447
NONLINEAR SEPARATION FOR CONSTRAINED MULTIOBJECTIVE OPTIMIZATION PROBLEMS AND APPLICATIONS TO PENALTY
FUNCTIONS
K. Guo* & S.Q. Feng
1College of Mathematics and Information, China West Normal University, Nanchong, Sichuan 637009, China
*E-mail address: [email protected] ABSTRACT
In this paper, we prove a nonlinear separation associated with a constrained multiobjective optimization problem in the image space, as well as the equivalence between the existence of nonlinear separation function and a saddle point condition for a generalized Lagrangian function associated with the given problem. As applications, we obtain some equivalent conditions on exact penalty methods for the given problem.
Keywords and Phrases: Image space analysis; Nonlinear separation; Multiobjective optimization; Penalty method.
2000 Mathematics Subject Classification: 90C29.
1. INTRODUCTION
The imgae of a constrained extremum problem was first developed in [7] by Giannessi. In recent years, there has been more and more vector variational inequalities and vector optimization problems are developed by means of the Image Space analysis (ISA)[6, 9, 10, 11, 14, 15]. The ISA is a powerful tool and a unifying scheme for studying constrained optimization problems, this approach can be applied to any kind of problems that can be expressed under the form of the impossibility of a parametric system. Many theoretical aspects results has been obtained by exploiting ISA, such as existence of optimal solutions, duality[1, 12, 17, 19], Lagrangian-type optimality condition[3, 6, 8, 16, 18], penalty methods[2, 4, 5] and so on. The optimality condition of the constrained extremum problem can be expressed under the form of the impossibility of a parametric system, which is reduced to the disjunction of two suitable subsets of the image space.
The purpose of this paper is to extend some results in [13] to constrained multiobjective optimization problem. The paper is organized as follows. In section 2, we recall the main definitions; In section 3, we recall basic properties of the ISA for constrained multiobjective optimization problem and discuss the image problem for constrained multiobjective optimization problem; In section 4, by virtue of the ISA, we characterize the (res,. regular, strongly regular) nonlinear separation and the saddle points of the generalized Lagrangian functions for constrained multiobjective optimization problem, and obtain the relation between the saddle points of the generalized Lagrangian function and the the Pareto efficient solution for the constrained multiobjective optimization problem. Also, we apply the above results to the augmented Lagrangian function; In section 5, we present some equivalent condition associated with exact penalty methods for constrained multiobjective optimization problem.
2. PRELIMINARIES
Let
R
s be thes
dimensional Euclidean space, wheres
is given positive integer. Denote by} , 1,
= 0, : ) , , ( :=
{
R
s:= x x
1 x
s x
i i s
and
R
s:= { x := ( x
1, , x
s)
: x
i> 0, i = 1, , s }
. For) R , , (
:= x
1x
l lx
, A nonempty subsetP R
s is said to be a cone iftP P
for allt 0
. A coneP R
l is said to be a convex cone ifP P P
. A coneP R
s is said to be a pointed cone if{0}
= ) ( P
P
. The closure ofP
is denoted bycl P
. ,
denotes the inner product,while} 0,
, : {
*
=
P x x
y R y
P
s
is the positive polar cone of a convex cone
P R
s.This work was supported by the Natural Science Foundation of China (60804065), the Key Project of Chinese Ministry of Education (211163), Sichuan Youth Science and Technology Foundation (2012JQ0032) and Students' Science and Technology Innovation Foundation of China West Normal University (42722050)
448
Let
P R
s such thatcl P
is a convex cone. We use the notationsR , ,
,
1 21 2 2 1
s
P
y y y P y y
y
R . ,
,
1 21 2 2 1
s
P
y y y P y y
y
In this paper, we consider the following constrained multiobjective optimization problem:
0}.
) ( : {
=
. .
) (
R {0}
\
l Rm
x g X x K x t s
x f min
(1)where
f : R
n R
m,g : R
n R
l andX
is a nonempty convex subset ofR
n.If
x K
satisfyingf x f x
mx K
R
( )
0, )
(
\{0} , thenx
is called a Pareto efficient solution of problem (1).3. ISA FOR CONSTRAINED MULTIOBJECTIVE OPTIMIZATION PROBLEM
In this section, we develop the image space analysis for constrained multiobjective optimization problem (1).
Observe that,
x K
solves problem (1), if and only if the system (in the unknownx
):
. 0, ) (
0, )
( ) (
R
{0}
R \
X x
x g
x f x f
l
m
(1)
is impossible.
