A Note on the Guignard Constraint Qualification and the
Guignard Regularity Condition in Vector Optimization
Giorgio Giorgi
Faculty of Economics, University of Pavia, Via S. Felice, Pavia, Italy Email: [email protected]
Received December 12, 2012; revised February 14, 2013; accepted February 21, 2013
Copyright © 2013 Giorgio Giorgi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
Some remarks are made on the use of the Abadie constraint qualification, the Guignard constraint qualifications and the Guignard regularity condition in obtaining weak and strong Kuhn-Tucker type optimality conditions in differentiable vector optimization problems.
Keywords: Constraint Qualifications; Regularity Conditions; Optimality Conditions; Vector Optimization Problems
1. Introduction
In discussing a gap between multiobjective optimization and scalar optimization (a gap first pointed out by Wang and Yang [1]), Aghezzaf and Hachimi [2] state that “in multiobjective optimization problems, many authors have derived the first-order and second-order necessary conditions under the Abadie constraint qualification, but never under the Guignard constraint qualification”. This deserves some comments. Indeed, some authors have proposed a suitable Guignard-Gould-Tolle constraint qualification (it would be better to speak of “Guignard- Gould-Tolle regularity condition”, as it involves also the objective function, besides the constraint functions) in
order to obtain a Karush-Kuhn-Tucker type multiplier rule for a Pareto optimization problem.
For example, Maeda [3] considers the following Pa- reto optimization problem
Min f x , subject tog x
0,m
,
where , are differentiable
functions, and introduces the following “generalized Guignard constraint qualification” for this problem
: n p
f g:n
0
0
1
, cl conv , p
i
i
C Q x T Q x
where x0 is a feasible vector,
n: 0, 0
Q x g x f x f x ;
: 0, 0 , 1, 2, , and
i n
k k
Q x g x f x f x k p ki ;
0
0
0
, n: i 0, 1, , ; j 0,
C Q x h f x h i p g x h jI x0 ;
0, i
T Q x is the Bouligand tangent cone or contin-gent cone to Qi at x0
0
(see Definition 2) and
Indeed, in the above constraint qualification (better: regularity condition) the inclusion means in fact equality.
0I x
j g: j
x 0 .
Jimenez and Novo [4], Giorgi, Jimenez and Novo [5], [6], Giorgi and Zuccotti [7] have given similar, but more general results. What is true is that the Guignard- Gould-Tolle theory cannot be transferred “sic et simplic-iter” from the scalar to the vector case. Indeed, it is well-
known that if we have a scalar optimization problem
Min f x ,xM n, with f :ϒn ϒ, f differentiable, and 0
x is a local solution of the said problem, then we have
0
,
f x T M x
0
eral, as Guignard claims that
0
,
f x P M x
0
where P M x
, 0
cl conv
T M x
, 0
tangent cone to
is the so-called
pseudo M at x0
. However, this greater generality is only apparent, as it is true that for any cone
C, it holds
cl conv
,C C
so we obtain
0
0
, ,
T M x P M x .
Relation (1) obviously is equivalent to the inconsis-te
for or, equivalently, for
ncy of the inequality
00
f x y
0
, ,
yT M x yP M x
, 0
.oblem (vop) Now, consider a vector optimization pr
Min f x ,xMn,
where f :ϒn ϒp is differentiable. When, as in the paper, the o
present rdering cone is p, (vop) is also known as Pareto optimization problem e recall some basic notions and definitions.
Definition 1.
. W
A vector x0 M
op) if t
is said to be a weak Pareto optimal point for (v here does not exist another vector
xM such that fk
x fk
x0 for all k1,, .p or 0A vect x is sai to optima
(vop) if the does not exist another vector d
e
to be a Pare l point for
r xM such
that fk
x fk
x0 for all k1,, ,p with fk
x
k
0f x for at least one index. A vector x0 M is a ely, a
eak Pareto optimal point (respectiv local Pareto optimal point) if the above definitions hold with respect to M B x
0,
,local w
where B x
0,
is a suitable neighborhood of x0.Definition 2.
