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ϵ-Pareto Optimality for Nondifferentiable Multiobjective Programming via Penalty Function
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
198, 248]261 Ž1996.
0080
e-Pareto Optimality for Nondifferentiable Multiobjective Programming via Penalty Function J. C. Liu Section of Mathematics, National Uni¨ ersity Preparatory School, National O¨ erseas Chinese Student Uni¨ ersity, Linkou, Taipei, Hsien, Taiwan, 24402, Republic of China Submitted by Koichi Mizukami Received January 9, 1995
Necessary and sufficient conditions without a constraint qualification for e-Pareto optimality of multiobjective programming are derived. The necessary Kuhn]Tucker condition suggests the establishment of a Wolf-type duality theorem for nondifferentiable, convex, multiobjective minimization problems. The generalized e-saddle point for Pareto optimality of the vector Lagrangian is studied. Q 1996 Academic Press, Inc.
1. INTRODUCTION Several authors have been interested recently in e-optimal solutions in nonlinear programming. For details, the readers are advised to consult w1]5x. Loridan w4x derived some properties of e-efficient points solution for vector minimization problems and used the Ekeland’s variational principle w6x to establish the e-Pareto optimality and e-quasi Pareto optimality. In w7x, Liu also adapted the same approach to obtain the e-duality theorem of nondifferentiable nonconvex multiobjective programming. Recently, several authors w8]12x have used an exact penalty function to transform the nonlinear scalar programming problem into an unconstrained problem and derived the e-optimality. In w14x, Yokoyama was concerned with the e-approximate solutions and extended some results of w13x to the vector minimization problems. Similar to w13x, Yokoyama transformed the vector problems into the unconstrained problems by using the exact penalty functions and showed the e-optimality criteria by estimating the penalty parameter in terms of e-approximate solutions for the associated dual problems. 248 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
e-PARETO
249
OPTIMALITY
In this paper, we are inspired to use the exact penalty function to transform the inequality multiobjective programming problem into a scalar unconstrained problem and to derive the Kuhn]Tucker conditions in which Lagrange multiplers of objective functions are one. Some definitions and notations are given in Section 2. In Section 3, we use scalar penalty functions to establish the necessary and sufficient conditions of e-Pareto solution. Using this result, we formulate a dual problem of the Wolfe-type for multiobjective programming. In Section 4, we give some relationships between the primal problem and the dual problem. The generalized e-Pareto saddle point of the vector Lagrangian is discussed in Section 5.
2. PRELIMINARIES We consider the following convex multiobjective programming problem:
Ž P.
minimize f Ž x . subject to g j Ž x . F 0,
j s 1, . . . , m,
where f s Ž f 1 , f 2 , . . . , f p . and each component function is a convex continuous real-valued function defined on R n and where g j are convex continuous real-valued functions defined on R n, 1 F j F m. We denote the feasible set x g R n < g j Ž x . F 0, 1 F j F m4 by F and assume the feasible set F is nonempty. Let e be an element of Rqp . We introduce the e feasible set Fe , p
½
Ý ei , 1 F j F m
Fe s x g R n g j Ž x . F
is1
5
.
p For convenience, let g s Ý is1 ei. To transform the problem ŽP. into a scalar unconstrained problem, we use the exact penalty function introduced by Zangwill w8x:
p
u Ž x, r . s
Ý
m
fi Ž x . q r
is1
Ý max Ž 0, g j Ž x . . , js1
where
r ) 0. The associated unconstrained problem in which
Ž ur .
minimize u Ž x, r .
250
J. C. LIU
is called a penalized problem with respect to the penalty parameter r . For convenience, let p Ž t . s max Ž 0, r t . . p Clearly, we have u Ž x, r . s Ý is1 f i Ž x . q Ý mjs1 pŽ g j Ž x ...
DEFINITION 2.1. A point x g R n is called an e-Pareto solution of ŽP. if x g F and there is no x g F such that f Ž x . F f Ž x . y e and f Ž x . / f Ž x . y e ,
in R p .
DEFINITION 2.2. A point x g R n is called an almost e-Pareto solution of ŽP. if x g Fe and there is no x g F such that f Ž x . F f Ž x . y e and f Ž x . / f Ž x . y e ,
in R p .
DEFINITION 2.3. Let a ) 0. A point x g R n is called an a-solution of the scalar problem Ž ur . if
u Ž x, r . F u Ž x, r . q a ,
for all x g R n .
DEFINITION 2.4. Let h: R n ª R j q`4 be a convex function, finite at x. The e-subdifferential of h at x is the set e hŽ x . defined by
e h Ž x . s x* g R n < h Ž y . G h Ž x . y e q ² x*, y y x : for any y g R n 4 .
3. NECESSARY AND SUFFICIENT CONDITIONS In this section, we present some Kuhn]Tucker conditions for e-Pareto optimality. THEOREM 3.1. If there exists r 0 such that x is a g-solution for Ž ur . for any r G r 0 , then x is an e-Pareto solution for ŽP. and there exist scalars e i G 0 Ž1 F i F p ., e jG 0 Ž1 F j F m., l jG 0 Ž1 F j F m. such that: p
Ži. 0 g
Ý e Žl j g j .Ž x .,
is1
js1
p
Žii.
m
Ý e i fi Ž x . q
j
m
Ý ei q
Ý
is1
js1
Ž1.
m
ej y g F
Ý l j g j Ž x . F 0. js1
Ž2.
e-PARETO
251
OPTIMALITY
Proof. If x is a g-solution of Problem Ž ur .,
u Ž x, r . F u Ž x, r . q g ,
for all x g R n .
Ž 3.
Clearly, p
u Ž x, r . s
for all x g F.
Ý fi Ž x . , is1
Thus, we have p
p
Ý f i Ž x . F u Ž x, r . F Ý f i Ž x . q g , is1
for all x g F.
Ž 4.
is1
If x f F, Ý mjs1 pŽ g j Ž x .. ) 0. Choose any feasible point ˆ x which is also in F and let
r ) max
½ž
p
p
m
Ý f i Ž ˆx . y Ý f i Ž x . is1
is1
Ý pŽ gj Ž x . . , r0
/
js1
5
.