Let
x K
. Define the mappingA
x: X R
m R
l by. )),
( ), ( ) ( (
= )
( x f x f x g x x X
A
x
(2)We can associate problem (1) with the following sets:
), (
= } ),
(
= ), ( ) (
= R : ) R
, {(
= u v
m lu f x f x v g x x X A
xX
x
R . {0}
R \
= 0}
0, R :
) R , {(
=
R \{0} Rl m
l m
l
m
u v
v
u
The set
x is the image associated with problem (1) atx K
, we callR
m R
l image space.It is easy to see that the system (1) is impossible if and only if
.
=
x (3)Consequently,
x K
solves (1) if and only if (3) is true.In general,to prove directly whther or not (3) holds is too difficult. This is because the image set is not convex even when the functions involved enjoy some convexity properties. To overcome this difficulty, we introduce a regularization of the image
x with respect to the conecl
, denoted by,R . ) R
(
=
} ),
( ),
( ) ( R :
) R , {(
=
=
R R
l m x
l m
l m x
x
X A
X x x g v x f x f u v
u cl
which is called the extended image associated with problem (1) at
x K
. Proposition 3.1x K
is a Pareto efficient solution of problem (1) if and only if,
=
x (4)or equivalently,
449
,
=
ux
(5)where
{0}.
{0}
R \
= 0}
= 0, R :
) R , {(
=
R \{0}
m m
l m
u
u v u v
Proof. It is clear that
x K
is a Pareto efficient solution of problem (1) if and only if (3) holds. We show (3) and (4) are equivalent firstly.It suffices to show (4) implies (3), since
x
x. To prove the reverse implication, ab absurdo, suppose that (4) does not hold. Then there exists( u ) , v
x
, i.e., x ~ X
such that~ ), (
~ ), ( )
(
RR
f x f x v g x
u
m l
and
0.
0,
R{0}
Rm\
v
lu
Observe that
R
m R
m\ {0} R
m\ {0}
. It follows that0 ( ) ( ~ ), 0 ( ~ )
{0} R
Rm\
f x f x
lg x
, which leadsto a contradiction, since
x =
. Now, we prove (4) and (5) are equivalent.Obviously, (4) implies (5), since
u
. Suppose to the contrary that (4) does not hold, i.e., ( u ) ˆ , v ˆ
x
. Since
, ˆ )
(0,
=
= ) (
=
x cl
x cl cl
x cl
xv cl
it follows that
( u ˆ , v ˆ ) (0, v ˆ ) = ( u ˆ ,0)
x . Again since( u ˆ ,0)
u , this leads a contradiction with
u=
x
.Consider the following image problem for problem (1):
R . , ) , (
{0}
R \
l x
m
v v
u
u max
(6)If
( u , v )
x, v R
l satisfying, ( , ) , R
{0}
R \
l x
m
u u v v
u
, thenu
is called a Pareto efficientsolution of (6).
Based on the extended image, we consider an equivlent image problem for problem (1):
0.
= , ) , (
{0}
R \
v v
u
u max
x m
(7)If
( u , v )
x, v = 0
satisfying \{0}, ( , ) , = 0
R
u u v v
u
m
x
, thenu
is called a Pareto efficient solutionof (7).
Proposition 3.2 Let
x K
andv = g ( x )
. Thenx K
is a Pareto efficient solution of problem (1) if and only if(i)
(0, v )
is a Pareto efficient solution of (6);or equivalently,
(ii)
(0,0)
is a Pareto efficient solution of (7).Proof. We first prove (1) and (i) are equivalent.
Suppose that
x K
is a Pareto efficient solution of problem (1). Thenx X
andv = g ( x ) R
l. We havex
xg x f x f
v ) = ( ( ) ( ), ( ))
(0,
, i.e.,(0, v )
is a feasible solution of (6). Since
x =
, it follows that(0, v )
is a Pareto efficient solution of (6). Vice versa, if(0, v )
is a Pareto efficient solution of (6), thenX
x
andv = g ( x ) R
l, i.e.,
x =
. Thusx K
is a Pareto efficient solution of problem (1).450 Now,we prove (1) and (ii) are equivalent.