Let Mn. The contingent cone at x0cl
Mor Bouligand tangent cone at x0 is:
0
n
n
0, : , ,
such that
n n
n n
T M x y y y t t
x t y M
0 ,
or, equivalently,
n
It is well-known that this cone is closed, but not nec-essarily convex (see, e.g., Aubin and Frankowska [9], Bazaraa and Shetty [10]).
0
0
, : , ,
such that .
n n
n n n
T M x y x M x x
x x y
0
,
It can be proved that if x0 is a local weak optimal point for (vop), then we have the relation (see, e.g., Bigi ([11], Corley [12], Giorgi and Zuccotti [13], Staib [14])
0
0
int p , , ,
f x v v T M x
(2)
i.e. the system
00, 1, , , i
f x v i p
for
(3)
has no solution
0
,
vT M x , i.e. it holds
0
0
Max k 0, , .
k 1, ,p
f x v v T M x
Λ
One may wonder if the system (3) (for ) is also inconsistent for
1
p
0
cl conv ,
v T M x , as it holds for
1
p . The ans d Yang
[1 (a misprint in the e
Castellani and Pa This
optimality conditions, for a Pareto
op uality and equality
co
Conditions: Abadie
Regularity Condition
ine ly, we
co
wer is: no, as shown by Wang an
] with a numerical example xample
has been corrected by ppalardo [15]).
is the “gap”, with reference to a result of Guignard to which the title of the paper of Wang and Yang in its turn makes reference.
This note is organized as follows.
In Section 2 we give short proofs of the weak Kuhn- Tucker type necessary
timization problem with both ineq
nstraints, under the Abadie constraint qualification, and of the strong Kuhn-Tucker type conditions, for the same problem, under a Guignard regularity condition.
In Section 3 we make some further comments on the said “gap” between scalar optimization problems and vector optimization problems.
The conclusions are briefly expounded in Section 4.
2. First Order Necessary
Constraint Qualification, Guignard
The feasible set of (vop) is, from now on, specified by quality and equality constraints. More precise nsider the problem
(vop)1 Minf x
,xM, where
: 0, 1, , 0, 1, , , n
i
j
M x g x i m;
h x j r
: n p, : n
f g
neighborhood of the feasib is continuously differen neighborhood of x0. We de
m
are differentiable at least in a le point x0, and
tiable at least i
: n r
h
n the same note by I x
0the index set of the active constraints g ii, 1,, ,m at x0M
, i.e.
0
0
: i 0 .I x i g x
0 : 0 0, ;
0, 1, ,
n i
0
0j
K x y g x y I x
j r
i
h x y
is called the linearizing cone at x0 for (vop)1. The cone
0
0 0
0
: 0, ;
0, 1, ,
n
j i
K x y g x y i I x
h x y j r
°
is called the weak linearizing cone at x0 or co of decreasing directions at
ne
0
x for (vop)1. Lemma 1. (see Giorgi [16])
Let x0M and let f, g and h verify th previous dif-ferentiability assumptions. Then we have:
0 0
e
1) T M x
,
K x
;2) if cobian matrix h x
0 has full rank, it holds
0 , 0
;the Ja
K° x T M x
3) if, moreover, K°
x0 , then it holds
0
0
cl K° x T M,x K x0 .
We are now ready to prove in a short way a Fritz John-type optimality condition for (vop)1.
Lemma 2.
Suppose that the Jacobian matrix h x
0 has full rank. Let x0M be a local weak Pareto optimal point for (vop)1. Then the system
0
0 0
, 0, 1, , , ,
i
00, ,
, 0, 1, ,
k
j ,
f x v k p
g x v
i I x
h x v j r
(4)
has no solution
Proof.
Apply Lemma 1 to the optimality condition (3). .
n
v
■
Theorem 1.