We then have the conclusion p
Ý
p
fi Ž ˆ x. s u Ž ˆ x, r . G u Ž x, r . y g )
is1
Ý f i Ž ˆx . y g . is1
This conclusion gives a contradiction and hence x g F. If x is not an e-Pareto solution of ŽP., then there exists x 1 g F such that fi Ž x1 . F fi Ž x . y ei , 1 F i F p, with at least one strict inequality. Therefore, we have p
p
Ý
fi Ž x1 . -
is1
Ý fi Ž x . y g , is1
which contradicts Ž4.. Thus x is an e-Pareto solution of ŽP.. With Ž3. and the result of w15x, we have p
0 g g
žÝ
m
fi Ž ?. q r
/
Ý p Ž g j Ž ?. . Ž x . .
is1
js1
Then, there exist scalars e iG 0 Ž1 F i F p ., e jG 0 Ž1 F j F m., a j G 0 Ž1 F j F m., and hj G 0 Ž1 F j F m. such that p
m
Ý ei q
Ý ej s g ,
is1
js1
p
0g
Ž 5.
m
Ý e i fi Ž x . q
Ý h Ž a j r g j . Ž x . ,
is1
js1
i
Ž 6.
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J. C. LIU
where
a j F 1, hj q p Ž g j Ž x . . y a j r g j Ž x . s e j
for j s 1, . . . , m.
Ž 7.
By Ž5. and Ž7., we have p
m
m
Ý e i q Ý hj y g F Ý a j r g j Ž x . F 0. is1
js1
js1
Finally, we obtain the results Ž1. and Ž2. by setting
lj s aj r , e j s hj ,
1 F j F m, 1 F j F m.
REMARK 3.1. If p s 1, then the necessary condition of Theorem 3.1 reduces to Theorem 4.1 of w13x. THEOREM 3.2. If x is a feasible solution of ŽP. and there exist e iG 0 Ž1 F i F p ., e jG 0 Ž1 F j F m., l jG 0 Ž1 F j F m. such that: p
Ži. 0 g
m
Ý e f i Ž x . q Ý e Žl j g j .Ž x ., i
is1 p
Žii.
Ž8.
j
js1 m
m
Ý e i q Ý e j y g F Ý l j g j Ž x . F 0. is1
js1
Ž9.
js1
Then, x is an e-Pareto solution of ŽP.. Proof. If x is a feasible solution of ŽP. and there exist e iG 0 Ž1 F i F p ., e jG 0 Ž1 F j F m., l jG 0 Ž1 F j F m. which satisfy relations Ž8. and Ž9., then there exist xUi g e i f i Ž x ., 1 F i F p, yUj g e j Ž l j g j .Ž x ., 1 F j F m, such that p
m
Ý xUi q Ý yUj s 0. is1
js1
By using the characterization of the e-subgradient, we obtain f i Ž x . G f i Ž x . q ² xUi , x y x : y e i ,
lj g j Ž x . G lj g j Ž x . q ²
yUj ,
x y x: y ej,
1 F i F p, 1 F j F m.
e-PARETO
253
OPTIMALITY
Thus, we have p
p
m
m
Ý fi Ž x . q Ý l j g j Ž x . y Ý ei y Ý e j is1
js1
is1
p
F
js1 p
m
Ý fi Ž x . q Ý l j g j Ž x . F Ý fi Ž x . , is1
js1
for all x g F.
is1
With Ž9., we have p
Ý
p
fi Ž x . F
is1
Ý fi Ž x . q g ,
for all x g F.
Ž 10 .
is1
If x is not an e-Pareto solution of ŽP., there exists x 1 g F such that fi Ž x1 . F fi Ž x . y ei ,
1 F i F p,
with at least one strict inequality. Therefore, we have p
p
Ý
fi Ž x1 . -
is1
Ý fi Ž x . y g , is1
which contradicts Ž10.. Thus x is an e-Pareto solution of ŽP.. THEOREM 3.3. If for sufficiently large r , x is a g-solution for Ž ur ., then x is an almost e-Pareto solution for ŽP. and there exist scalars e iG 0 Ž1 F i F p ., e jG 0 Ž1 F j F m., l jG 0 Ž1 F j F m. such that: p
Ž i. 0 g
m
Ý e fi Ž x . q Ý e Ž l j g j . Ž x . , i
is1
p
Ž ii .
j
js1
m
m
Ý e i q Ý e j y g F Ý l j g j Ž x . F 0. is1
js1
js1
Proof. If x is a g-solution of Problem Ž ur ., p
p
Ý f i Ž x . F u Ž x, r . F Ý f i Ž x . q g , is1
is1
for all x g F.
254
J. C. LIU
Since p
inf
p
Ý fi Ž x . F Ý fi Ž x . ,
xgR n is1
is1
we have p
m
r
p
inf Ý f i Ž x . y Ý p Ž g j Ž x . . F xgF
js1
is1
inf n
xgR
Ý fi Ž x . q g . is1
Let p
r0 Ž e . s g
inf
Ý
xgF is1
p
f i Ž x . y inf n xgR
Ý fi Ž x . q g is1
Then, there exists r F r 0 Ž e . such that pŽ gj Ž x. . F g ,
1 F j F m.
Hence, we have x g Fe . This concludes the proof of the theorem. 4. e-DUALITY THEOREM OF THE WOLFE TYPE The result of Theorem 3.1 is used to formulate such a dual problem of the Wolfe type for multiobjective programming as follows: maximize L Ž x, l . < Ž x, l . g FD 4 ;
Ž D. here
p
½
m < FD s Ž x, l . g R n = Rq 0g
m
Ý e fi Ž x . q Ý e Ž l j g j . Ž x . , i
j
is1 p
m
js1
m
Ý e i q Ý e j y g F Ý l j g j Ž x . F 0, is1
js1
js1
5
e i G 0, 1 F i F p, e j G 0, 1 F j F m ,
e-PARETO
255
OPTIMALITY
and the vector Lagrangian function LŽ x, l. is defined by L Ž x, l . s f Ž x . q ²² l , g Ž x . :: s
¦
m
f 1 Ž x . q Ž 1rp .
m
;
Ý l j g j Ž x . , . . . , f p Ž x . q Ž 1rp . Ý l j g j Ž x . js1
js1
,
for all x g R n, l j g R, 1 F j F m. m DEFINITION 4.1. A point Ž x, l. g R n = Rq is called an e-Pareto solution of ŽD. if Ž x, l. g FD and there is no Ž x, l. g FD , such that
m
f i Ž x . q Ž 1rp .
