presented by Sakawa and Yana [15], or by the
linearization method of Kettani and Oral [2].
4.1. Application of modified subgradient method to fuzzy linear programming problems.
For applying the subgradient method [6] to the problem (3.18)(3.21), we first formulate it with
equality constraints by using slack variables y 0 and y i , i = 1, mT + m + 2 . Then, we can be written
as
(4.1)
max l
(4.2)
n
g 0 ( X , l , y 0 ) = l ( z 1 - z 2 ) - å c j X j + z 2 + y 0 = 0
j = 1
(4.3)
n + mT
ˆ ij ( l ) X j - b i + y i = 0 , i = 1, mT + m + 2
g i ( X , l , y i ) = å a
j =1
(4.4)
X j ³ 0 , y 0 , y i ³ 0 , 0 £ l £ 1 , j = 1 , n + mT , i = 1, mT + m + 2 .
where y = ( y 0 ,..., y n )
*)
7
International Congress Industry and Appplied Mathemaic (ICIAM07) ElvetiaZurich
Switzerland 1620 July 2007
For this problem the set S can be defined as
S = {( X , p , l ) X ³ 0 , y ³ 0 , 0 £ l £ 1 } .
Since max l = - min( -l ) and g = ( g 0 ,..., g mT + m + 2 ) the augmented Lagrangian associated with
the problem (4.1)(4.4) can be written in the form
1
2
é
ù 2
2
üï mT +m +2 n +mT
n
êìï
æ ˆ
ö ú
L ( x , u , c ) = -l + c êíl ( z 1 - z 2 ) - å c X + z 2 + y 0 ý +
å ç å a
ij ( l) X j - b
i + y i ÷ ú
j j
j
1
=
è
ø ú
i =1
ïþ
j = 1
êïî
ë
û
ö mT +m +2 æ n +mT
æ
n
ˆ ij ( l) X j - b i + y i ö÷.
- m ç l ( z - z ) - å c X +z + y ÷ å u i ç å a
0 ç 1 2
j j 2 0 ÷
j =1
ø
è
1
=
i
j = 1
ø
è
The modified subgradient method may be applied to the problem (4.1)(4.4) in the following way:
1
1
1
1
1
Initialization Step. Choose a vector ( u 0 , u 1 ,..., u mT + m + 2 , c ) with c ³ 0 . Let k = 1 , and go to main
step.
Main Step.
k
k
k
k
Step 1 . Given ( u 0 , u 1 ,..., u mT + m +2 , c ) ; solve the following subproblem :
1
2 ù 2
2
é
ì
ü
ö
æ
n
ú
êï
ï mT +m +2 ç n + mT
ˆ ( l ) X - b + y ÷ ú
min - l + c êíl ( z 1 - z 2 ) - å c X + z 2 + y 0 ý +
å
å a
÷
ç
j j
ij
j
i
i
i =1 è j = 1
ïþ
j = 1
êïî
ø ú
û
ë
ö
ö mT +2 æ n + mT
æ
n
ˆ ( l ) X - b + y ÷ .
- u ç l ( z - z ) - å c X +z + y ÷ - å u ç å a
0 ç 1 2
j j 2 0 ÷ i =1 i ç
ij
j i i ÷
j = 1
ø
è j = 1
ø
è
( X , y , l ) Î S .
k
k
k
k
k
k
k
k
k
k
Let ( X , y , l ) be any solution. If g ( X , y , l ) , then stop; ( u 0 , u 1 ,..., u mT , c ) is a solution
k
k
to dual problem, ( X , l ) is a solution to problem (3.18)(3.21). Otherwise, go to Step 2.
Step 2 . Let
n
æ
ö
u 0 k + 1 = u 0 k - h k çç l ( z 1 - z 2 ) - å c j x j + z 2 + y 0 ÷÷
j =1
è
ø
n
æ
ö
ˆij ( l ) X j - b i + y i ÷ , i = 1, mT + m + 2
u i k + 1 = u i k - h k çç å a
÷
è j =1
ø
k + 1
k
k
k
k
k k
c = c + ( h + e ) g ( X , y , l )
k
k
k
k
where h and e are positive scalar stepsizes and h > e > 0 , replace k by k + 1; and
repeat Step 1.