If x0M is a local weak Pareto optimal point for (vop)1, then there exist vectors p,m, and
zero, such that ,
r
not all
0 m
0 r 0 k fk x i g xi
(5)1 1 1
0, p
j j
k i j
h x
00, 1, , , ig xi i
m (6)
m
0, 1, , ; 0, 1, , .
k k p i i
(
Proof.
If the gradients , are linearly
pendent, just set
7)
0 , 1, , jh x j r
0, 0
de- and choo
such that r
se ve
dients are linearly Lemma 2,
e theor Mangasarian [17]) and
set-ting
a nonzero ctor r
0 10. j j
j
h x
If the above gra independent, the
thesis follows from applying the Motzkin al-ternativ em (see, e. g.,
for all iI x
0 . ucker-ty 0i
-Kuhn-T pe optim ity conditions f (vop)1, by means of the Abadie constraint qualification
(a
■
or
A Karush al
cq), has been obtained by Lin [18] and by Singh [19]; however, their proofs work only if x0 is a weak Pareto optimal point for (vop)1, and not a local weak Pareto op-timal point. The flaw is corrected in Marusciac [20]; s ee also the errata corrige of Singh [19], who, however, does not justify his rectification; see also the paper of Wang [21], more specific on this point. We give here a simple proof of the result of Wang.
Definition 3.
The constraint set M satisfies the Abadie constraint qualification (acq)at x0M if
0 0
, .
K x T M x
Due to relation 1) of Lemma 1, the (acq) can be writ-ten also as an inclusion
0 0
, .
K x T M x
Singh [19] claims that his version, as an inclusion, of the (acq) is more general than the one proposed by M
Let
arusciac [20] as an equality. Lemma 3.
0
x M
, and supp
be a local weak Pareto optimal point for (vop)1 ose that the (acq) holds at 0
x . Then the system
00, 1, , ,
k
0
00, ,
00, 1, , ,
i
j
f x v k p
Λ
g x v i I x
h x v j r
Λ
has no so
(8)
lution
Proof.
Suppose, on the contrary, that the above solution
. n
v
system has a
n
v . Then, the (acq) implies that v T M x
, 0
; thanks t on (3) it will holdleast one in k, contradicting the
o system (8) the Motzkin theorem of the al-te
o relati dex
v 0 for at
ns of the ■ 0
k
f x
relatio tem.
Applying t
rnative, we get the following (weak) Karush-Kuhn- Tucker-type multiplier rule for (vop)1.
Theorem 2.
Let x0M be a local weak Pareto optimal point for
(
vop)1 and let the (acq) hold at x0. Then there ex t is vectors p,m, and , with 0, such that (5), (6) and (7) hold.As the cone K x
0 is polyhedric, the (acq) implies that T M x
, 0
is a convex cone. If we substitute
0
x with ,
M the closure of its convex hull, we obtain
th t q
Gould and Tolle
e Guignard-Gould-Tolle constrain ualification (Guig-
nard [8] [22])
0
0
0
, , ,
cl convK x T M x P M x (9)
but, ready remarked in the Introduction, this more general constraint qualification does not enable to obtain, for p1, the multiplier rule of Theorem 2, unless to make further assumptions o
as al
n or on the
ob unction. See the ne
If we want to obtain a strong Karush-Kuhn-Tucker
0
,
T M x
xt Section 3. jective f
type optimality condition for (vop)1, i.e. a multiplier rule (5), (6) and (7), where 0 in (5), we have to impose som dition, where also the objective function is con-sidered. We prefer, in this case, of regularity conditions, instead of constraint qualificati
e con
to speak
ons. The con-dition considered by Maeda [3] and reported in the In-troduction of the present paper, is therefore a regularity condition. For other regularity conditions for (vop)1 and their relationships, see also Jimenez and Novo [4] and Giorgi and Zuccotti [7].
We now consider a slightly modified version of the Guignard regularity condition proposed by the above authors. See also Bigi [11].