Ý
m
l j g j Ž x . G f i Ž x . q Ž 1rp .
js1
Ý l j g j Ž x . q ei ,
1 F i F p,
js1
with at least one strict inequality. THEOREM 4.1 ŽDuality.. If there exists r 0 such that x is a g-solution for Ž ur . for any r G r 0 , then x is an e-Pareto solution for ŽP. and there exist m scalars l g Rq such that Ž x, l. is an e-Pareto solution of ŽD., for all l j G l j, 1 F j F m. Proof. With Theorem 3.1, we conclude that Ž x, l. is a feasible solution m of ŽD.. Let Ž x, l. g R n = Rq be any feasible solution of ŽD.. Then, there U exist x i g e i f i Ž x ., 1 F i F p, yUj g e j Ž l j g j .Ž x ., 1 F j F m, such that p
m
Ý xUi q Ý yUj s 0, is1 p
js1
m
m
Ý e i q Ý e j y g F Ý l j g j Ž x . F 0. is1
js1
js1
By using the characterization of the e-subgradient, we obtain f i Ž x . G f i Ž x . q ² xUi , x y x : y e i ,
lj g j Ž x . G lj g j Ž x . q ²
yUj ,
x y x: y ej,
1 F i F p, 1 F j F m.
Thus, we have p
p
m
m
p
m
Ý fi Ž x . q Ý l j g j Ž x . G Ý fi Ž x . q Ý l j g j Ž x . y Ý ei y Ý e j is1
js1
is1
js1
is1
p
G
Ý fi Ž x . y g is1 p
G
Ý is1
m
fi Ž x . q
Ý lj g j Ž x . y g . js1
js1
256
J. C. LIU
Since l j G l j and g j Ž x . F 0, for all 1 F j F m, m
m
lj g j Ž x . G
Ý
Ý lj g j Ž x . .
js1
js1
We then deduce that p
m
fi Ž x . q
Ý is1
Ý lj g j Ž x . q g js1
p
G
Ý
m
fi Ž x . q
is1
Ý lj gj Ž x . ,
for all Ž x, l . g FD .
Ž 11 .
js1
If Ž x, l. is not an e-Pareto solution of the dual problem ŽD., there exists Ž x*, l*. g FD such that m
f i Ž x* . q Ž 1rp .
Ý lUj g j Ž x*. js1 m
G f i Ž x . q Ž 1rp .
Ý l j g j Ž x . q ei ,
1 F i F p,
js1
with at least one strict inequality. Thus, we have p
Ý is1
m
f i Ž x* . q
Ý js1
p
lUj g j Ž x* . )
Ý is1
m
fi Ž x . q
Ý lj g j Ž x . q g js1
which contradicts Ž11.. THEOREM 4.2 ŽConverse Duality.. Let x be a feasible solution of ŽP.. If Ž x, l. is a feasible solution of ŽD., x is an e-Pareto solution of ŽP.. Proof. It follows from Theorem 3.2. 5. VECTOR LAGRANGIAN AND ITS e-PARETO SADDLE POINT In this section, we consider the e-Pareto saddle point of the vector Lagrangian function.
e-PARETO
257
OPTIMALITY
m DEFINITION 5.1. A point Ž x, l. g R n = Rq is called an e-Pareto saddle point of the vector Lagrangian LŽ x, l. if the following conditions hold:
Ž i.
L Ž x, l . sŽ p. L Ž x, l . q e ,
m for all l g Rq ;
Ž ii .
L Ž x, l . sŽ p. L Ž x, l . q e ,
for all x g R n .
m That is to say, there exist neither l g Rq nor x g R n such that:
m
Ž i.
Ý lj g j Ž x .
f i Ž x . q Ž 1rp .
js1 m
G f i Ž x . q Ž 1rp .
Ý l j g j Ž x . q ei ,
1 F i F p,
js1
with at least one strict inequality, m
Ž ii .
Ý l j g j Ž x . q ei
f i Ž x . q Ž 1rp .
js1 m
F f i Ž x . q Ž 1rp .
Ý lj g j Ž x . ,
1 F i F p,
js1
with at least one strict inequality. THEOREM 5.1. If there exists r 0 such that x is a g-solution for Ž ur . for any m r G r 0 , then x is an e-Pareto solution for ŽP. and there exist scalars l g Rq such that Ž x, l. is an e-Pareto-saddle point of the ¨ ector Lagrangian. m Proof. With Theorem 3.1, there exist l g Rq such that
p
0g
m
Ý e fi Ž x . q Ý e Ž l j g j . Ž x . , i
j
is1 p
m
m
Ý e i q Ý e j y g F Ý l j g j Ž x . F 0. is1
Ž 12 .
js1
js1
Ž 13 .
js1
Then, there exist xUi g e i f i Ž x ., 1 F i F p, yUj g e j Žl j g j .Ž x ., 1 F j F m, such that p
Ý is1
xUi q
m
Ý yUj s 0. js1
258
J. C. LIU
By using the characterization of the e-subgradient, we obtain f i Ž x . G f i Ž x . q ² xUi , x y x : y e i ,
lj g j Ž x . G lj g j Ž x . q ²
yUj ,
1 F i F p,
x y x: y ej,
1 F j F m.
Thus, we have p
p
m
p
m
m
Ý fi Ž x . q Ý l j g j Ž x . G Ý fi Ž x . q Ý l j g j Ž x . y Ý ei y Ý e j is1
js1
is1
js1
is1
js1
p
G
Ý fi Ž x . y g is1 p
G
m
Ý fi Ž x . q Ý l j g j Ž x . y g , is1
for all x g R n .
js1
Ž 14 . Assume that there is an x* g R n such that m
f i Ž x* . q Ž 1rp .
