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Duality in disjunctive linear fractional programming
101
Region Development Authority, Bombay-400051, India
I.M. S T A N C U - M I N A S I A N Economic Cybernetics Department, Bucharest-7.1131, Romania
Abstract: In this note a dual problem is formulated for a given class c f disjunctive linear fractiona~ programming problems. This result generalizes to fractional programming ~he duality theorem of disjunctive linear programming originated by Balas. Two examples are given to illustrate the result. Keywords: Fractional programming, disjunctive constraints, duality theory
1. Introduction
The study of disjunctive linear programming problems has received much attention in the last decade and many papers are now available which deal with its computational and theoretical aspects. A good account of these developments can be found in the papers of Balas [2], Jeroslow [4,5] and the references there. Recently, Patkar and Stancu-Minasian [6,7] studied a few aspects of disjunctive finear fractional programming and in this paper we present duality results for a class of such problems. This development is motivated by the work of Balas [1]. The plan of the paper is as follows. SectiorL 2 contains the notations and definitions. The formulation of a dual for the given class of problems and the proof of the duafity theorem are presented in Section 3. The last section deals with two numerical examples which illustrate the results.
2. Notations and definitions
The disjunctive linear fractional programming problem under consideration is; (V)
~mm~'.e
F ( x ) = ~x + co dx + d o '
subject 1:o V ( Ahx >t bh; x >10}, hEQ
where A" is an m × n matrix for all h ~ Q, c, d are I × n vectors, b h is an m × 1 vector for all h ~ Q, x is
We are thankful to Prof. R.G. Jeroslow and the referee for their helpful comments to improve presentation of the paper. The typing assistance by Mrs. J.l. Luis is gratefully acknowledged. Received August 1984; revised September 1984 North, Holland European Journal of Operational Research 21 (1985) 101-105 0377.2217/85/$3.30 © 1985, Elsevier Science Publishers B.V. (North-Holland)
subject ~:o
~< c, - u sb h +doz~co; u h 0},
where it is clear that (D) is feasible if there exists z, and for each h ~ Q, there exists (u h, z) = (u~ ..... u~, z) satisfies the corresponding bracketed system. Let
Xh={x:Ahx>~b;x~O},
such that
Xh={x:ahx>~O;x>~O},
U,~= {(u~', z): uhAt'+dz<~O,-uhbh+doz~O;
uh~O}
and
Q * = { h ~ Q : Xh#-f3 },
Q * * = { h ~ Q : Uh#-'6}.
w e ~ ~sume the following regularity condition:
(RC1)
(Q* ~,~, Q\Q** ~'6) =~ Q*\Q** #'6;
i.e., if (P) is feasible and (D) is infeasible, then there exists h ~ Q such that Xh ~'6 and U, =,6. 3. Dualily r~ult Now we consider the steps by wlfich the problem (D) is formulated as the dual of the (P) and establish a duality theorem. By r.he one-to-one variable tr,'msformafion [3], (TR)
y=xt,
t=l/(dx+do}>O ,
the problem (P) becomes: Minimize
H( y, t) = cy + Cot, { A;'y - bht >i0,
(I)
subject to
V
}
~dy+dot>~l,
hEO| - d y - d ° t > ~ - l ' ~y .:~ 0, t > O
or
Minimize
Y
H(y, t)=(c, c.) t
(
(n) subject to
V ~
d
do
-d "do ~y'~ 0, .~>0
L,j ....
! i i~
103
~sing his result the
(U~,!..,Uhm, V, W ) [ u1 ] .... [_~j'
Maximize
(.f,.:.,.~, ~, ~)>i0 that
)
is;.