Suppose that
x K
is a Pareto efficient solution of problem (1). Then we have
x
u=
from proposition 3.1. Sincex K
, one hasx X , v = g ( x ) R
l, it follows that(0, v ) = ( f ( x ) f ( x ), g ( x ))
x. Notice that(0, v ) cl
. Then(0, v ) (0, v ) = (0,0)
x cl =
x, i.e.,(0,0)
is a feasible solution of (7). Suppose(0,0)
is not a Pareto efficient of (6). Then y X
such that( ) 0
y
Rlg
,) ( ) (
0
Rm\{0}f x f y
. It follows that
0 ( ) ( ) ( ) ( )
{0} R
Rm\
f x f y
mf x f y
. Letu = f ( x ) f ( y )
,this leads a contradiction, since
( u ,0)
x
u . Vice versa, if(0,0)
is a Pareto efficient of (8), then) R (
=
, v g x
lX
x
, i.e.,
x
u=
. It follows thatx K
is a Pareto efficient solution of problem (1).Proposition 3.3 Let
x K
. If there existsx ~ K
such thatf x
mf x x K
( ),
~ )
(
R , then0}
=
~ ), ( ) ( 0
: ) , {(
= u v
Ru
Rf x f x v
cl
u m mx
.Proof. If there exists
~ x K
such thatf x
mf x x K
( ),
~ )
(
R , then. 0, ) (
~ ), ( ) ( )
( )
( x f x
Rf x f x g x
Rx X
f
m
l
(8)
Consequently,
( ) ( ~ ), ( , ) , R
R
l x
m
f x f x u v v
u
. We declare thatR . , ) , (
~ ), ( )
R
(
l x
m
f x f x u v v
u
(9)In fact, if
( u , v )
x, v R
l, then there existsy ˆ X
such that( ) ( ˆ ), 0 ( ˆ )
R R
R
f x f y v g y
u
m l l
.It follows that
y ˆ K
, and from (8), we have~ ), ( ) ( ˆ )
( )
(
RR
f x f y f x f x
u
m
m
which implies that (9) holds. Since
= {( , ) R R : 0, = 0}
R
v
u v
u
cl
u m l m
, from (9) we can obtain0}
=
~ ), ( ) ( 0
: ) , {(
= u v
Ru
Rf x f x v
cl
u m mx
.4. NONLINEAR SEPARATION AND SADDLE POINTS FOR CONSTRAINED MULTIOBJECTIVE OPTIMIZATION PROBLEM
In order to prove (3), we will show
x and lie in two disjoint level sets of a suitable nonlinear separation function by ISA. Let us introduce the class of functions : R
m R
l ( R
m ) R
1, defined by:, R ,
),
; ( ,
= ) ,
; ,
(
u v u v
m (1)where
: R
l R
1,
is a given parameter set.Definition 4.1 Given a parameter set
. We say that
x and are nonlinearly separable if and only if there exist R
m,
and ( u , v ; , ) = , u ( v ; ) 0
, such that:, ) , ( 0, )
; ( ,
= ) ,
; ,
( u v u v u v
(2)
. ) , ( 0, )
; ( ,
= ) ,
; ,
( u v u v u v
x
(3)Moreover, if
R
m\ {0}
, then the separation is regular; if R
m, the separation is strongly regular.
451
Lemma 4.1 Suppose there exist
R
m,
such that (2). Then R
m and ( v ; ) 0, v R
l.Proof. Ab absurdo, if
= (
1,
2, ,
m)
R
m, then there existsi {1,2, , m }
such that
i< 0
. Let0}
<
: {
0
= i
iI
andv R
l be fixed, and{ u
k} R
m be a sequence of points. Denote by( u )
k i thei
thcomponent of
u
k,
0 0
I ifi 0,
I ifi
= , )
( k
u
k ithen
0
{0}
R m\
u
k
. u
k
, ifk
. It follows that{( u
k, v )}
and
( , ; , ) = lim [ , ( ; )] =
lim
k u
kv
k u
k v
, a contradiction from (2). Therefore R
m . Now, we prove ( v ; ) 0, v R
l. Letv R
l be fixed, and{ u
k} R
m be a sequence of points such that.
± µ 0,
{0}
0,
R \
u k
u
k m
k
Then,
{( u
k, v )}
and ( v ; ) = lim
k ( u
k, v ; , ) 0
.In view of Lemma 4.1, we can restrict our analysis to the class of functions
( u , v ; , )
such that
R
m,
and ( v ; ) 0, v R
l. This class of functions will be called separation functions. It is clear that when = 0
, the existence of a nonlinear separation can not guarantee the
x =
. However, ifR
m
, i.e., the separation is strongly regular, strict inequality holds in (2), then (3) holds. Consequently, the following proposition is clear.Proposition 4.1 Suppose
x K
. If
x and are strongly regular nonlinearly separable, thenx K
is aPareto efficient solution of problem (1).
Proof. Suppose
x K
. If
x and are nonlinearly separable, then there exist R
m,
such that (2) and (3) hold. Since this separation is strongly regular, then R
m, i.e., the strict inequality holds in (2). Therefore,
=
x , i.e.,x K
is a Pareto efficient solution of (1).In the following, without loss of generality, we assume
R
m, denoted the strongly regular separation functions by:).