Ý l j g j Ž x*. q e i js1 m
F f i Ž x . q Ž 1rp .
Ý lj g j Ž x . ,
1 F i F p,
js1
with at least one strict inequality, we obtain p
Ý
p
m
f i Ž x* . q
is1
Ý
l j g j Ž x* . q g -
js1
m
Ý fi Ž x . q Ý l j g j Ž x . , is1
js1
which contradicts Ž14.. This gives the first condition of the definition for e-Pareto saddle point. With Ž13., we deduce that p
m
m
Ý l j g j Ž x . q g G Ý e i q Ý e j G 0. js1
is1
js1
m Since x g F, Ý mjs1 l j g j Ž x . F 0, for all l g Rq . Thus, we have
m
Ý js1
m
lj g j Ž x . q g G
Ý lj g j Ž x . js1
m for all l g Rq .
e-PARETO
259
OPTIMALITY
Therefore, we obtain p
m
Ý fi Ž x . q Ý l j g j Ž x . q g is1
js1 p
G
m
Ý fi Ž x . q Ý l j g j Ž x . , is1
m for all l g Rq .
Ž 15 .
js1
m If there is a l* g Rq such that
m
f i Ž x . q Ž 1rp .
Ý lUj g j Ž x . js1 m
G f i Ž x . q Ž 1rp .
Ý l j g j Ž x . q ei ,
1 F i F p,
js1
with at least one strict inequality, we obtain p
m
p
js1
is1
m
Ý f i Ž x . q Ý lUj g j Ž x . ) Ý f i Ž x . q Ý l j g j Ž x . q g is1
js1
which contradicts Ž15.. This completes the proof. THEOREM 5.2. If Ž x, l. is an e-Pareto saddle point of the ¨ ector Lagrangian L and g j Ž x . F g j Ž x ., 1 F j F m, for all x g F, then x is an almost e-Pareto solution of ŽP.. Proof. If Ž x, l. is an e-Pareto saddle point of the vector Lagrangian L, m there is no l g Rq such that m
f i Ž x . q 1rp
m
Ý l j g j Ž x . G f i Ž x . q 1rp Ý l j g j Ž x . q e i , js1
1 F i F p,
js1
Ž 16 . with at least one strict inequality. If x f Fe , g k Ž x . ) g ) ei for some k and all i. Thus, we have g k Ž x . q lk g k Ž x . ) lk g k Ž x . q ei for all i.
260
J. C. LIU
Choose l k s 1 q l k and l j s l j for all j / k for LŽ x, l.; we obtain m
f i Ž x . q 1rp
m
Ý l j g j Ž x . s f i Ž x . q 1rp Ž 1 q lk . g k Ž x . q 1rp Ý l j g j Ž x . js1
j/k m
) f i Ž x . q 1rp
Ý l j g j Ž x . q ei js1 m
s f i Ž x . q 1rpl k g k Ž x . q 1rp
Ý l j g j Ž x . q ei j/k
for all i which contradicts Ž16.. We conclude that x g Fe . Now, we use the other inequality for an e-Pareto saddle point. Then, there is no x g R n such that m
f i Ž x . q Ž 1rp .
m
Ý l j g j Ž x . G f i Ž x . q Ž 1rp . Ý l j g j Ž x . q e i , js1
1 F i F p,
js1
with at least one strict inequality. From gj Ž x. F gj Ž x. ,
1 F j F m,
for all x g F ,
we conclude that there is no x g F such that fi Ž x . F fi Ž x . y ei ,
1 F i F p,
with at least one strict inequality. This concludes the proof of the theorem.
ACKNOWLEDGMENTS The author is thankful to Dr. Koichi Mizukami of Hirosaki University for his comments and suggestions on an earlier version of the paper, especially on the proof of Theorem 5.2.
REFERENCES 1. S. S. Kutateladze, Convex e-programming, So¨ iet. Math. Dokl 20 Ž1979., 391]393. 2. P. Loridan, Necessary conditions for e-optimality, Math. Programming Study 19 Ž1982., 140]152. 3. P. Loridan and J. Morgan, Penalty functions in e-programming and e-minimax problems, Math. Programming 26 Ž1983., 213]231. 4. P. Loridan, e-Solution in vector minimization problems, J. Optim. Theory Appl. 43 Ž1984., 265]267.