4.2. The algorithm of the fuzzy decisive set method
This method is based on the idea that, for a fixed value of
*
l ; the problems (3.18)(3.21) is
linear programming problems. Obtaining the optimal solution l to the problems (3.18)(3.21) is
equivalent to determining the maximum value of l so that the feasible set is nonempty. Bellow is
presented the algorithm [6] of this method for the problem (3.18)(3.21).
8
*)
International Congress Industry and Appplied Mathemaic (ICIAM07) ElvetiaZurich
Switzerland 1620 July 2007
Algorithm
Step 1. Set l = 1 and test whether a feasible set satisfying the constraints of the problem (3.18)(3.21)
exists or not using phase one of the simplex method. If a feasible set exists, set l = 1: Otherwise, set
l L = 0
R
and l = 1 and go to the next step.
L
L
R
R
Step 2. For the value of l = ( l + l ) / 2 ; update the value of l and l using the bisection
method as follows :
l L = l
l R = l
if feasible set is nonempty for
l
if feasible set is empty for l .
Consequently, for each l , test whether a feasible set of the problem (3.18)(3.21) exists or
*
not using phase one of the Simplex method and determine the maximum value l satisfying the
constraints of the problem (3.18)(3.21).
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
Bellman, R.E., Zadeh,L.A., Decisionmaking in fuzzy environment, Management Science 17
(1970), B141B164.
Birge J.R. and Louveaux F.V., Introduction to Stochastic Programming, SpringerVerlag, New
York, 1997.
Dentcheva, D. and Ruszczynski, A., “Frontiers of stochastically nondominated portfolio”
Operations research and financial engineering, Princeton University, ORFE 001,2002.
Dentcheva, D. and Ruszczynski, A., Portfolio Optimization with Stochastic Dominance
Constraints, May 12, 2003
Dentcheva, D. and Ruszczynski, A, Optimization with stochastic dominance constraints, Siam
J. Optim.Society For Industrial And Applied Mathematics Vol. 14, No. 2, Pp. 548–566,2003.
Gasimov, R. N., Yenilmez K., Solving fuzzy linear programming problems with linear
membership functions, Turk J Math. 26 , 375 396, 2002.
Hadar J. and Russell W., Rules for ordering uncertain prospects, Amer. Econom. Rev., 59, pp.
25–34, 1969.
Hanoch G. and Levy H., The efficiency analysis of choices involving risk, Rev. Econom.Stud.,
36 (1969), pp. 335–346.
Kettani, O., Oral, M.: Equivalent formulations of nonlinear integer problems for eficient
0ptimization, Management Science Vol. 36 No. 1 115119, 1990.
Klir, G.J., Yuan, B.: Fuzzy Sets and Fuzzy LogicTheory and Applications, PrenticeHall Inc.,
574p, 1995.
Markowitz H. M., Mean–Variance Analysis in Portfolio Choice and Capital Markets,
Blackwell, Oxford, 1987.
Negoita, C.V.: Fuzziness in management, OPSA/TIMS, Miami 1970.
Ruszczynski A. and Vanderbei R. J., Frontiers of stochastically nondominated portfolios,
Operations Research and Financial Engineering, Princeton University, ORFE0201, 2002
Rockafellar R.T. and Wets R.J.B., Stochastic convex programming: Basic duality , Pacific
J.Math., 62 (1976), pp. 173195.
Sakawa, M., Yana, H.: Interactive decision making for multiobjective linear fractional
programming problems with fuzy parameters, Cybernetics Systems 16 (1985) 377397.
Tanaka, H., Asai, K.: Fuzzy linear programming problems with fuzzy numbers, Fuzzy Sets and
Systems 13 (1984) 110
Tanaka, H., Okuda, T., Asai, K.: On fuzzy mathematical programming, J. Cybernetics 3 (1984)
3746.
Uryasev S. and Rockafellar R.T., Conditional valueatrisk: Optimization approach, in
Stochastic Optimization: Algorithms and Applications (Gainesville, FL, 2000), Appl. Optim. 54,
Kluwer Academic, Dordrecht, The Netherlands, 2001, pp. 411–435.
