A Note on the Guignard Constraint Qualification and the Guignard Regularity Condition in Vector Optimization ()
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 “GuignardGould-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 Pareto optimization problem
![](https://www.scirp.org/html/20-7401310\7cd8a50c-e75f-4343-9a1c-b6d61f70abdf.jpg)
where
,
are differentiable functions, and introduces the following “generalized Guignard constraint qualification” for this problem
![](https://www.scirp.org/html/20-7401310\6aa80c27-c0d0-4528-a6c5-a3ba5ce4410d.jpg)
where
is a feasible vector,
![](https://www.scirp.org/html/20-7401310\7109e9c4-ea45-4de9-9aac-943391c56f20.jpg)
![](https://www.scirp.org/html/20-7401310\f172634d-e587-429c-9012-ae3b4a6054fb.jpg)
![](https://www.scirp.org/html/20-7401310\30c2d038-0b98-4c2b-8519-fb1ce18cb111.jpg)
is the Bouligand tangent cone or contingent cone to
at
(see Definition 2) and
Indeed, in the above constraint qualification (better: regularity condition) the inclusion means in fact equality.
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 GuignardGould-Tolle theory cannot be transferred “sic et simpliciter” from the scalar to the vector case. Indeed, it is wellknown that if we have a scalar optimization problem
![](https://www.scirp.org/html/20-7401310\815977cb-de2d-45df-8d05-079bedd31ba9.jpg)
with
, f differentiable, and
is a local solution of the said problem, then we have
(1)
(A* is the negative polar cone of the cone A). The result obtained by Guignard [8] seems at first sight more general, as Guignard claims that
![](https://www.scirp.org/html/20-7401310\8801c5fa-aa7e-4fe3-abb9-0d0a295316d0.jpg)
where
is the so-called pseudotangent cone to
at
. However, this greater generality is only apparent, as it is true that for any cone
, it holds
![](https://www.scirp.org/html/20-7401310\9c3bcc94-7264-4b94-8696-2b3b41f9989d.jpg)
so we obtain
![](https://www.scirp.org/html/20-7401310\1d3c0447-4755-4bda-b900-731e23972ed6.jpg)
Relation (1) obviously is equivalent to the inconsistency of the inequality
![](https://www.scirp.org/html/20-7401310\88d203f5-5472-4787-9f3b-c11abd6bfe1c.jpg)
for
or, equivalently, for ![](https://www.scirp.org/html/20-7401310\d69f4224-7976-494d-9649-5c048d9e6bb1.jpg)
Now, consider a vector optimization problem (vop)
![](https://www.scirp.org/html/20-7401310\ee9cd1d4-74dc-4a27-9f89-2e8eab1b9d76.jpg)
where
is differentiable. When, as in the present paper, the ordering cone is
(vop) is also known as Pareto optimization problem. We recall some basic notions and definitions.
Definition 1.
A vector
is said to be a weak Pareto optimal point for (vop) if there does not exist another vector
such that
for all
A vector
is said to be a Pareto optimal point for (vop) if there does not exist another vector
such that
for all
with
for at least one index. A vector
is a local weak Pareto optimal point (respectively, a local Pareto optimal point) if the above definitions hold with respect to
where
is a suitable neighborhood of ![](https://www.scirp.org/html/20-7401310\bd03ce71-14f1-49c9-ac7a-e8dd814e8d3f.jpg)
Definition 2.
Let
The contingent cone at
or Bouligand tangent cone at
is:
![](https://www.scirp.org/html/20-7401310\da9e47c8-7997-4e0c-a355-047b8fc91a55.jpg)
or, equivalently,
![](https://www.scirp.org/html/20-7401310\4dc35ec4-9f9a-41d2-93ad-80064356e177.jpg)
It is well-known that this cone is closed, but not necessarily convex (see, e.g., Aubin and Frankowska [9], Bazaraa and Shetty [10]).