Definition 4. Let x0M
; then the set
0 n
0
: k 0, 1, ,F x v x v k Λ p
is the cone of descent directions for f at 0
x . Given any
1, ,
s P Λ p , the set
0
0, , 0
:
s k k
M xM f x f x kP ks x
is the set of all feasible points, which do not allow x0 to en
be a weak minimum point for (vop)1 wh the compo-nent fs is drop
on 5. Let
ped from f. Definiti
0
x M Then the (modified) Guignard-Goul - Tolle regularity condition (ggtrc) holds at
d 0
x if
ose that
0 0
0
1
cl conv , . p
s s
F x K x T M x
Lemma 4.
Supp x0M is a local weak Pareto optimal point for (vop)1 and that (ggtrc) holds at x0. Then for each k P
1,Λ,p
the following system
0
0
0 , 0, , 0, , s
k
i
j
f x v
0
0 ,, 0, ,
, 0, 1, , ,
f x v k P k s
h x v j r
Λ
g x v i I x
(10)
has no solution
Proof.
On the contrary suppose that there exists
. n
v
sP such that (10) has a solution. Therefore, (ggtrc) implies that
0
cl conv , .
v T M x
weak Pareto optimal point f
Since the def
or (vop)1 implies that inition of local
0
x is point of the scalar function
a local minimum fs over
s
M , then we get the inequality
e assump
0 0 s
f x v
, in co
n- tion.
vious lemma, which states the impossibility of
p
rush-Theorem 3.
tradiction with th ■
The pre
systems, enables us to obtain a strong Ka Kuhn- Tucker-type multiplier rule for (vop)1.
Suppose that x0M is a local weak Pareto op mal point for (vop)1 and that (ggtrc) holds at 0.
ti
x Th there ts vectors ,
en exis p m and , h k 0
r
wit , for each k P
1, ,Λ p
, such that (5), (6) and (7) hold.Proof.
The proof is immediate, by applying the Motzkin theorem of the alternative to system (10 sP
■ ), for each an the resulting multipliers.
p)1
y Giorgi, Jim d Novo nd by
d Zuccotti [7], b
tio for local Pareto optimal points and not for th
7 ha
d summing up
We note that Maeda [3] has proposed a slightly weaker condition than (ggtrc), generalized to (vo by Jimenez
and Novo [4], b enez an [5,6], a
Giorgi an ut this weaker regularity condi-n “works”
e weak ones. Finally, we remark that (generalizing some approaches of Bigi and Pappalardo [23]) Maciel, Santos and Sottosanto [24] and Giorgi and Zuccotti [ ] ve studied the following question: which regularity condition for (vop)1 is both necessary and sufficient to obtain that k 0,kP, for all triplets of multipliers
, ,
which satisfy relations (5), (6) and (7)? Let us denote by FJ x
0 the above set of multipliers, i.e. the set of all Fritz John multipliers for (vop)1, and let us in-troduce the following generalization of the Mangasar-ian-Fromovitz constraint qualification.Definition 6.
Let x0M , then the Mangasarian-Fromovitz regu-larity condition (mfrc) for (vop)1 is satisfied at x0 if:
vectors
1) the hj
x0 ,j1,, ,r are linearly inde-pendent;2) for all s P
1, ,Λ p
the system
0, 0, , ,
k
0
0
0, 0, ,
, 0, 1, , ,
i
j
f x v k P k s
g x v i I x
h x v j r
Λ
has solution
the fo (see Maciel,
Santos and Sottosanto [24], Giorgi and Zuccotti [7]). Theorem 4.
Let
. n
v
proved
It can be llowing result
0
x M and let . Then in (vop)1 we
have
0
FJ x
0, , FJ x
and only if (mfrc) holds at x0. given
f th ons t qua usly
In this section we have a simple and short proof of the Fritz John type and o e weak Kuhn-Tucker type necessary optimality conditi for the problem (vop)1, under the Abadie constrain lification. This completes what previo proved by Singh [19]. We
eda [3
ween
and improves
give also a short proof of the strong Kuhn-Tucker type conditions for (vop)1, under a Guignard regularity condi-tion. This improves what previously proved by Ma
].