Zimmermann, H.J.: Fuzzy mathematical programming, Comput. & Ops. Res. Vol. 10 No 4
(1983) 291298.
Su o
9
*)
International Congress Industry and Appplied Mathemaic (ICIAM07) ElvetiaZurich
Switzerland 1620 July 2007
Supian SUDRADJAT and Vasile PREDA
Departement of Mathematics, Faculty of Mathematics and Natural Sciences Padjadjaran University,
Bandung, Indonesia Jl. Raya Bandung Sumedang km.21 Bandung 40600 Indonesia, email :
adjat03@yahoo.com
Faculty of Mathematics and Computer Science Bucharest University Romania 14 Academiei Str.,
Sector 1, Bucharest 010014 email : preda@fmi.unibuc.ro
Abstract.
We consider stochastic optimization problems involving stochastic dominance constraints. We develop portfolio optimization
model involving stochastic dominance constrains using fuzzy decisions and we concentrate on fuzzy linear programming
problems with only fuzzy technological coefficients and application/implementation of modified subgradient method to fuzy
linear programming problems.
AMS subject classifications. Primary, 90C15, 90C29, 90C46, 90C48, 90C70;
Secondary, 46N10, 60E15, 91B06
1. Introduction
The problem of optimizing a portfolio of finitely many assets is a classical problem in theoretical and
computational finance. Since the seminal work of Markowitz [11, 2] it is generally agreed that portfolio
performance should be measured in two distinct dimensions: the mean describing the expected return,
and the risk which measures the uncertainty of the return. As one of theoretical approach to the
portfolio selection problem is that of stochastic dominance (see [18, 7]).
The rest of the paper is organized in the following manner. Section 1 provides some newly obtained
results on stochastic dominance, motivation for posing the portfolio selection problem in the fuzzy
decisions theory framework. Section 2 describes the formulation of the portfolio selection problems.
Section 3, consider an overview of portfolio problem fuzzy technological coefficient. Section 4, give a
solution of defuzzificated problems.
2. Portfolio problem
Let R ,...,
R n be random returns of assets 1 ,..., n . We assume that E [ R j ] 0 we look for a compromise between the mean and the risk.
Here,
The second approach is to select a certain utility function u : R ® R and to formulate
the following optimization problem
max E u (R ( x ) ) .
(2.3)
x Î X
[
]
It is usually required that the function u(∙) is concave and nondecreasing, thus representing preferences
of a riskaverse decision maker.
In this paper the portfolio optimization, we shall consider stochastic dominance relations between
random returns in (2.1) if to avoid placing the decision vector, x, in a subscript expression, we shall
simply write
F ( h ; x ) = F R ( x ) ( h ) and F2 ( h ; x ) = F 2 R ( x ) ( h ) .
We say that portfolio x dominates portfolio y under the FSD rules, if
F ( R ( x ); h ) £ F ( R ( y ); h ) for all h Î R ,
where at least one strict inequality holds. Similarly, we say that x dominates y under the SSD rules
( R ( x ) f SSD R ( y )) , if
F2 ( R ( x ); h ) £ F 2 ( R ( y ); h ) for all h Î R ,
with at least one inequality strict. Recall that the individual returns R j have finite expected values and
thus the function F 2 ( R ( x ); × h ) is well defined.
Stochastic dominance relations are of crucial importance for decisions theory. It is known that
R ( x ) f FSD R ( y ) if and only if
(2.4)
E[ u ( R ( x ))] ³ E [ u ( R ( y ))]
for any nondecreasing function u(∙) for which these expected values are finite. Furthermore,
R ( x ) f SSD R ( y ) if and only if (2.4) holds true for every nondecreasing and concave u(∙) for which
these expected values are finite [4, 5, 8].
A portfolio x is called SSDefficient (or FSDefficient) in a set of portfolios X if there is no y Î X
such that ( R ( y ) f SSD R ( x )) (or ( R ( y ) f FSD R ( x )) ).
We shall focus our attention on the SSD relation, because of its consistency with riskaverse
preferences: if ( R ( y ) f SSD R ( x )) , then portfolio x is preferred to y by all riskaverse decision
makers.