e-PARETO
OPTIMALITY
261
5. J. J. Strodiot, V. H. Nguyen, and N. Heukemes, e-Optimal solutions in nondifferentiable convex programming and some related questions, Math. Programming 25 Ž1983., 307]328. 6. I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 Ž1974., 324]353. 7. J. C. Liu, e-Duality theorem of nondifferentiable nonconvex multiobjective programming, J. Optim. Theory Appl. 69 Ž1991., 153]167. 8. W. I. Zangwill, Nonlinear programming via penalty functions, Management Sci. 13 Ž1967., 344]358. 9. D. P. Bertsekas, Necessary and sufficient conditions for a penalty method to be exact, Math. Programming 9 Ž1975., 87]99. 10. S. P. Han and O. L. Mangasarian, Exact penalty functions in nonlinear programming, Math. Programming 17 Ž1979., 251]269. 11. S. P. Han and O. L. Mangasarian, A dual differentiable exact penalty function, Math. Programming 25 Ž1983., 293]306. 12. O. L. Mangasarian, Sufficiency of exact penalty minimization, SIAM J. Control Optim. 23 Ž1985., 30]37. 13. K. Yokoyama, e-Optimality criteria for convex programming problems via exact penalty functions, Math. Programming 56 Ž1992., 233]243. 14. K. Yokoyama, e-Optimality criteria for vector minimization problems via exact penalty functions, J. Math. Anal. Appl. 187 Ž1994., 296]305. 15. J. B. Hiriart Urruty, e-Subdifferential calculus, in ‘‘Convex Analysis and Optimization,’’ Research Notes in Mathematics Series, Vol. 57 ŽJ. P. Aubin and R. B. Vinter, Eds.., pp. 43]92, Pitman, Boston, MA, 1982. 16. P. Wolfe, A duality theorem for nonlinear programming, Quart. Appl. Math. 19 Ž1961., 239]244. 17. H. C. Lai and C. P. Ho, Duality theorem of nondifferentiable convex multiobjective programming, J. Optim. Theory Appl. 50 Ž1986., 407]420. 18. J. C. Liu, Duality for nondifferentiable multiobjective programming without a constraint, preprint. 19. T. Tanino and Y. Sawaragi, Duality theorem in multiobjective programming, J. Optim. Theory Appl. 27 Ž1979., 509]529. 20. J. Jahn, Duality in vector optimization, Math. Programming 25 Ž1983., 343]353. 21. R. R. Egudo and M. A. Hanson, Multiobjective duality with invexity, J. Math. Anal. Appl. 126 Ž1987., 469]477. 22. T. Weir, Duality for nondifferentiable multiple objective fractional programming problems, Utilitas Math. 36 Ž1989., 53]64. 23. T. Weir, Proper efficiency and duality for vector valued optimization problems, J. Austral. Math. Soc. Ser. A 43 Ž1987., 21]35. 24. B. Mond, I. Hsian, and M. V. Durga Prassad, Duality for a class of nondifferentiable multiobjective programs, Utilitas Math. 39 Ž1991., 3]19. 25. R. N. Kual, S. K. Suneja, and M. K. Srivastava, Optimality criteria and duality in multiple-objective optimization involving generalized invexity, J. Optim. Theory Appl. 80 Ž1994., 465]482. 26. J. W. Nieuwenhuis, Some minimax theorems in vector-valued functions, J. Optim. Theory Appl. 40 Ž1983., 463]475.
198, 248]261 Ž1996.
0080
e-Pareto Optimality for Nondifferentiable Multiobjective Programming via Penalty Function J. C. Liu Section of Mathematics, National Uni¨ ersity Preparatory School, National O¨ erseas Chinese Student Uni¨ ersity, Linkou, Taipei, Hsien, Taiwan, 24402, Republic of China Submitted by Koichi Mizukami Received January 9, 1995
Necessary and sufficient conditions without a constraint qualification for e-Pareto optimality of multiobjective programming are derived. The necessary Kuhn]Tucker condition suggests the establishment of a Wolf-type duality theorem for nondifferentiable, convex, multiobjective minimization problems. The generalized e-saddle point for Pareto optimality of the vector Lagrangian is studied. Q 1996 Academic Press, Inc.
1. INTRODUCTION Several authors have been interested recently in e-optimal solutions in nonlinear programming. For details, the readers are advised to consult w1]5x. Loridan w4x derived some properties of e-efficient points solution for vector minimization problems and used the Ekeland’s variational principle w6x to establish the e-Pareto optimality and e-quasi Pareto optimality. In w7x, Liu also adapted the same approach to obtain the e-duality theorem of nondifferentiable nonconvex multiobjective programming. Recently, several authors w8]12x have used an exact penalty function to transform the nonlinear scalar programming problem into an unconstrained problem and derived the e-optimality. In w14x, Yokoyama was concerned with the e-approximate solutions and extended some results of w13x to the vector minimization problems. Similar to w13x, Yokoyama transformed the vector problems into the unconstrained problems by using the exact penalty functions and showed the e-optimality criteria by estimating the penalty parameter in terms of e-approximate solutions for the associated dual problems. 248 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
e-PARETO
249
OPTIMALITY
In this paper, we are inspired to use the exact penalty function to transform the inequality multiobjective programming problem into a scalar unconstrained problem and to derive the Kuhn]Tucker conditions in which Lagrange multiplers of objective functions are one. Some definitions and notations are given in Section 2. In Section 3, we use scalar penalty functions to establish the necessary and sufficient conditions of e-Pareto solution. Using this result, we formulate a dual problem of the Wolfe-type for multiobjective programming. In Section 4, we give some relationships between the primal problem and the dual problem. The generalized e-Pareto saddle point of the vector Lagrangian is discussed in Section 5.
2. PRELIMINARIES We consider the following convex multiobjective programming problem:
Ž P.
minimize f Ž x . subject to g j Ž x . F 0,
j s 1, . . . , m,
where f s Ž f 1 , f 2 , . . . , f p . and each component function is a convex continuous real-valued function defined on R n and where g j are convex continuous real-valued functions defined on R n, 1 F j F m. We denote the feasible set x g R n < g j Ž x . F 0, 1 F j F m4 by F and assume the feasible set F is nonempty. Let e be an element of Rqp . We introduce the e feasible set Fe , p
½
Ý ei , 1 F j F m
Fe s x g R n g j Ž x . F
is1
5
.
p For convenience, let g s Ý is1 ei. To transform the problem ŽP. into a scalar unconstrained problem, we use the exact penalty function introduced by Zangwill w8x:
p
u Ž x, r . s
Ý
m
fi Ž x . q r
is1
Ý max Ž 0, g j Ž x . . , js1
where
r ) 0. The associated unconstrained problem in which
Ž ur .
minimize u Ž x, r .
250
J. C. LIU
is called a penalized problem with respect to the penalty parameter r . For convenience, let p Ž t . s max Ž 0, r t . . p Clearly, we have u Ž x, r . s Ý is1 f i Ž x . q Ý mjs1 pŽ g j Ž x ...
DEFINITION 2.1. A point x g R n is called an e-Pareto solution of ŽP. if x g F and there is no x g F such that f Ž x . F f Ž x . y e and f Ž x . / f Ž x . y e ,
in R p .
DEFINITION 2.2. A point x g R n is called an almost e-Pareto solution of ŽP. if x g Fe and there is no x g F such that f Ž x . F f Ž x . y e and f Ž x . / f Ž x . y e ,
in R p .