Maximize (w)
[ uhAh+d(v-w)<~c, subject to
Substituting
(v)
V -- IV),
z = v -
A I - u h b h + d o ( v - w ) < ~ c o , I. h~Q ~ u h, v, w>~O w
so that z is unrestricted in sign, this problem reduces
Maximize
G(z ) = z ,
subject to
A -uhb h + doz <~co, , h~2 uh.) 0
to
which is just the problem (D). Since the problem (P) and (I) (or (II)) are q-equivalent [3], and the problem (V) (or (Iii) or (IV)) is the dual of (i) [1], it follows that the problem (D) is the dual of (P). It is to be observed that the dual problem (D) is an ordinary linear programming problem and would offer a relative computational simplicity. In order to prove the main duality theorem, the regularity condition (RC1) must be expressed in terms of the problems (I) and (V). Let
Yh= ((y, t): Ahy-b~t>~O, d y + d o t = l ; y>~O, t>O}, ~,=((y,t):
Ahy-bht>lO, dy+dot=O; y>~O, t>O}
and
Q*** = (h s Q: r~.Z}. We shall consider the following regularity condition: (RC2)
(Q*** 4=~, Q\Q** , ~ )
~
Q***\Q** ,,0,
i.e., if (I) is feasible and (D) Js infeasible then there exists h ~ Q such that Yh4=,(~and Uh =;~, Now we show that the con~tions (RC1) and (RC2) are equivalent. ~ m m a 3;1. If the regularity condition (RC1)holds, then the regularity condition (RC2) also holds, and
conversely, Preof2 The p r ~ f follows from the fact that the mapping (TR) from the constraint set of (P) onto the
We
P r ~ L The proof fohows from the Lemma 3.1 and the Theorem 2.1 of the paper of Balas [1].
4. Numerical exam#es
The following two examples iilustr~te the results: Exam#e 1. Minimize subject to
F(x) --. (2x a + x 2 + 1)/(3xa + 5x2 + 3),
Xl +X2 ~<3' l v XI>~0, X2~>0
2XI--X2<~4'~V~ 3x2<~2' . Xl>~O, X2>~O] kXl>10, X2>~0
The optimal sol udor~ is x ° -- (0, 2) with F(x °) = 3/13 [6]. The corresponding dual problem is: MaxhrAze
G(z)=~,
subject to
- u ~ + 3z ~< 2.
- ul - u~ + 5~ ~ 1,
2u I + 3u~ 4- 3z <~ 1, u~ - 2u~ + 3z ~<2, - 2 u ~ + u ~ + 5 z ~ 1, u~ + 4 u ,2 + 3z ~< 1, - u 3 + 3z ~< 2, - 3 u ~ + 5 z < 1,
4ul3 + 2u23 + 3z ~< 1,
u~'~O,
i=1,2;k=1,2,3.
The optimal solution is u'1 = 2/'13, u~ = 1/13, u 2 3 = 2/39 and z = 3,/13. It is clear that there is no duality gap. Exam#e 2. Minimize
subject to
F ( x ) = (3x~ - 4x2 + 1 ) / ( x ~ + x2 + 3),
xa>10, x 2 ~ 0
J
~ x i > / 0 , x2>/0
J"
fractionalprogramming
105
esponding dual problem is: M _
G ( z ) = z, --ui~" u 2 ~ z ~ 3, _1 "- ul
--
U1 2
+z~< - 4 ,
9
-u~- u~+z~<-4, 2u~ + 3z .<<1,
uk>~O, i=k=1,2. This dual problem is infeasible. To_is is due to the fact that the regularity condition (RC1) is violated because Q* = { 1 }, Q \ Q ** = { 2 } , X 2 = ~ and U2=,~ but Q * \ Q * * = , O .
References [1] Balas, E., "A note on duality in disjunctive programming", Journal of Optimization Theory and Applications 21 (1977) 52"~-528. [2] Balas, E., "Disjunctive programming", Annals of Discrete Mathematics 5 (1979) 3-51. [3] Craven, B.D., and Mond, B., "The dual of a fractional linear program", Journal of Mathematical Analysis and Applications 42 (1973) 507-512. [4] Jeroslow, R.G., "Cutting plane theory: disjunctive methods", Annals of Discrete Mathematics 1 (1977) 293-330. [5] Jeroslow, R.G., "Cutting plane for complementary constra2ms", SIAM Journal of Control t6 (1978) 56-62. [6] Patkar, V., and Stancu-MJnasian, LM., "On disjunctive linear fractionet programming", Economic Computation and Economic Cybernetics Studies and Research 17 (!982) 87-96. [7] Patkar, V., and Stancu-Minasian, I.M., "Parametric algorithm foz ~olving a class of disjunctive linear fractional programs", Bulletin Mat,~bmatique de la Socibtb des Sciences Mathbmatiques de la Re#u~lique Socialiste de Roumanie, to appear.