It can be proved that if
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])
(2)
i.e. the system
(3)
has no solution for
, i.e. it holds
![](https://www.scirp.org/html/20-7401310\561105a3-c07d-4eb1-8a24-164ba9e5b3fb.jpg)
One may wonder if the system (3) (for
) is also inconsistent for
, as it holds for
. The answer is: no, as shown by Wang and Yang [1] with a numerical example (a misprint in the example has been corrected by Castellani and Pappalardo [15]). This 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 KuhnTucker type necessary optimality conditions, for a Pareto optimization problem with both inequality and equality constraints, 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 Conditions: Abadie Constraint Qualification, Guignard Regularity Condition
The feasible set of (vop) is, from now on, specified by inequality and equality constraints. More precisely, we consider the problem
(vop)1 ![](https://www.scirp.org/html/20-7401310\1be41676-e3d2-421d-9da1-eb64a306393f.jpg)
where
![](https://www.scirp.org/html/20-7401310\8212fe6e-a5de-48ff-af83-3b4fee5ef27c.jpg)
are differentiable at least in a neighborhood of the feasible point x0, and
is continuously differentiable at least in the same neighborhood of x0. We denote by
the index set of the active constraints
at
, i.e.
![](https://www.scirp.org/html/20-7401310\4ba2e12c-d009-49c4-9af0-ba2bf143e7a6.jpg)
Let
; the cone
![](https://www.scirp.org/html/20-7401310\0ca9b086-dfb9-4c54-9021-65f1b420e855.jpg)
is called the linearizing cone at
for (vop)1.
The cone
![](https://www.scirp.org/html/20-7401310\c1888fed-357b-4001-8735-a5b10917f077.jpg)
is called the weak linearizing cone at
or cone of decreasing directions at
for (vop)1.
Lemma 1. (see Giorgi [16])
Let
and let f, g and h verify the previous differentiability assumptions. Then we have:
1) ![](https://www.scirp.org/html/20-7401310\064d2242-781e-4024-b3f6-cdfe85ab0dfe.jpg)
2) if the Jacobian matrix
has full rank, it holds ![](https://www.scirp.org/html/20-7401310\2aa14286-b7de-4f46-97d0-aa9a78a6d176.jpg)
3) if, moreover,
then it holds
![](https://www.scirp.org/html/20-7401310\d41f3004-49ba-437f-86fe-a62299669187.jpg)
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
has full rank. Let
be a local weak Pareto optimal point for (vop)1. Then the system
(4)
has no solution ![](https://www.scirp.org/html/20-7401310\8837405e-5053-4d81-a5e8-c1ad6d95f8b6.jpg)
Proof.
Apply Lemma 1 to the optimality condition (3). ![](https://www.scirp.org/html/20-7401310\6086dafb-7fba-4fca-be4e-1fd714486db0.jpg)
Theorem 1.
If
is a local weak Pareto optimal point for (vop)1, then there exist vectors
and
not all zero, such that
(5)
(6)
(7)
Proof.
If the gradients
are linearly dependent, just set
and choose a nonzero vector
such that
![](https://www.scirp.org/html/20-7401310\b25d5195-6219-4554-8b73-22968ad2b402.jpg)
If the above gradients are linearly independent, the thesis follows from Lemma 2, applying the Motzkin alternative theorem (see, e. g., Mangasarian [17]) and setting
for all
![](https://www.scirp.org/html/20-7401310\2f736a1c-a339-46f1-a57b-6d175d1ecd23.jpg)
A Karush-Kuhn-Tucker-type optimality conditions for (vop)1, by means of the Abadie constraint qualification (acq), has been obtained by Lin [18] and by Singh [19]; however, their proofs work only if
is a weak Pareto optimal point for (vop)1, and not a local weak Pareto optimal point. The flaw is corrected in Marusciac [20]; see 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
if
![](https://www.scirp.org/html/20-7401310\cc96edd4-9336-457c-a65f-651d0b588b97.jpg)
Due to relation 1) of Lemma 1, the (acq) can be written also as an inclusion
![](https://www.scirp.org/html/20-7401310\60127731-a9d3-43aa-9612-13b99a899c9a.jpg)
Singh [19] claims that his version, as an inclusion, of the (acq) is more general than the one proposed by Marusciac [20] as an equality.
Lemma 3.