DEFINITION 2.3. Let a ) 0. A point x g R n is called an a-solution of the scalar problem Ž ur . if
u Ž x, r . F u Ž x, r . q a ,
for all x g R n .
DEFINITION 2.4. Let h: R n ª R j q`4 be a convex function, finite at x. The e-subdifferential of h at x is the set e hŽ x . defined by
e h Ž x . s x* g R n < h Ž y . G h Ž x . y e q ² x*, y y x : for any y g R n 4 .
3. NECESSARY AND SUFFICIENT CONDITIONS In this section, we present some Kuhn]Tucker conditions for e-Pareto optimality. THEOREM 3.1. If there exists r 0 such that x is a g-solution for Ž ur . for any r G r 0 , then x is an e-Pareto solution for ŽP. and there exist scalars e i G 0 Ž1 F i F p ., e jG 0 Ž1 F j F m., l jG 0 Ž1 F j F m. such that: p
Ži. 0 g
Ý e Žl j g j .Ž x .,
is1
js1
p
Žii.
m
Ý e i fi Ž x . q
j
m
Ý ei q
Ý
is1
js1
Ž1.
m
ej y g F
Ý l j g j Ž x . F 0. js1
Ž2.
e-PARETO
251
OPTIMALITY
Proof. If x is a g-solution of Problem Ž ur .,
u Ž x, r . F u Ž x, r . q g ,
for all x g R n .
Ž 3.
Clearly, p
u Ž x, r . s
for all x g F.
Ý fi Ž x . , is1
Thus, we have p
p
Ý f i Ž x . F u Ž x, r . F Ý f i Ž x . q g , is1
for all x g F.
Ž 4.
is1
If x f F, Ý mjs1 pŽ g j Ž x .. ) 0. Choose any feasible point ˆ x which is also in F and let
r ) max
½ž
p
p
m
Ý f i Ž ˆx . y Ý f i Ž x . is1
is1
Ý pŽ gj Ž x . . , r0
/
js1
5
.
We then have the conclusion p
Ý
p
fi Ž ˆ x. s u Ž ˆ x, r . G u Ž x, r . y g )
is1
Ý f i Ž ˆx . y g . is1
This conclusion gives a contradiction and hence x g F. If x is not an e-Pareto solution of ŽP., then there exists x 1 g F such that fi Ž x1 . F fi Ž x . y ei , 1 F i F p, with at least one strict inequality. Therefore, we have p
p
Ý
fi Ž x1 . -
is1
Ý fi Ž x . y g , is1
which contradicts Ž4.. Thus x is an e-Pareto solution of ŽP.. With Ž3. and the result of w15x, we have p
0 g g
žÝ
m
fi Ž ?. q r
/
Ý p Ž g j Ž ?. . Ž x . .
is1
js1
Then, there exist scalars e iG 0 Ž1 F i F p ., e jG 0 Ž1 F j F m., a j G 0 Ž1 F j F m., and hj G 0 Ž1 F j F m. such that p
m
Ý ei q
Ý ej s g ,
is1
js1
p
0g
Ž 5.
m
Ý e i fi Ž x . q
Ý h Ž a j r g j . Ž x . ,
is1
js1
i
Ž 6.
252
J. C. LIU
where
a j F 1, hj q p Ž g j Ž x . . y a j r g j Ž x . s e j
for j s 1, . . . , m.
Ž 7.
By Ž5. and Ž7., we have p
m
m
Ý e i q Ý hj y g F Ý a j r g j Ž x . F 0. is1
js1
js1
Finally, we obtain the results Ž1. and Ž2. by setting
lj s aj r , e j s hj ,
1 F j F m, 1 F j F m.
REMARK 3.1. If p s 1, then the necessary condition of Theorem 3.1 reduces to Theorem 4.1 of w13x. THEOREM 3.2. If x is a feasible solution of ŽP. and there exist e iG 0 Ž1 F i F p ., e jG 0 Ž1 F j F m., l jG 0 Ž1 F j F m. such that: p
Ži. 0 g
m
Ý e f i Ž x . q Ý e Žl j g j .Ž x ., i
is1 p
Žii.
Ž8.
j
js1 m
m
Ý e i q Ý e j y g F Ý l j g j Ž x . F 0. is1
js1
Ž9.
js1
Then, x is an e-Pareto solution of ŽP.. Proof. If x is a feasible solution of ŽP. and there exist e iG 0 Ž1 F i F p ., e jG 0 Ž1 F j F m., l jG 0 Ž1 F j F m. which satisfy relations Ž8. and Ž9., then there exist xUi g e i f i Ž x ., 1 F i F p, yUj g e j Ž l j g j .Ž x ., 1 F j F m, such that p
m
Ý xUi q Ý yUj s 0. is1
js1
By using the characterization of the e-subgradient, we obtain f i Ž x . G f i Ž x . q ² xUi , x y x : y e i ,
lj g j Ž x . G lj g j Ž x . q ²
yUj ,
x y x: y ej,
1 F i F p, 1 F j F m.
e-PARETO
253
OPTIMALITY
Thus, we have p
p
m
m
Ý fi Ž x . q Ý l j g j Ž x . y Ý ei y Ý e j is1
js1
is1
p
F
js1 p
m
Ý fi Ž x . q Ý l j g j Ž x . F Ý fi Ž x . , is1
js1
for all x g F.
is1
With Ž9., we have p
Ý
p
fi Ž x . F
is1
Ý fi Ž x . q g ,
for all x g F.