The starting point for our model is the assumption that a reference random return Y having a finite
expected value is available. It may be an index or our current portfolio. Our intention is to have the
return of the new portfolio, R (x ) , preferable over Y. Therefore, we introduce the following
optimization problem [4]:
(2.10)
max f ( x )
Subject to R ( x ) f ( 2 ) Y ,
(2.11)
x Î X
(2.9)
2
*)
International Congress Industry and Appplied Mathemaic (ICIAM07) ElvetiaZurich
Switzerland 1620 July 2007
Here f : X ® R is a concave continuous functional. In particular, we may use
f ( x ) = E[ R ( x )]
and this will still lead to nontrivial solutions, due to the presence of the dominance constraint
Proposition 2.1 [4] Assume that Y has a discrete distribution with realizations y i , i = 1 , m . Then
relation (2.10) is equivalent to
(2.12)
E [( y i - R ( x )) + ] £ E [( y i - Y ) + ], "i = 1 , m .
Let us assume now that the returns have a discrete joint distribution with realizations r jt , t = 1, T ,
j = 1, n , attained with probabilities p t , t = 1, T . The formulation of the stochastic dominance
relation (2.10) resp. (2.12) simplifies even further. Introducing variables s it representing shortfall of
R(x) below yi in realization t, i = 1, m and t = 1, T , we obtain the following result.
Proposition 2.2 The problem (2.9)(2.11) is equivalent to the problem:
max f ( x )
(2.13)
(2.14)
Subject to -
n
år
j =1
jt
x j - s it £ - y i , i = 1, m , t = 1, T ,
T
å p s £ F ( Y ; y ) , i = 1, m ,
(2.15)
t = 1
t it
2
i
s it ³ 0 , i = 1, m , t = 1, T
(2.16)
n
å x £ 1
(2.17)
j =1
j
n
- å x j £ -1
(2.18)
j =1
x j ³ 0 ,
(2.19)
j = 1, n ,
and problema (2.13)–(2.19) can be written as
n
max j ( X ) = max å c j X j
(2.20)
j = 1
(2.21)
Subject to
n + mT
åa
j =1
(2.22)
ij
X j £ b i , i = 1, mT + m + 2 ,
X j ³ 0 , j = 1 , n + mT ,
Where X = ( x 1 ,..., x n , s 11 ,..., s 1 T , s 21 ,..., s 2 T ,..., s m 1 ,..., s mT )
ì- r ij , j = 1 , n , i = Km +1 , ( K +1 ) m , K = 0 , ( T -1 )
ï
a ij = í -1 , i = Km +1 , ( K +1 ) m , K = 0 , ( T -1 ) , and j = n +1 , n +T ( i -1 ) +1
ï0
, otherwise
î
ì1 , j = 1 , n , i = mT + 1
a ij = í
î0 , otherwise
3
*)
International Congress Industry and Appplied Mathemaic (ICIAM07) ElvetiaZurich
Switzerland 1620 July 2007
ì- 1 , j = 1 , n , i = mT + 2
a ij = í
î 0 , otherwise
ìp
, j = n +T ( K -1 ) +1 , n +TK , K =1 , m , i = mT + 3 , mT + m + 2
a ij = í j - n -T ( K -1 )
, otherwise
î 0
In the next section we extended this result to fuzzy decisions theory.
3. Portfolio problems with fuzzy technological coefficients
In this section presents an approach to portfolio selection using fuzzy decisions theory.
We consider a linear programming problem (2.20) – (2.22) with fuzzy technological coefficients [13].
n
max j ( X ) = max å c j X j
(3.1)
j = 1
(3.2)
Subject to
n + mT
å a~ X
j =1
ij
j
£ b i , i = 1, mT + m + 2 ,
X j ³ 0 , j = 1 , n + mT .
(3.3)
~ is a fuzzy number for any i and j.
Assumption 3.1. a
ij
In this case we consider the following membership functions:
(i) 1. For i = Km + 1, ( K + 1 ) m , K = 0 , ( T - 1 ) and j = 1, n
ì 1
if t < - r ji ,
ï
m a ( t ) = í( -r ij + d ij - t ) / d ij if - r ij £ t < -r ij + d ij ,
ï 0
if t ³ - r ij + d ij .