Let
be a local weak Pareto optimal point for (vop)1, and suppose that the (acq) holds at
. Then the system
(8)
has no solution ![](https://www.scirp.org/html/20-7401310\e6b010e3-c724-4087-9332-5322682aab79.jpg)
Proof.
Suppose, on the contrary, that the above system has a solution
. Then, the (acq) implies that
; thanks to relation (3) it will hold
for at least one index k, contradicting the relations of the system. ![](https://www.scirp.org/html/20-7401310\b325ca01-297b-489d-9b89-fa79d628487a.jpg)
Applying to system (8) the Motzkin theorem of the alternative, we get the following (weak) Karush-KuhnTucker-type multiplier rule for (vop)1.
Theorem 2.
Let
be a local weak Pareto optimal point for
vop)1 and let the (acq) hold at
. Then there exist vectors
and
, with
, such that (5), (6) and (7) hold.
As the cone
is polyhedric, the (acq) implies that
is a convex cone. If we substitute
with the closure of its convex hull, we obtain the Guignard-Gould-Tolle constraint qualification (Guignard [8] Gould and Tolle [22])
(9)
but, as already remarked in the Introduction, this more general constraint qualification does not enable to obtain, for
, the multiplier rule of Theorem 2, unless to make further assumptions on
or on the objective function. See the next Section 3.
If we want to obtain a strong Karush-Kuhn-Tucker type optimality condition for (vop)1, i.e. a multiplier rule (5), (6) and (7), where
in (5), we have to impose some condition, where also the objective function is considered. We prefer, in this case, to speak of regularity conditions, instead of constraint qualifications. The condition considered by Maeda [3] and reported in the Introduction 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
; then the set
![](https://www.scirp.org/html/20-7401310\784c098b-14cc-4c5d-8bc5-8171cbc48db0.jpg)
is the cone of descent directions for f at
. Given any
, the set
![](https://www.scirp.org/html/20-7401310\b63fa8e6-de53-4615-ac66-1b1a2a508c51.jpg)
is the set of all feasible points, which do not allow
to be a weak minimum point for (vop)1 when the component
is dropped from f.
Definition 5.
Let
Then the (modified) Guignard-GouldTolle regularity condition (ggtrc) holds at
if
![](https://www.scirp.org/html/20-7401310\fd2b3637-9ec3-4e31-8a59-a350dc44459e.jpg)
Lemma 4.
Suppose that
is a local weak Pareto optimal point for (vop)1 and that (ggtrc) holds at
Then for each
the following system
(10)
has no solution ![](https://www.scirp.org/html/20-7401310\047f9005-ca4e-4779-9f17-d54138db5a8b.jpg)
Proof.
On the contrary suppose that there exists
such that (10) has a solution. Therefore, (ggtrc) implies that
Since the definition of local weak Pareto optimal point for (vop)1 implies that
is a local minimum point of the scalar function
over
, then we get the inequality
, in contradiction with the assumption. ![](https://www.scirp.org/html/20-7401310\984a6de8-b612-4384-b24c-01f7bd558dc5.jpg)
The previous lemma, which states the impossibility of p systems, enables us to obtain a strong Karush-KuhnTucker-type multiplier rule for (vop)1.
Theorem 3.
Suppose that
is a local weak Pareto optimal point for (vop)1 and that (ggtrc) holds at
Then there exists vectors
and
with
, for each
such that (5), (6) and (7) hold.
Proof.
The proof is immediate, by applying the Motzkin theorem of the alternative to system (10), for each
and summing up the resulting multipliers. ![](https://www.scirp.org/html/20-7401310\dc2a3f65-9ec1-4317-a2e4-613e6af44c22.jpg)
We note that Maeda [3] has proposed a slightly weaker condition than (ggtrc), generalized to (vop)1 by Jimenez and Novo [4], by Giorgi, Jimenez and Novo [5,6], and by Giorgi and Zuccotti [7], but this weaker regularity condition “works” for local Pareto optimal points and not for the weak ones. Finally, we remark that (generalizing some approaches of Bigi and Pappalardo [23]) Maciel, Santos and Sottosanto [24] and Giorgi and Zuccotti [7] have studied the following question: which regularity condition for (vop)1 is both necessary and sufficient to obtain that
, for all triplets of multipliers
which satisfy relations (5), (6) and (7)? Let us denote by
the above set of multipliers, i.e. the set of all Fritz John multipliers for (vop)1, and let us introduce the following generalization of the Mangasarian-Fromovitz constraint qualification.
