1. Introduction
Convex programming deals with the minimization of a convex objective function over a convex set usually described by convex constraint functions. In the past various attempts have been made to weaken the convexity hypothesis [2-4] by replacing convex objective as well as constraint functions with more general ones and thus exploring the extent of optimality conditions applicability.
As a breakthrough to this, Lassere [1] showed that as far as KKT optimality conditions are concerned, the convexity (or any of its generalization) of the constraint functions can be replaced by the convexity of the feasible set described by the constraints. More precisely, Lassere considered the following convex optimization problem (CP):
(CP) minimize ![](https://www.scirp.org/html/10-1040268\aaf9bd4a-8cd9-4ca2-99ec-82bac89e4053.jpg)
subject to
![](https://www.scirp.org/html/10-1040268\46c3d2f9-bd58-4de0-b257-cd4f6ffc9b6d.jpg)
![](https://www.scirp.org/html/10-1040268\b1a1e284-9139-4b31-890e-e260790d8f55.jpg)
where
is a differentiable convex function and the feasible set
![](https://www.scirp.org/html/10-1040268\bfd720ce-98fa-4713-9d21-5ccb6e663cc8.jpg)
is a convex set while the
:
are differentiable but not necessarily convex functions. To prove the necessity and sufficiency of KKT conditions in this framework Lassere considered the following non-degeneracy condition (ND1): For all
,
, whenever
and
(ND1)
He showed that if the Slater constraint qualification1 and the above non-degeneracy condition (ND1) hold, then a feasible point x* of (CP) is a global minimizer if and only if it is a KKT point, that is,
,
and
,
(KKT1)
for some non-negative vector
.
This work of Lassere [1] has been carried forward to the non-smooth case by Dutta and Lalitha [5]. They considered the same problem (CP) with the only difference being that the function f is a non-differentiable convex function and the convex set
is described by local Lipschitz constraint functions
which are not necessarily differentiable or convex. In terms of Dutta and Laltha [5] a point
is said to be a KKT point for the problem (CP) if there exist scalars
, such that
![](https://www.scirp.org/html/10-1040268\f61df48a-af6d-49de-ba72-182ec83c3cdb.jpg)
and
![](https://www.scirp.org/html/10-1040268\1e510b39-0c2f-4a3b-a2d5-2eb77d82e04f.jpg)
(KKT2)
where
![](https://www.scirp.org/html/10-1040268\cf2792fb-ee95-4010-b03d-9be9cf690d0e.jpg)
denotes the sub-differential of f at x* and
![](https://www.scirp.org/html/10-1040268\fde59d36-81c3-4914-916d-37ab73ac55f2.jpg)
denotes the Clarke sub-differential of the function
at x*.
Further, Dutta and Lalitha [5] introduced the following non-smooth version (ND2) of Lassere’s non-degeneracy condition:
For all ![](https://www.scirp.org/html/10-1040268\a05b6526-c2ec-443c-92a1-3bac15632020.jpg)
(ND2)
In this modified setting Dutta and Lalitha [5] concluded that if each
is assumed to be regular in the sense of Clarke [6] and if the Slater constraint qualification and the non-degeneracy condition (ND2) hold, then a feasible point x* is a global minimizer of f over
if and only if it is a KKT point.
The overall aim of this paper is to extend Lassere’s [1] results to a vector optimization problem over cones.
2. Preliminaries and Problem Formulation
We consider the following vector optimization problem (VOP) over cones:
(VOP) K – minimize ![](https://www.scirp.org/html/10-1040268\c7c1f4e3-9fef-44d8-bd44-031efe143ce5.jpg)
subject to ![](https://www.scirp.org/html/10-1040268\4313c164-a172-49a9-9a39-58bc900bc2ee.jpg)
where
and
are differentiable functions, K and Q are closed convex cones with non-empty interiors in Rp and Rm respectively.
Let
be the set of feasible solutions of (VOP).
The positive dual cone K* and the strict positive dual cone
of K are respectively defined as
![](https://www.scirp.org/html/10-1040268\5d8cd3ca-e0e0-406e-a02a-18260ef00abb.jpg)
and
.