î
ij
2. For i = Km + 1, ( K + 1 ) m , K = 0 , ( T - 1 ) and j=n+T(iKm1)+K+1
ì1
if t < -1
ï
m a ( t ) = í( -1 + d ij - t ) / d ij if - 1 £ t < -1 + d ij ,
ï0
if t ³ -1 + d ij ,
î
ij
(ii) For i = mT + 3, mT + m + 2 , K = 1, m and j = n + T ( K - 1 ) + 1 , n + TK
ì
1
if t < p
ï
m a ( t ) = í( p j - n -T ( K -1 ) + d ij - t ) / d ij if p
ï
0
if t ³ p
î
,
£ t < p
+ d ij ,
j - n -T ( K -1 )
ij
j - n -T ( K -1 )
j - n -T ( K -1 )
+ d ij ,
j - n -T ( K -1 )
where t Î R and d ij > 0 for all i = 1, mT + m + 2 , K = 0, ( T - 1 ) and j = 1 , n + mT .
4
*)
International Congress Industry and Appplied Mathemaic (ICIAM07) ElvetiaZurich
Switzerland 1620 July 2007
For defuzzification of this problem, we first fuzzify the objective function. This is done by
calculating the lower and upper bound of the optimal values first. The bounds of the optimal
values z l and z u are obtained by solving the standard linear programming problems
(3. 4)
(3. 5)
z1 = max j ( X )
Subject to
n + mT
åa
j =1
ij
X j £ b i , i = 1, mT + m + 2
(3.6)
X j ³ 0 , j = 1 , n + mT
and
(3.7)
z 2 = max j ( X )
(3.8)
Subject to
n + mT
å aˆ X
j =1
(3.9)
ij
j
£ b i , i = 1, mT + m + 2
X j ³ 0 , j = 1 , n + mT
where
ì- r ij + d ij , j =1 , n , i = Km +1 , ( K +1 ) m and K = 0 , ( T -1 )
ï
ˆij = í -1 + d ij , i = Km +1 , ( Km +1 ) m , j = n +1 , n + T ( i -1 ) +1 , and K = 0 , ( T -1 )
a
ï d
, otherwise
î ij
ìï1 + d ij , j = 1 , n , , i = mT + 1
ˆij = í
a
, otherwise
ïî d ij
ìï - 1 + d ij , j = 1 , n , i = mT + 2
ˆ ij = í
a
, otherwise
ïîd ij
The objective function takes values between z 1 and z 2 while technological coefficients vary
between a ij and aij + d ij . Let zl = min( z 1 , z 2 ) and z u = max( z 1 , z 2 ) . Then z l and z u are
called the lower and upper bounds of the optimal values, respectively.
Assumption 3.2. The linear crisp problem (3.4)(3.6) and (3.7)(3.9) have finite optimal values. In this
n
case the fuzzy set of optimal values, G, which is subset of R , is defined as [10]
n
ì
0
if
c j X j < z l
ï
å
=
j
1
ï
n
ï n
(3.10) m G ( X ) = í ( å c j X j - z l ) /( z u - z l ) if z l £ å c j X j £ z u
j =1
ï j =1
n
ï
if å c j X j ³ z u
ï1
j =1
î
n + mT
The fuzzy set of the ith constraint, C i , which is a subset of R
, is defined by:
(i) 1. For i = Km + 1, ( K + 1 ) m and K = 0, ( T - 1 )
5
*)
International Congress Industry and Appplied Mathemaic (ICIAM07) ElvetiaZurich
Switzerland 1620 July 2007
(3.11)
n
ì
, b i < -år ji X j
ï0
j = 1
ï
n
n
n
n
ï
m C ( X ) = í ( b i + år ij X j ) / åd ij X j , - år ij X j £ b i < å( -r ij + d ij ) X j
j =1
j =1
j =1
j =1
ï
n
ï
, b i ³ å( -r ji + d ij ) X j
ï1
j =1
î
i
2. For i = Km + 1, ( K + 1 ) m and K = 0, ( T - 1 )
n ( i , K )
ì
b
X j
0
,
<
ï
å
i
j
n
=
+
1
ï
n ( i , K )
n ( i , K )
n ( i , K )
n
ï
(3.12) m C ( X ) = í( b i + å X j ) / åd ij X j , - å X j £ b i < å( -1 + d ij ) X j
j =n +1
j =1
j =n +1
j =n +1
ï
n ( i , K )
ï
1
, b i ³ å( -1 + d ij ) X j
ï
j =n +1
î
i
where n(i,K)=n+T(iKm1)+K+1
(ii) For i = mT + 3, mT + m + 2 , and K = 1, m
n +TK
ì
<
b
p j - n -T ( K -1 ) X j ,
0
,
ï
å
i
=
+
-1 )
j
n
T
K
(
ï
n +TK
n +TK
n +TK
n +TK
ï
£
<
p
X
d
X
p
X
b
)
/
,
( p j -n -T ( K -1 ) +d ij ) X j ,
(3.13) m C ( X ) = í( b
å
å
å
å
i
j -n -T ( K -1 ) j
ij j
j -n -T ( K -1 ) j
i
=
+
=
+
=
+
=
+
-1 )
j
n
T
K
j
n
T
K
j
n
T
K
j
n
T
K
(
1
)
(
1
)
(
1
)
(
ï
n +TK
ï
, b i ³ å( p j -n -T ( K -1 ) +d ij ) X j .