Ž 10 .
is1
If x is not an e-Pareto solution of ŽP., there exists x 1 g F such that fi Ž x1 . F fi Ž x . y ei ,
1 F i F p,
with at least one strict inequality. Therefore, we have p
p
Ý
fi Ž x1 . -
is1
Ý fi Ž x . y g , is1
which contradicts Ž10.. Thus x is an e-Pareto solution of ŽP.. THEOREM 3.3. If for sufficiently large r , x is a g-solution for Ž ur ., then x is an almost e-Pareto solution for ŽP. and there exist scalars e iG 0 Ž1 F i F p ., e jG 0 Ž1 F j F m., l jG 0 Ž1 F j F m. such that: p
Ž i. 0 g
m
Ý e fi Ž x . q Ý e Ž l j g j . Ž x . , i
is1
p
Ž ii .
j
js1
m
m
Ý e i q Ý e j y g F Ý l j g j Ž x . F 0. is1
js1
js1
Proof. If x is a g-solution of Problem Ž ur ., p
p
Ý f i Ž x . F u Ž x, r . F Ý f i Ž x . q g , is1
is1
for all x g F.
254
J. C. LIU
Since p
inf
p
Ý fi Ž x . F Ý fi Ž x . ,
xgR n is1
is1
we have p
m
r
p
inf Ý f i Ž x . y Ý p Ž g j Ž x . . F xgF
js1
is1
inf n
xgR
Ý fi Ž x . q g . is1
Let p
r0 Ž e . s g
inf
Ý
xgF is1
p
f i Ž x . y inf n xgR
Ý fi Ž x . q g is1
Then, there exists r F r 0 Ž e . such that pŽ gj Ž x. . F g ,
1 F j F m.
Hence, we have x g Fe . This concludes the proof of the theorem. 4. e-DUALITY THEOREM OF THE WOLFE TYPE The result of Theorem 3.1 is used to formulate such a dual problem of the Wolfe type for multiobjective programming as follows: maximize L Ž x, l . < Ž x, l . g FD 4 ;
Ž D. here
p
½
m < FD s Ž x, l . g R n = Rq 0g
m
Ý e fi Ž x . q Ý e Ž l j g j . Ž x . , i
j
is1 p
m
js1
m
Ý e i q Ý e j y g F Ý l j g j Ž x . F 0, is1
js1
js1
5
e i G 0, 1 F i F p, e j G 0, 1 F j F m ,
e-PARETO
255
OPTIMALITY
and the vector Lagrangian function LŽ x, l. is defined by L Ž x, l . s f Ž x . q ²² l , g Ž x . :: s
¦
m
f 1 Ž x . q Ž 1rp .
m
;
Ý l j g j Ž x . , . . . , f p Ž x . q Ž 1rp . Ý l j g j Ž x . js1
js1
,
for all x g R n, l j g R, 1 F j F m. m DEFINITION 4.1. A point Ž x, l. g R n = Rq is called an e-Pareto solution of ŽD. if Ž x, l. g FD and there is no Ž x, l. g FD , such that
m
f i Ž x . q Ž 1rp .
Ý
m
l j g j Ž x . G f i Ž x . q Ž 1rp .
js1
Ý l j g j Ž x . q ei ,
1 F i F p,
js1
with at least one strict inequality. THEOREM 4.1 ŽDuality.. If there exists r 0 such that x is a g-solution for Ž ur . for any r G r 0 , then x is an e-Pareto solution for ŽP. and there exist m scalars l g Rq such that Ž x, l. is an e-Pareto solution of ŽD., for all l j G l j, 1 F j F m. Proof. With Theorem 3.1, we conclude that Ž x, l. is a feasible solution m of ŽD.. Let Ž x, l. g R n = Rq be any feasible solution of ŽD.. Then, there U exist x i g e i f i Ž x ., 1 F i F p, yUj g e j Ž l j g j .Ž x ., 1 F j F m, such that p
m
Ý xUi q Ý yUj s 0, is1 p
js1
m
m
Ý e i q Ý e j y g F Ý l j g j Ž x . F 0. is1
js1
js1
By using the characterization of the e-subgradient, we obtain f i Ž x . G f i Ž x . q ² xUi , x y x : y e i ,
lj g j Ž x . G lj g j Ž x . q ²
yUj ,
x y x: y ej,
1 F i F p, 1 F j F m.
Thus, we have p
p
m
m
p
m
Ý fi Ž x . q Ý l j g j Ž x . G Ý fi Ž x . q Ý l j g j Ž x . y Ý ei y Ý e j is1
js1
is1
js1
is1
p
G
Ý fi Ž x . y g is1 p
G
Ý is1
m
fi Ž x . q
Ý lj g j Ž x . y g . js1
js1
256
J. C. LIU
Since l j G l j and g j Ž x . F 0, for all 1 F j F m, m
m
lj g j Ž x . G
Ý
Ý lj g j Ž x . .
js1
js1
We then deduce that p
m
fi Ž x . q
Ý is1
Ý lj g j Ž x . q g js1
p
G
Ý
m
fi Ž x . q
is1
Ý lj gj Ž x . ,
for all Ž x, l . g FD .
Ž 11 .
js1
If Ž x, l. is not an e-Pareto solution of the dual problem ŽD., there exists Ž x*, l*. g FD such that m
f i Ž x* . q Ž 1rp .
Ý lUj g j Ž x*. js1 m
G f i Ž x . q Ž 1rp .
Ý l j g j Ž x . q ei ,
1 F i F p,
js1
with at least one strict inequality. Thus, we have p
Ý is1
m
f i Ž x* . q
Ý js1
p
lUj g j Ž x* . )
Ý is1
m
fi Ž x . q
Ý lj g j Ž x . q g js1
which contradicts Ž11.. THEOREM 4.2 ŽConverse Duality.. Let x be a feasible solution of ŽP.. If Ž x, l. is a feasible solution of ŽD., x is an e-Pareto solution of ŽP.. Proof. It follows from Theorem 3.2. 5. VECTOR LAGRANGIAN AND ITS e-PARETO SADDLE POINT In this section, we consider the e-Pareto saddle point of the vector Lagrangian function.
e-PARETO
257
OPTIMALITY
m DEFINITION 5.1. A point Ž x, l. g R n = Rq is called an e-Pareto saddle point of the vector Lagrangian LŽ x, l. if the following conditions hold:
Ž i.
L Ž x, l . sŽ p. L Ž x, l . q e ,
m for all l g Rq ;
Ž ii .