Region Development Authority, Bombay-400051, India
I.M. S T A N C U - M I N A S I A N Economic Cybernetics Department, Bucharest-7.1131, Romania
Abstract: In this note a dual problem is formulated for a given class c f disjunctive linear fractiona~ programming problems. This result generalizes to fractional programming ~he duality theorem of disjunctive linear programming originated by Balas. Two examples are given to illustrate the result. Keywords: Fractional programming, disjunctive constraints, duality theory
1. Introduction
The study of disjunctive linear programming problems has received much attention in the last decade and many papers are now available which deal with its computational and theoretical aspects. A good account of these developments can be found in the papers of Balas [2], Jeroslow [4,5] and the references there. Recently, Patkar and Stancu-Minasian [6,7] studied a few aspects of disjunctive finear fractional programming and in this paper we present duality results for a class of such problems. This development is motivated by the work of Balas [1]. The plan of the paper is as follows. SectiorL 2 contains the notations and definitions. The formulation of a dual for the given class of problems and the proof of the duafity theorem are presented in Section 3. The last section deals with two numerical examples which illustrate the results.
2. Notations and definitions
The disjunctive linear fractional programming problem under consideration is; (V)
~mm~'.e
F ( x ) = ~x + co dx + d o '
subject 1:o V ( Ahx >t bh; x >10}, hEQ
where A" is an m × n matrix for all h ~ Q, c, d are I × n vectors, b h is an m × 1 vector for all h ~ Q, x is
We are thankful to Prof. R.G. Jeroslow and the referee for their helpful comments to improve presentation of the paper. The typing assistance by Mrs. J.l. Luis is gratefully acknowledged. Received August 1984; revised September 1984 North, Holland European Journal of Operational Research 21 (1985) 101-105 0377.2217/85/$3.30 © 1985, Elsevier Science Publishers B.V. (North-Holland)
subject ~:o
~< c, - u sb h +doz~co; u h 0},
where it is clear that (D) is feasible if there exists z, and for each h ~ Q, there exists (u h, z) = (u~ ..... u~, z) satisfies the corresponding bracketed system. Let
Xh={x:Ahx>~b;x~O},
such that
Xh={x:ahx>~O;x>~O},
U,~= {(u~', z): uhAt'+dz<~O,-uhbh+doz~O;
uh~O}
and
Q * = { h ~ Q : Xh#-f3 },
Q * * = { h ~ Q : Uh#-'6}.
w e ~ ~sume the following regularity condition:
(RC1)
(Q* ~,~, Q\Q** ~'6) =~ Q*\Q** #'6;
i.e., if (P) is feasible and (D) is infeasible, then there exists h ~ Q such that Xh ~'6 and U, =,6. 3. Dualily r~ult Now we consider the steps by wlfich the problem (D) is formulated as the dual of the (P) and establish a duality theorem. By r.he one-to-one variable tr,'msformafion [3], (TR)
y=xt,
t=l/(dx+do}>O ,
the problem (P) becomes: Minimize
H( y, t) = cy + Cot, { A;'y - bht >i0,
(I)
subject to
V
}
~dy+dot>~l,
hEO| - d y - d ° t > ~ - l ' ~y .:~ 0, t > O
or
Minimize
Y
H(y, t)=(c, c.) t
(
(n) subject to
V ~
d
do
-d "do ~y'~ 0, .~>0
L,j ....
! i i~
103
~sing his result the
(U~,!..,Uhm, V, W ) [ u1 ] .... [_~j'
Maximize
(.f,.:.,.~, ~, ~)>i0 that
)
is;.