ï 1
j =n +T ( K -1 )
î
i
By using the definition of the fuzzy decisions proposed by Bellman and Zadeh [1], we have
m D ( X ) = min( mG ( X ), min ( mC j ( X ))) .
j
i.e.
max ( m D ( X )) = max min( mG ( X ), min ( mC ( X ))) .
x ³ 0
j
x ³ 0
j
Consequently, the problem (3.1)(33) becomes to the following optimization problem
(3.14)
max l
m G ( X ) ³ l
(3.15)
(3.16)
m C ( X ) ³ l , i = 1 , mT + m + 2 ,
(3.17)
X j ³ 0 , 0 £ l £ 1 , j = 1, mT .
i
By using (3.10) and (3.11)(3.13), we obtain the following theorem.
Theorem 3.1 The problem (3.1)(3.3) is reduced to one of the following crisp
problems :
6
*)
International Congress Industry and Appplied Mathemaic (ICIAM07) ElvetiaZurich
Switzerland 1620 July 2007
max l
(3. 18)
(3.19)
n
l ( z 1 - z 2 ) - å c j X j + z 2 £ 0
j = 1
(3.20)
n + mT
å a ˆ ( l ) X - b £ 0 , i = 1, mT + m + 2 ,
j =1
(3.21)
ij
j
i
X j ³ 0 , 0 £ l £ 1 , j = 1, mT .
where
ì- r ij + ld ij , j =1 , n , i = Km +1 , ( K +1 ) m , and K = 0 , ( T -1 ) ,
ï
aˆ ij ( l) = í -1 + ld ij , i = Km +1 , ( K +1 ) m , K = 0 , ( T -1 ) and j = n +1 , T ( i -1 ) +1 ,
ï ld
, otherwise,
î ij
ìï1 + ld ij , j = 1 , n , i = mT + 1 ,
aˆ ij ( l ) = í
, otherwise ,
ïî ld ij
ìï- 1 + ld ij , j = 1 , n , i = mT + 2 ,
aˆ ij ( l ) = í
, otherwise,
ïî ld ij
ìï p
+ ld ij , j = n + T ( K - 1 ) + 1 , n + TK , i = mT + 3 , mT + m + 2
aˆ ij ( l ) = í j - n -T ( K -1 )
, otherwise .
ïîld ij
Notice that, the constraints in problem (3.18)(3.21) containing the cross product term l X
j
are not
convex. Therefore the solution of this problem requires the special approach adopted for solving
general nonconvex optimization problem.
4. Solution of defuzzificated problems
In this section, we present the modified subgradient method [6] and use it for solving the defuzzificated
problems (3.18)(3.21) for nonconvex constrained problems and can be applied for solving a large class
of such problems.
Notice that, the constraints in problem (3.18)(3.21) generally are not convex. These problems may be
solved either by the fuzzy decisive set method, which is presented by Sakawa and Yana [15], or by the
linearization method of Kettani and Oral [2].