; ( ,
= ) ,
; ,
(
u v u v
(4)
Theorem 4.1 Let
x K
. Consider the class of strongly regular separation functions (4) that fulfill the following assumptions:R , ,
= )
;
inf ( v v
l
(5)
R . 0,
= )
;
inf ( v v
l
(6)
Then,
. , sup
= )]
; ( ,
inf [ sup
R ) , ) (
, (
u v
u
v l v x x u
v u
(7)
Proof. Since (5) and (6) hold, for every fixed
u R
m, we have
, , ifv R .
R , ifv
= , )]
; ( ,
inf [
ll
v u
u
(8)452
Notice that
( u , v ) = ( f ( x ) f ( x ), g ( x )) = (0, g ( x )) K
x andv R
l , sincex K
. Taking the supremum in (8) leads to (7).Remark 4.1 When the separation function
( , ; , )
is linear, i.e., ( v ; ) = , v , ( R
l)
*= R
l, (2) and (3) are obvious.
Now, we consider the generalized Lagrangian function associated with problem (1), defined by
).
);
( ( ) ( ,
= ) ,
;
(
x f x g x
L
(9)
For a given separation function
( u , v ; , ) = , u ( v ; )
, we shall present the equivalence between a nonlinear separation for problem (1) and the existence of saddle points of the generalized Lagrangian function (9).Obviously, when
( v ; ) = , v
and = m = 1
, (9) collapses to the standard Lagrangian function.Theorem 4.2 Suppose that (5) and (6) hold. Then the sets
x and
are nonlinearly separable andx K
, if and only if there exists( , ) R
m
such that( x , )
is a saddle point forL
( x ; , )
onX
. i.e.,. )
, ( ), ,
; ( ) ,
; ( ) ,
;
( x
x
x x X
Proof. Sufficiency. Suppose there exists
( , ) R
m
such that( x , )
is a saddle point forL
( x ; , )
on X
, i.e.,. )
, ( ), );
( ( ) ( , ) );
( ( ) ( , ) );
( ( ) (
,
f x g x f x g x f x g x x X
(10)Let
v = g ( x )
. From above we have. ,
) , ( ),
; ( ,
) , ( )
;
(
v v u v u v
x (11)We now prove
x K
. It suffices to showv R
l. Ab absudo, supposev R
l. Then by (5), it follows that
( , ) =
inf
v
. This contradicts the first inequality in (11). Thusv R
l. Again from the first inequality in (11) and (6), it follows that).
, ( ) , inf (
=
0
v v
(12)
Since
v R
l, one has ( v , ) 0
. This associated with (12) implies:0.
= ) , (
v
(13)Therefore, from above and the second inequality in (11) it follows that
. ) , ( ), ,
; , (
= )
; ( ,
0 u v u v u v
x (14)It is clear that
( u , v ; , ) 0, ( u , v )
, which yields
x and are nonlinearly separable.Necessity. Suppose
x K
and
x and are nonlinearly separable. Let( u , v ) = (0, g ( x ))
be the image of the pointsx
through the mappingA
x, associated with (14). We can obtain ( v , ) 0
. Sincev R
l and from (6), it follows that ( v , ) 0
, i.e., (13) holds. Again from (6), one has (v , ) 0,
, i.e., (11) holds. Consequently, (10) is true.Remark 4.2 Similarly, we can prove the following results. Suppose that (5) and (6) hold. Then the sets
x and are regular (res., strongly regular) nonlinearly separable and
x K
, if and only if there exists
R
\ {0}
) ,
(
m (res.,( , ) R
m
) such that( x , )
is a saddle point for
( x ; , )
onX
.Specially, when the separation function is strongly regular, from proposition 4.1, it follows that the existence of Pareto efficient solution of problem (1) can be obtained by the saddle point condition of the generalized Lagrangian function.
It is well known the generalized Lagrangian function contains many special Lagrange-type function as its special case,
453
such as the classic Lagrangian function, the exponential-type Lagrangian function and the augmented Lagrangian function and so on. Next, we will introduce an example associated with generalized Lagrangian function, which is called the augmented Lagrangian function
ˆ : X R
m R
l R
, defined by:)], ( ,
inf [ )
( ,
= ) ,
; ˆ (
IR 0 ) (
z c z x
f x
z l x g
(15)where
c > 0
is a real number and : R
l R
is a function satisfying0.