Maximize (w)
[ uhAh+d(v-w)<~c, subject to
Substituting
(v)
V -- IV),
z = v -
A I - u h b h + d o ( v - w ) < ~ c o , I. h~Q ~ u h, v, w>~O w
so that z is unrestricted in sign, this problem reduces
Maximize
G(z ) = z ,
subject to
A -uhb h + doz <~co, , h~2 uh.) 0
to
which is just the problem (D). Since the problem (P) and (I) (or (II)) are q-equivalent [3], and the problem (V) (or (Iii) or (IV)) is the dual of (i) [1], it follows that the problem (D) is the dual of (P). It is to be observed that the dual problem (D) is an ordinary linear programming problem and would offer a relative computational simplicity. In order to prove the main duality theorem, the regularity condition (RC1) must be expressed in terms of the problems (I) and (V). Let
Yh= ((y, t): Ahy-b~t>~O, d y + d o t = l ; y>~O, t>O}, ~,=((y,t):
Ahy-bht>lO, dy+dot=O; y>~O, t>O}
and
Q*** = (h s Q: r~.Z}. We shall consider the following regularity condition: (RC2)
(Q*** 4=~, Q\Q** , ~ )
~
Q***\Q** ,,0,
i.e., if (I) is feasible and (D) Js infeasible then there exists h ~ Q such that Yh4=,(~and Uh =;~, Now we show that the con~tions (RC1) and (RC2) are equivalent. ~ m m a 3;1. If the regularity condition (RC1)holds, then the regularity condition (RC2) also holds, and
conversely, Preof2 The p r ~ f follows from the fact that the mapping (TR) from the constraint set of (P) onto the
We
P r ~ L The proof fohows from the Lemma 3.1 and the Theorem 2.1 of the paper of Balas [1].
4. Numerical exam#es
The following two examples iilustr~te the results: Exam#e 1. Minimize subject to
F(x) --. (2x a + x 2 + 1)/(3xa + 5x2 + 3),
Xl +X2 ~<3' l v XI>~0, X2~>0
2XI--X2<~4'~V~ 3x2<~2' . Xl>~O, X2>~O] kXl>10, X2>~0
The optimal sol udor~ is x ° -- (0, 2) with F(x °) = 3/13 [6]. The corresponding dual problem is: MaxhrAze
G(z)=~,
subject to
- u ~ + 3z ~< 2.
- ul - u~ + 5~ ~ 1,
2u I + 3u~ 4- 3z <~ 1, u~ - 2u~ + 3z ~<2, - 2 u ~ + u ~ + 5 z ~ 1, u~ + 4 u ,2 + 3z ~< 1, - u 3 + 3z ~< 2, - 3 u ~ + 5 z < 1,
4ul3 + 2u23 + 3z ~< 1,
u~'~O,
i=1,2;k=1,2,3.
The optimal solution is u'1 = 2/'13, u~ = 1/13, u 2 3 = 2/39 and z = 3,/13. It is clear that there is no duality gap. Exam#e 2. Minimize
subject to
F ( x ) = (3x~ - 4x2 + 1 ) / ( x ~ + x2 + 3),
xa>10, x 2 ~ 0
J
~ x i > / 0 , x2>/0
J"
fractionalprogramming
105
esponding dual problem is: M _
G ( z ) = z, --ui~" u 2 ~ z ~ 3, _1 "- ul
--
U1 2
+z~< - 4 ,
9
-u~- u~+z~<-4, 2u~ + 3z .<<1,
uk>~O, i=k=1,2. This dual problem is infeasible. To_is is due to the fact that the regularity condition (RC1) is violated because Q* = { 1 }, Q \ Q ** = { 2 } , X 2 = ~ and U2=,~ but Q * \ Q * * = , O .
References [1] Balas, E., "A note on duality in disjunctive programming", Journal of Optimization Theory and Applications 21 (1977) 52"~-528. [2] Balas, E., "Disjunctive programming", Annals of Discrete Mathematics 5 (1979) 3-51. [3] Craven, B.D., and Mond, B., "The dual of a fractional linear program", Journal of Mathematical Analysis and Applications 42 (1973) 507-512. [4] Jeroslow, R.G., "Cutting plane theory: disjunctive methods", Annals of Discrete Mathematics 1 (1977) 293-330. [5] Jeroslow, R.G., "Cutting plane for complementary constra2ms", SIAM Journal of Control t6 (1978) 56-62. [6] Patkar, V., and Stancu-MJnasian, LM., "On disjunctive linear fractionet programming", Economic Computation and Economic Cybernetics Studies and Research 17 (!982) 87-96. [7] Patkar, V., and Stancu-Minasian, I.M., "Parametric algorithm foz ~olving a class of disjunctive linear fractional programs", Bulletin Mat,~bmatique de la Socibtb des Sciences Mathbmatiques de la Re#u~lique Socialiste de Roumanie, to appear.