4.1. Application of modified subgradient method to fuzzy linear programming problems.
For applying the subgradient method [6] to the problem (3.18)(3.21), we first formulate it with
equality constraints by using slack variables y 0 and y i , i = 1, mT + m + 2 . Then, we can be written
as
(4.1)
max l
(4.2)
n
g 0 ( X , l , y 0 ) = l ( z 1 - z 2 ) - å c j X j + z 2 + y 0 = 0
j = 1
(4.3)
n + mT
ˆ ij ( l ) X j - b i + y i = 0 , i = 1, mT + m + 2
g i ( X , l , y i ) = å a
j =1
(4.4)
X j ³ 0 , y 0 , y i ³ 0 , 0 £ l £ 1 , j = 1 , n + mT , i = 1, mT + m + 2 .
where y = ( y 0 ,..., y n )
*)
7
International Congress Industry and Appplied Mathemaic (ICIAM07) ElvetiaZurich
Switzerland 1620 July 2007
For this problem the set S can be defined as
S = {( X , p , l ) X ³ 0 , y ³ 0 , 0 £ l £ 1 } .
Since max l = - min( -l ) and g = ( g 0 ,..., g mT + m + 2 ) the augmented Lagrangian associated with
the problem (4.1)(4.4) can be written in the form
1
2
é
ù 2
2
üï mT +m +2 n +mT
n
êìï
æ ˆ
ö ú
L ( x , u , c ) = -l + c êíl ( z 1 - z 2 ) - å c X + z 2 + y 0 ý +
å ç å a
ij ( l) X j - b
i + y i ÷ ú
j j
j
1
=
è
ø ú
i =1
ïþ
j = 1
êïî
ë
û
ö mT +m +2 æ n +mT
æ
n
ˆ ij ( l) X j - b i + y i ö÷.
- m ç l ( z - z ) - å c X +z + y ÷ å u i ç å a
0 ç 1 2
j j 2 0 ÷
j =1
ø
è
1
=
i
j = 1
ø
è
The modified subgradient method may be applied to the problem (4.1)(4.4) in the following way:
1
1
1
1
1
Initialization Step. Choose a vector ( u 0 , u 1 ,..., u mT + m + 2 , c ) with c ³ 0 . Let k = 1 , and go to main
step.
Main Step.
k
k
k
k
Step 1 . Given ( u 0 , u 1 ,..., u mT + m +2 , c ) ; solve the following subproblem :
1
2 ù 2
2
é
ì
ü
ö
æ
n
ú
êï
ï mT +m +2 ç n + mT
ˆ ( l ) X - b + y ÷ ú
min - l + c êíl ( z 1 - z 2 ) - å c X + z 2 + y 0 ý +
å
å a
÷
ç
j j
ij
j
i
i
i =1 è j = 1
ïþ
j = 1
êïî
ø ú
û
ë
ö
ö mT +2 æ n + mT
æ
n
ˆ ( l ) X - b + y ÷ .
- u ç l ( z - z ) - å c X +z + y ÷ - å u ç å a
0 ç 1 2
j j 2 0 ÷ i =1 i ç
ij
j i i ÷
j = 1
ø
è j = 1
ø
è
( X , y , l ) Î S .
k
k
k
k
k
k
k
k
k
k
Let ( X , y , l ) be any solution. If g ( X , y , l ) , then stop; ( u 0 , u 1 ,..., u mT , c ) is a solution
k
k
to dual problem, ( X , l ) is a solution to problem (3.18)(3.21). Otherwise, go to Step 2.
Step 2 . Let
n
æ
ö
u 0 k + 1 = u 0 k - h k çç l ( z 1 - z 2 ) - å c j x j + z 2 + y 0 ÷÷
j =1
è
ø
n
æ
ö
ˆij ( l ) X j - b i + y i ÷ , i = 1, mT + m + 2
u i k + 1 = u i k - h k çç å a
÷
è j =1
ø
k + 1
k
k
k
k
k k
c = c + ( h + e ) g ( X , y , l )
k
k
k
k
where h and e are positive scalar stepsizes and h > e > 0 , replace k by k + 1; and
repeat Step 1.
4.2. The algorithm of the fuzzy decisive set method
This method is based on the idea that, for a fixed value of
*
l ; the problems (3.18)(3.21) is
linear programming problems. Obtaining the optimal solution l to the problems (3.18)(3.21) is
equivalent to determining the maximum value of l so that the feasible set is nonempty. Bellow is
presented the algorithm [6] of this method for the problem (3.18)(3.21).