3. Again on the “Gap” bet
Scalar
Problems and Vecto P
r roblems
We have described in the Introduction the “gap” occur-ring between scalar and vector optimization problems, generated by the use of the classical Guignard-Gould- Tolle constraint qualification. We have also mentioned this “gap” in the previous section. An obvious sufficient condition to remove this “gap” is that the cone T M x
, 0
is convex; this condition has been envisaged by Wang and Yang [1], who, how did not go furtdiscussion. It is well-kn at
ge
ever, own th
her in the is convex ontin-order local
0
,
T M x
st-when M is a convex set. As the structure of the c
nt cone T M x
, 0
, as of all the other fircone approximations, depends only from the structure of
M arouund x0 (see, e.g., Elster and Thierfelder [25]), we can state that T M x
, 0
is convex also when M is locally convex at x0, i.e. there exists a neighborhood of0
x , N x
0 , such that MN x
0 is a convex set. This is no longer true when M is only star-shapedat x0,i.e. x0
1
xM, x M, contrary to what stated in Giorgi and Guerraggio [26] and to what seems stated in Palata [27] and in Staib [14].If M is star-shaped at x0M it holds that
0
0
, , ,
T M x A where
0
0
, : : ,
h that ,
0, , , 0
n n
A M x v
M
M x0 suc
x v
is the cone of the attainable directions or Kuhn-Tuck r cone a
e t x0cl
M . So, whe n M is star-shaped at 0x M , the (acq) and the Kuhn-Tucker constraint qualification (ktcq)
0 0
,
K x A M x
coincide (see Bazaraa and Shetty [10], who, however, do not recognize that the cone
0
,
A M x
if T M
is closed).
It is also well-known equals the
Clarke tangent cone
0
,
then
that
0
n
n
t
0
,x
,
n: 0 , n , nTC M x v x x x M
v x
such that n n , ,
n
v t v M n
0
,
M x is convex, being TC M x
, 0
always closed and convex.T
M as
In this case Pen fies the
set tangentially regular at
ot [28] quali 0
.
x We fo [9
llow the
treatm n and Frankowska
subset
ent of Aubi ].
Definition 7.
We say that a closed M n is sleek at 0
x M if the multivalued map xT M
,x0
,xM, is lower semicontinuous at x0.Fran Theorem 5. (Aubin and
Let M be a closed subset of and kowska [9])
n
0
x M . If M
is sleek at x0, then the con cone the Clarke tangent cone
tly
tingent 0 ,
M x
,
T M x
coincide an
0d c and
on-
TC
sequen
,x0
is convex.Another possibility to remove the “gap” is suggested by Castellani and Pappalardo [15], impose that the objective function f is convexlike ad refe
T M
who
r a result of Li
an s can be proved directly in the
fo Defin
is convexlike on the non-em
d Wang [29]. Thi llowing way.
ition 8.
A function f :np
pty set Xn if for any x x1, 2X and any
0,1
there exists 3
x X such that
1
2 3
1
p.f x x
f x f
Note that in the above definition 3 ,
x which usually depends from 1 2
,
x x and , is not required to be the convex combination of 1
x and 2 .
x Note, moreover, that if p1, then any real-valued function is convexlike. Obviously, all convex functions f : n p
are con-ve
t cond For o xlike; this is o
n
nly a sufficien ther
suf-fic s see Elste
Komiya umar [32]
ns o nction tion
A straigh seque ition 8 is the
following characterization. Theorem 6.
l ition.
ient conditio r and Nehse [30]. See also
Hayashi and [31] and Jeyak for
appli-catio f convexlike fu s to optimiza problems. tforward con nce of Defin
The function : n p
f is convex ike on X n
if and only if the set
pf X is convex.
Theorem 7.
Supp hat f is convexlike on n.