We begin by defining the notion of a KKT point in terms of (VOP).
Definition 2.1: A point
is said to be a KKT-point if there exist
and
such that
and
.
For the problem (VOP), the solutions are defined in the following sense:
Definition 2.2 [7]: A point
is called 1) a weak minimum of (VOP) if for all ![](https://www.scirp.org/html/10-1040268\214c7ea9-4009-4b66-a0b6-0767b8712b16.jpg)
;
2) a Pareto-minimum of (VOP) if for all ![](https://www.scirp.org/html/10-1040268\a4c48066-fdd2-4de7-8f71-a7eaf049ed46.jpg)
;
3) a Strong minimum of (VOP) if for all ![](https://www.scirp.org/html/10-1040268\08b589a8-cdf3-439f-84a2-e40b7d5c2fa2.jpg)
.
Let
denote the set of weak minimum solutions of (VOP).
The forthcoming optimality and duality results are based on suitable generalized convexity assumptions over cones, thus we recall some known definitions in the literature.
Definition 2.3 [8,9]: A function
is said to be
1) K-convex at a point
if for every ![](https://www.scirp.org/html/10-1040268\a5e47724-9745-415c-b510-dd5ba6157c2f.jpg)
.
2) K-pseudoconvex at
if for every ![](https://www.scirp.org/html/10-1040268\7bedc06d-e459-4479-8bf3-784950c8a0ba.jpg)
.
3) strongly K-pseudoconvex at
if for every ![](https://www.scirp.org/html/10-1040268\3d2bbe88-eaa7-496b-a58f-d338ab43e0f9.jpg)
.
4) strictly K-pseudoconvex at
if for every ![](https://www.scirp.org/html/10-1040268\d99baf22-70d8-480e-a8db-7c3bb3e88c82.jpg)
.
If f is K-convex (K-pseudoconvex, strongly K-pseudoconvex, strictly K-pseudoconvex) at every
then f is said to be K-convex (K-pseudoconvex, strongly K-pseudoconvex, strictly K-pseudoconvex) on Rn.
On the lines of Jahn [10] we define Slater-type cone constraint qualification as follows:
Definition 2.4: The problem (VOP) is said to satisfy Slater-type cone constraint qualification at
if there exists
such that
.
Note that if g is Q-convex at x* and the problem (VOP) satisfies Slater constraint qualification, that is, there exists
such that
, then (VOP) satisfies Slater-type cone constraint qualification at x*.
Also, it is worth noticing that following the steps of Lassere [1] and Dutta and Lalitha [5] we can define the analogous non-degeneracy condition (ND3) for (VOP) as follows:
For all
,
, whenever
and
.
But if we assume that Slater-type cone constraint qualification holds at a point
, then there exists
such that
![](https://www.scirp.org/html/10-1040268\6d213a4a-702c-49cc-81d6-860fc646efc1.jpg)
which means that for all
for which
, we have
which itself implies that
and hence the nondegeneracy condition holds.
Thus in the paper, we shall extend Lassere’s [1] results to the vector optimization problem (VOP) over cones but, unlike Lassere, to prove our results we need to assume only Slater-type cone constraint qualification at a point.
3. Optimality Conditions
In this section we prove several classical optimality results by taking generalized convexity assumptions over cones on the objective function and assuming the feasible set to be convex and with no convexity type restriction on the constraint function. It is clear that if the constraint function g in (VOP) is Q-convex then the feasible set F is convex, so we begin by exemplifying the fact that F can be convex without g being Q-convex.
Example 3.1: Consider
defined as
![](https://www.scirp.org/html/10-1040268\27cf88f9-da16-4452-9ca7-268d1980c0b1.jpg)
and
.
Here g is not Q-convex, because if we take
and
then
.
But the feasible set
is convex. We have the following lemma.
Lemma 3.1: If the feasible set F of (VOP) is convex then
(1)
.
Proof: Let F be convex and suppose
satisfy
.