8
*)
International Congress Industry and Appplied Mathemaic (ICIAM07) ElvetiaZurich
Switzerland 1620 July 2007
Algorithm
Step 1. Set l = 1 and test whether a feasible set satisfying the constraints of the problem (3.18)(3.21)
exists or not using phase one of the simplex method. If a feasible set exists, set l = 1: Otherwise, set
l L = 0
R
and l = 1 and go to the next step.
L
L
R
R
Step 2. For the value of l = ( l + l ) / 2 ; update the value of l and l using the bisection
method as follows :
l L = l
l R = l
if feasible set is nonempty for
l
if feasible set is empty for l .
Consequently, for each l , test whether a feasible set of the problem (3.18)(3.21) exists or
*
not using phase one of the Simplex method and determine the maximum value l satisfying the
constraints of the problem (3.18)(3.21).
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
Bellman, R.E., Zadeh,L.A., Decisionmaking in fuzzy environment, Management Science 17
(1970), B141B164.
Birge J.R. and Louveaux F.V., Introduction to Stochastic Programming, SpringerVerlag, New
York, 1997.
Dentcheva, D. and Ruszczynski, A., “Frontiers of stochastically nondominated portfolio”
Operations research and financial engineering, Princeton University, ORFE 001,2002.
Dentcheva, D. and Ruszczynski, A., Portfolio Optimization with Stochastic Dominance
Constraints, May 12, 2003
Dentcheva, D. and Ruszczynski, A, Optimization with stochastic dominance constraints, Siam
J. Optim.Society For Industrial And Applied Mathematics Vol. 14, No. 2, Pp. 548–566,2003.
Gasimov, R. N., Yenilmez K., Solving fuzzy linear programming problems with linear
membership functions, Turk J Math. 26 , 375 396, 2002.
Hadar J. and Russell W., Rules for ordering uncertain prospects, Amer. Econom. Rev., 59, pp.
25–34, 1969.
Hanoch G. and Levy H., The efficiency analysis of choices involving risk, Rev. Econom.Stud.,
36 (1969), pp. 335–346.
Kettani, O., Oral, M.: Equivalent formulations of nonlinear integer problems for eficient
0ptimization, Management Science Vol. 36 No. 1 115119, 1990.
Klir, G.J., Yuan, B.: Fuzzy Sets and Fuzzy LogicTheory and Applications, PrenticeHall Inc.,
574p, 1995.
Markowitz H. M., Mean–Variance Analysis in Portfolio Choice and Capital Markets,
Blackwell, Oxford, 1987.
Negoita, C.V.: Fuzziness in management, OPSA/TIMS, Miami 1970.
Ruszczynski A. and Vanderbei R. J., Frontiers of stochastically nondominated portfolios,
Operations Research and Financial Engineering, Princeton University, ORFE0201, 2002
Rockafellar R.T. and Wets R.J.B., Stochastic convex programming: Basic duality , Pacific
J.Math., 62 (1976), pp. 173195.
Sakawa, M., Yana, H.: Interactive decision making for multiobjective linear fractional
programming problems with fuzy parameters, Cybernetics Systems 16 (1985) 377397.
Tanaka, H., Asai, K.: Fuzzy linear programming problems with fuzzy numbers, Fuzzy Sets and
Systems 13 (1984) 110
Tanaka, H., Okuda, T., Asai, K.: On fuzzy mathematical programming, J. Cybernetics 3 (1984)
3746.
Uryasev S. and Rockafellar R.T., Conditional valueatrisk: Optimization approach, in
Stochastic Optimization: Algorithms and Applications (Gainesville, FL, 2000), Appl. Optim. 54,
Kluwer Academic, Dordrecht, The Netherlands, 2001, pp. 411–435.
Zimmermann, H.J.: Fuzzy mathematical programming, Comput. & Ops. Res. Vol. 10 No 4
(1983) 291298.
Su o
9
*)
International Congress Industry and Appplied Mathemaic (ICIAM07) ElvetiaZurich
Switzerland 1620 July 2007