L Ž x, l . sŽ p. L Ž x, l . q e ,
for all x g R n .
m That is to say, there exist neither l g Rq nor x g R n such that:
m
Ž i.
Ý lj g j Ž x .
f i Ž x . q Ž 1rp .
js1 m
G f i Ž x . q Ž 1rp .
Ý l j g j Ž x . q ei ,
1 F i F p,
js1
with at least one strict inequality, m
Ž ii .
Ý l j g j Ž x . q ei
f i Ž x . q Ž 1rp .
js1 m
F f i Ž x . q Ž 1rp .
Ý lj g j Ž x . ,
1 F i F p,
js1
with at least one strict inequality. THEOREM 5.1. If there exists r 0 such that x is a g-solution for Ž ur . for any m r G r 0 , then x is an e-Pareto solution for ŽP. and there exist scalars l g Rq such that Ž x, l. is an e-Pareto-saddle point of the ¨ ector Lagrangian. m Proof. With Theorem 3.1, there exist l g Rq such that
p
0g
m
Ý e fi Ž x . q Ý e Ž l j g j . Ž x . , i
j
is1 p
m
m
Ý e i q Ý e j y g F Ý l j g j Ž x . F 0. is1
Ž 12 .
js1
js1
Ž 13 .
js1
Then, there exist xUi g e i f i Ž x ., 1 F i F p, yUj g e j Žl j g j .Ž x ., 1 F j F m, such that p
Ý is1
xUi q
m
Ý yUj s 0. js1
258
J. C. LIU
By using the characterization of the e-subgradient, we obtain f i Ž x . G f i Ž x . q ² xUi , x y x : y e i ,
lj g j Ž x . G lj g j Ž x . q ²
yUj ,
1 F i F p,
x y x: y ej,
1 F j F m.
Thus, we have p
p
m
p
m
m
Ý fi Ž x . q Ý l j g j Ž x . G Ý fi Ž x . q Ý l j g j Ž x . y Ý ei y Ý e j is1
js1
is1
js1
is1
js1
p
G
Ý fi Ž x . y g is1 p
G
m
Ý fi Ž x . q Ý l j g j Ž x . y g , is1
for all x g R n .
js1
Ž 14 . Assume that there is an x* g R n such that m
f i Ž x* . q Ž 1rp .
Ý l j g j Ž x*. q e i js1 m
F f i Ž x . q Ž 1rp .
Ý lj g j Ž x . ,
1 F i F p,
js1
with at least one strict inequality, we obtain p
Ý
p
m
f i Ž x* . q
is1
Ý
l j g j Ž x* . q g -
js1
m
Ý fi Ž x . q Ý l j g j Ž x . , is1
js1
which contradicts Ž14.. This gives the first condition of the definition for e-Pareto saddle point. With Ž13., we deduce that p
m
m
Ý l j g j Ž x . q g G Ý e i q Ý e j G 0. js1
is1
js1
m Since x g F, Ý mjs1 l j g j Ž x . F 0, for all l g Rq . Thus, we have
m
Ý js1
m
lj g j Ž x . q g G
Ý lj g j Ž x . js1
m for all l g Rq .
e-PARETO
259
OPTIMALITY
Therefore, we obtain p
m
Ý fi Ž x . q Ý l j g j Ž x . q g is1
js1 p
G
m
Ý fi Ž x . q Ý l j g j Ž x . , is1
m for all l g Rq .
Ž 15 .
js1
m If there is a l* g Rq such that
m
f i Ž x . q Ž 1rp .
Ý lUj g j Ž x . js1 m
G f i Ž x . q Ž 1rp .
Ý l j g j Ž x . q ei ,
1 F i F p,
js1
with at least one strict inequality, we obtain p
m
p
js1
is1
m
Ý f i Ž x . q Ý lUj g j Ž x . ) Ý f i Ž x . q Ý l j g j Ž x . q g is1
js1
which contradicts Ž15.. This completes the proof. THEOREM 5.2. If Ž x, l. is an e-Pareto saddle point of the ¨ ector Lagrangian L and g j Ž x . F g j Ž x ., 1 F j F m, for all x g F, then x is an almost e-Pareto solution of ŽP.. Proof. If Ž x, l. is an e-Pareto saddle point of the vector Lagrangian L, m there is no l g Rq such that m
f i Ž x . q 1rp
m
Ý l j g j Ž x . G f i Ž x . q 1rp Ý l j g j Ž x . q e i , js1
1 F i F p,
js1
Ž 16 . with at least one strict inequality. If x f Fe , g k Ž x . ) g ) ei for some k and all i. Thus, we have g k Ž x . q lk g k Ž x . ) lk g k Ž x . q ei for all i.
260
J. C. LIU
Choose l k s 1 q l k and l j s l j for all j / k for LŽ x, l.; we obtain m
f i Ž x . q 1rp
m
Ý l j g j Ž x . s f i Ž x . q 1rp Ž 1 q lk . g k Ž x . q 1rp Ý l j g j Ž x . js1
j/k m
) f i Ž x . q 1rp
Ý l j g j Ž x . q ei js1 m
s f i Ž x . q 1rpl k g k Ž x . q 1rp
Ý l j g j Ž x . q ei j/k
for all i which contradicts Ž16.. We conclude that x g Fe . Now, we use the other inequality for an e-Pareto saddle point. Then, there is no x g R n such that m
f i Ž x . q Ž 1rp .
m
Ý l j g j Ž x . G f i Ž x . q Ž 1rp . Ý l j g j Ž x . q e i , js1
1 F i F p,
js1
with at least one strict inequality. From gj Ž x. F gj Ž x. ,
1 F j F m,
for all x g F ,
we conclude that there is no x g F such that fi Ž x . F fi Ž x . y ei ,
1 F i F p,
with at least one strict inequality. This concludes the proof of the theorem.
ACKNOWLEDGMENTS The author is thankful to Dr. Koichi Mizukami of Hirosaki University for his comments and suggestions on an earlier version of the paper, especially on the proof of Theorem 5.2.
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