Definition 6.
Let
, then the Mangasarian-Fromovitz regularity condition (mfrc) for (vop)1 is satisfied at
if:
1) the vectors
are linearly independent;
2) for all
the system
![](https://www.scirp.org/html/20-7401310\1ddd7322-75c0-4785-ae5e-73eaaef09add.jpg)
has solution ![](https://www.scirp.org/html/20-7401310\cc88157a-2d3f-4195-9ab5-64144fc46fe4.jpg)
It can be proved the following result (see Maciel, Santos and Sottosanto [24], Giorgi and Zuccotti [7]).
Theorem 4.
Let
and let
. Then in (vop)1 we have
with
for all
, if and only if (mfrc) holds at ![](https://www.scirp.org/html/20-7401310\7362ae86-8ade-4c66-9452-242656336c25.jpg)
In this section we have given a simple and short proof of the Fritz John type and of the weak Kuhn-Tucker type necessary optimality conditions for the problem (vop)1, under the Abadie constraint qualification. This completes and improves what previously proved by Singh [19]. We give also a short proof of the strong Kuhn-Tucker type conditions for (vop)1, under a Guignard regularity condition. This improves what previously proved by Maeda [3].
3. Again on the “Gap” between Scalar Problems and Vector Problems
We have described in the Introduction the “gap” occurring between scalar and vector optimization problems, generated by the use of the classical Guignard-GouldTolle constraint qualification. We have also mentioned this “gap” in the previous section. An obvious sufficient condition to remove this “gap” is that the cone
is convex; this condition has been envisaged by Wang and Yang [1], who, however, did not go further in the discussion. It is well-known that
is convex when M is a convex set. As the structure of the contingent cone
, as of all the other first-order local cone approximations, depends only from the structure of M arouund
(see, e.g., Elster and Thierfelder [25]), we can state that
is convex also when M is locally convex at
, i.e. there exists a neighborhood of
,
, such that
is a convex set. This is no longer true when M is only star-shaped at
, i.e.
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
it holds that
where
![](https://www.scirp.org/html/20-7401310\48b17416-81e4-4413-9951-f42a68370b3f.jpg)
is the cone of the attainable directions or Kuhn-Tucker cone at
So, when
is star-shaped at
, the (acq) and the Kuhn-Tucker constraint qualification (ktcq)
![](https://www.scirp.org/html/20-7401310\54ea0427-e909-4d35-980f-d714f50bc859.jpg)
coincide (see Bazaraa and Shetty [10], who, however, do not recognize that the cone
is closed).
It is also well-known that if
equals the Clarke tangent cone
![](https://www.scirp.org/html/20-7401310\af5f2e07-c016-49ee-b6af-e6e7ab1fde89.jpg)
then
is convex, being
always closed and convex. In this case Penot [28] qualifies the set M as tangentially regular at
We follow the treatment of Aubin and Frankowska [9].
Definition 7.
We say that a closed subset
is sleek at
if the multivalued map
is lower semicontinuous at
.
Theorem 5. (Aubin and Frankowska [9])
Let M be a closed subset of
and
. If
is sleek at
, then the contingent cone
and the Clarke tangent cone
coincide and consequently
is convex.
Another possibility to remove the “gap” is suggested by Castellani and Pappalardo [15], who impose that the objective function f is convexlike ad refer a result of Li and Wang [29]. This can be proved directly in the following way.
Definition 8.