M If 0
ose t x M
is a weak Pareto optimal point for ere exists a nonnegative nonzero vector
(vop), then th p
such that 0
x is a minimum point of the scalar problem
Min f x ,xM .
Proof.
n
By assumptio f x
0 f M
int
p . ption on f imThe con-vexlikeness assum plies that
int
pf M is co refore the usual
tio plies the e
nvex; the
xistence of a nonzero
separa-vector n theorem im
p
and of a scalar such that
0
f x f x d
for all dint
M and for all xM. Therefor con-sidering 0e,
xx, we obtain 0; given any dint
p
and considering td as t0, we get
0f x f x
and therefore the thesis follows.
proof of the above ressult we use th
■
We note that in the
e property that f M
int
p
weaker condition is equivalent that f
M (see Li and Wang [29]). The “sca-larization theorem” just proved allows us to state the
result (recall that 1 the Guign
for p
is a convex set; this is
subconvexlike on
lowing ld-
Tolle theory is consistent). Theorem 8.
ose that conve
on M
:np
ard-Gou
Supp :np
is xlike (or subcon-vexlike) . If 0
f
x M is a weak Pareto optimal point for (vop), then
0
Max k 0,
k P
f x v
0
.
v T x
Similarly, with reference to (vop)1, we can assert the following result.
Theorem 9. Suppose that 0
cl conv M,
x M is a weak Pareto optimal point for (vop) and that 1 f :n
nvexlike) on
p
is convexlike (or sub-co
ose th ation
M. Supp alific
at the Guignard-Gould- Tolle constraint qu (9) holds at x0. Then, there exist multipliers p, 0,m and r such that (5), (6) and (7) hold.
Another condition which allows to apply to (vop)1 the (ggtcq) and which entails the objective function f is given in the following theorem.
Theorem 10.
Let x0M be a weak Pareto optimal point for (vop)1; suppose that x0 verifies the (ggtcq) and that there exists a nonnegative vector p,0, such that
0
f x T M,x0
.
Then x0 verifies the conditions (5), (6) and (7).
Proof.
The (ggtcq) can be equival
ently written in the form
0
0, ,
M x K x or
T T M x, 0
K x
0
. Onthe other hand, being K x
0 a polyhedral cone, deter- mined by the vectors g xi
0 ,
0
iI x and hj
x0 ,1, olar will be given by
0
0 0
,
r
i j
,
, its p
j r
0
1
i j
j i I x
00, .
i
K x g h x
x
i I x
Therefore, being f x
0
T M x
, 0
, we can write
0 0
0
,
0, ,
r
i j j
0
0 j1i I x
i i
f x g x h
i I x
xi.e. the conditions (5), (6) and (7), by choosing
0 i for iI x
0 .We remark tha
t if
■ 0
x M
n, for e
is a weak Pareto o point for (vop)1, the ach convex cone S
pyimal , with
0
,
ST M x , there exists a nonnegative vector p, 0
, such that f x
0 Spply the classical theorem
when
, The proof is left on separation
and obtain also the result stated in the previous theorem. In this section we have investigated the reasons for the existence of a gap between a scalar programming prob-lem and a multiobjective programming probprob-lem (vop)1,
ssica
- ou ve
o o
n
si
The a esent paper is
hn-Tucker t
O
EFERENCES
to the of con-reader (a
vex sets). Note th we obtain the
result already observed at the be of this Section at S T
M x, 0ginning
gap which has its origins in the use of the cla l Guignard G ld-Tolle constraint qualification. We ha given some proposals t rem ve the said gap.
4. Co clu ons
im of the pr twofold. On one side we
present simple and brief optimality conditions of the Fritz John type and of the Ku ype (both weak and strong) for a Pareto multiobjective programming problem with both inequality and equality constraints.
n the other side we investigate the existence of a gap, originated by the use of the classical Guignard constraint qualification, between a scalar nonlinear programming problem and the Pareto problem described above.
The author thanks an anonymous referee for his sug-gestions.
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