= (0) {0},
= )
R
( z
min arg
z lActually, to show the augmented Lagrangian function (15) is a special case of the generalized Lagrangian function (9), we only should assume
= R
l and)], ( ,
[ sup
= )
; (
R
z c z v
lv z
(16)
where
c
and
are the same as in (15). Then (9) reduces to)].
( ,
sup [ )
( ,
= ) );
( ( ) ( ,
= ) ,
; (
) R (
z c z x
f x
g x
f x
x l g z
Letting
=
, allows (15).If we consider the augmented Lagrangian function (15), then (5) and (6) always hold.
Lemma 4.2 Suppose
= R
l, ( v ; )
is defined by (16). Then (5) and (6) are true.Proof. Since
, z c ( z ) , z , ( z , ) R
l .
, it follows thatR . ) , ( , sup , )]
( ,
sup [
= )
; (
R R
l lv
z lv
z
v z z
c z
v
If
R
l, then, R ,
= , sup
R
l lv
z
v v
z
and it follows that
R , ,
= inf ,
)
; inf (
R
l l
v v
v
which proves (5). If
v R
l, then (v ; ) 0,
, and so0.
)
;
inf (
v
Again since
( v ; ) , v , ( v , ) R
l R
l, we haveinf
Rl , v = 0
, v R
l. As a consequence,0, )
; inf ( )
; inf (
R
v
v
l
which implies (6) is true.
The following is an immediate consequence of Theorem 4.2 and Lemma 4.2.
Corollary 4.1 Suppose
= R
l and the separation function is ( u , v ; , ) = , u ( v , )
, where) , (
v
is defined by (16). Then, the sets
x and are nonlinearly separable andx K
if and only if there exists( , ) R
m
such that( x , )
is a saddle point for the augmented Lagrangian ˆ ( x ; , )
onX
.
5. SEPARATION AND PENALTY METHODS
In order to prove (3), we introduce the strongly regular separation function to obtain saddle point condition. We always assume
R
m. Consider the following extremum problem:, . . )]
; ( ) ( ,
[ f x x s t x K
min
(1)454
where
: R
n R
,
is a parameter set. , f ( x ) ( x ; )
is called a penalty function for Problem (1).If there exists
such that a solution of (1), sayx
, is a Pareto efficient solution of problem (1), then we say that , f ( x ) ( x ; )
is an exact penalty function for problem (1) atx
.Proposition 5.1 Suppose that
x K
and 5 and (6) hold. If0, ) ,
; , ( inf sup
) , (
u v
v x
u K
(2) then
x
is a Pareto efficient solution of problem (1).Proof. Notice that
x
is a Pareto efficient solution of problem (1) is equivalent to
x =
. Ab absudo, suppose there exists( u ~ , v ~ )
x
. Sincex K
and (5) and (6) hold, we have, 0,
~ >
, )
~ ;
~ ( , ) ,
; , sup (
) , (
u v u v u
v x
u K
Thus it follows that
0,
~ >
, ) ,
; , ( inf sup
) , (
u v u
v x u
K
a contradiction with (2).
The following statements are related with the existence of an exact penalty function at
x K
.Denote by
( x , B ) = inf { P x b P
2: b B }
the distance from a pointx R
l to a setB R
l. Consider the function
: X R
\ {0} R
, defined by:R ).
), ( ( ) ( ,
= )
;
( x f x g x
l
(3)
Theorem 5.1 Let
x K
. The following statements are equivalent:(i)there exists
= R
\ {0}
such that,, ) , ( 0, ) ,
; ,
( u v u v K
x
where
( u , v ; , ) = , u ( v ; ) = , u ( v , R
l),
,(ii)
L
( x ; )
defined by (3) is an exact penalty function for problem (1) atx
,(iii)there exists
= R
\ {0}
such that. R ),
), ( ( ) ( ) (
, f x f x g x
l x X
Proof. We first show (i) implies (ii). From (i), it follows that
( v ; ) = ( v , R
l)
. Since (5), (6) and (2) is true, we can obtainx
is a Pareto efficient solution of problem (1) from the Proposition 5.1. Sincev
xu v
u , ; , ) 0, ( , ) K
(
, it is easy to knowx
is a solution of (1). Again from the difinition of exact penalty function atx
, we obtain (ii) holds.Now, we show (ii) implies (iii). Since
( x ; )
is an exact penalty function for problem(1) atx
, thereexists
R
\ {0}
such that. R ),
), ( ( ) ( , R )
), ( ( ) (
, f x g x
l f x g x
l x X
Again since
x
is a Pareto efficient solution of problem (1), we haveg ( x ) R
l, it follows that. R ),
), ( ( ) ( ) (
, f x f x g x
l x X
(iii)
(i) is obvious.455 6. REFERENCES
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