A function
is convexlike on the nonempty set
if for any
and any
there exists
such that
![](https://www.scirp.org/html/20-7401310\cf91a6a3-8afe-427c-b2e3-91f9fbcef113.jpg)
Note that in the above definition
which usually depends from
and
is not required to be the convex combination of
and
Note, moreover, that if
, then any real-valued function is convexlike. Obviously, all convex functions
are convexlike; this is only a sufficient condition. For other sufficient conditions see Elster and Nehse [30]. See also Hayashi and Komiya [31] and Jeyakumar [32] for applications of convexlike functions to optimization problems.
A straightforward consequence of Definition 8 is the following characterization.
Theorem 6.
The function
is convexlike on
if and only if the set
is convex.
Theorem 7.
Suppose that f is convexlike on
If
is a weak Pareto optimal point for (vop), then there exists a nonnegative nonzero vector
such that
is a minimum point of the scalar problem
![](https://www.scirp.org/html/20-7401310\43f4112a-ad37-47dc-998f-2e1b124c6e91.jpg)
Proof.
By assumption
The convexlikeness assumption on f implies that
is convex; therefore the usual separation theorem implies the existence of a nonzero vector
and of a scalar
such that
![](https://www.scirp.org/html/20-7401310\4f101e7d-307a-4717-bf4a-baefbc841e7e.jpg)
for all
and for all
Therefore, considering
, we obtain
; given any ![](https://www.scirp.org/html/20-7401310\1d4c99c0-21c4-4260-b86b-8ad7f34f133a.jpg)
and considering td as
we get ![](https://www.scirp.org/html/20-7401310\c9fea4e6-59ec-4d1f-bdae-8089399a662b.jpg)
and therefore the thesis follows. ![](https://www.scirp.org/html/20-7401310\f59a9d1f-9574-463f-881b-63137406e6ce.jpg)
We note that in the proof of the above ressult we use the property that
is a convex set; this weaker condition is equivalent that
is subconvexlike on M (see Li and Wang [29]). The “scalarization theorem” just proved allows us to state the following result (recall that for
the Guignard-GouldTolle theory is consistent).
Theorem 8.
Suppose that
is convexlike (or subconvexlike) on M. If
is a weak Pareto optimal point for (vop), then
![](https://www.scirp.org/html/20-7401310\a5c0bc18-17d0-44d8-83de-2373c89c8754.jpg)
Similarly, with reference to (vop)1, we can assert the following result.
Theorem 9.
Suppose that
is a weak Pareto optimal point for (vop)1 and that
is convexlike (or subconvexlike) on M. Suppose that the Guignard-GouldTolle constraint qualification (9) holds at
. Then, there exist multipliers
and
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
be a weak Pareto optimal point for (vop)1; suppose that x0 verifies the (ggtcq) and that there exists a nonnegative vector
, such that
![](https://www.scirp.org/html/20-7401310\44dfa639-127e-4d6f-a778-90da45a1820b.jpg)
Then
verifies the conditions (5), (6) and (7).
Proof.
The (ggtcq) can be equivalently written in the form
or
On the other hand, being
a polyhedral cone, determined by the vectors
,
and
,
, its polar will be given by
![](https://www.scirp.org/html/20-7401310\371e3fe5-5be5-460f-97a7-4980c6a5de49.jpg)
Therefore, being
, we can write
![](https://www.scirp.org/html/20-7401310\1bd91774-bc9d-4b10-8f94-5b7e8acc06f4.jpg)
i.e. the conditions (5), (6) and (7), by choosing
for
![](https://www.scirp.org/html/20-7401310\82273dd3-59f5-429a-8cde-0dbb7345891e.jpg)
We remark that if
is a weak Pareto opyimal point for (vop)1, then, for each convex cone S, with
, there exists a nonnegative vector
,
, such that
, The proof is left to the reader (apply the classical theorem on separation of convex sets). Note that when
we obtain the result already observed at the beginning of this Section 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 problem and a multiobjective programming problem (vop)1, gap which has its origins in the use of the classical Guignard-Gould-Tolle constraint qualification. We have given some proposals to remove the said gap.
4. Conclusions
The aim of the present paper is twofold. On one side we present simple and brief optimality conditions of the Fritz John type and of the Kuhn-Tucker type (both weak and strong) for a Pareto multiobjective programming problem with both inequality and equality constraints. On 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 suggestions.