Assume that
. (2)
Now, for
, we have
![](https://www.scirp.org/html/10-1040268\4bffa328-4ae8-4044-b7d8-ea29357fbcd7.jpg)
where
.
This implies that.
![](https://www.scirp.org/html/10-1040268\6a630f15-396c-419d-8496-ee51794d05ef.jpg)
Using (2) together with
for a sufficiently small,
, we get
. (3)
Since F is convex, therefore
, that is,
so that
.
This contradicts (3). Hence the result.
The above lemma plays a pivotal role throughout the rest of the paper, thus we illustrate it by means of an example.
Example 3.2: Consider
and Q as defined in Example 3.1. Then we have already seen that g is not Q-convex whereas the feasible set F is convex.
Now, if we take
, then
if and only if
, and for this choice of m,
![](https://www.scirp.org/html/10-1040268\72355a89-8026-42da-9313-d02539363f16.jpg)
Also, for any other
, there does not exist any
for which
.
Hence the lemma holds.
The following theorem serves the main purpose of the paper.
Theorem 3.1: Consider a feasible solution x* of the vector optimization problem (VOP) and assume that Slater-type cone constraint qualification holds at x*. If f is K-convex at x* and the feasible set F is convex then x* is a weak minimum of (VOP) if and only if it is a KKTpoint.
Proof: Let
be a weak minimum of (VOP). By Lemma 1 [11], there exist
and
not both zero such that
(4)
and
. (5)
If possible, let
, then
so that from (4), we get
. (6)
Since Slater-type cone constraint qualification holds at x*, there exists
such that
which gives that
.
This together with (5) implies
which contradicts (6). Therefore
.
Since the inequality (4) holds for every
, we conclude that
(7)
and
. (8)
Hence x* is a KKT-point.
Conversely, let
be a KKT-point, that is, there exist
and
such that (7) and (8) hold.
Suppose x* is not a weak minimum of (VOP), so there exists
such that
. (9)
Since f is K-convex at x*,
. (10)
By (9) and (10),
which implies
.
This, by (7), gives
.
But this contradicts Lemma 3.1 as
.
Hence
is a weak minimum for (VOP).
Theorem 3.2: Let f be K-pseudoconvex at
and suppose that F is convex. Further assume that Slater-type cone constraint qualification holds at x*. Then x* is a weak minimum of (VOP) if and only if it is a KKT-point.
Proof: Proof follows on similar lines as Theorem 3.1.
Now we obtain sufficient optimality conditions for (VOP).
Theorem 3.3: Let f be K-convex at
and the feasible set F be convex and suppose that there exist
and
such that (7) and (8) hold. Then
is a Pareto minimum of (VOP).
Proof: Let if possible,
be not a Pareto minimum of (VOP). Then there exists
such that
. (11)
Since f is K-convex at
we have
.
Using (11), we get
.
Since
we have
.
Now proceeding as in the converse part of Theorem 3.1, we get a contradiction to Lemma 3.1. Hence
is a Pareto minimum of (VOP).
We now give an example to illustrate Theorem 3.3.
Example 3.3: Consider the problem
(VOP) K-Minimize ![](https://www.scirp.org/html/10-1040268\52c96736-a5f6-45ce-8441-463b6f672034.jpg)
Subject to ![](https://www.scirp.org/html/10-1040268\4d935d91-123e-460c-846b-716f16fc992c.jpg)
where
and Q are as defined in Example 3.1 and
and K are given by
.
Then, as shown in Example 3.1, g is not Q-convex. while the feasible set
of (VOP) is convex. Also f is K-convex at
.
It can be seen that for
,
and
.
Thus by Theorem 3.3,
is a Pareto minimum of (VOP).
Remark 3.1: Example 3.3 describes a vector optimization problem in which a Pareto minimum is obtained by applying Theorem 3.3 whereas it is impossible to do so using Lassere’s [1] results.
Theorem 3.4: Let f be strictly K-pseudoconvex at
and the feasible set F be convex and suppose that there exist
and
such that (7) and (8) hold. Then
is a Pareto minimum of (VOP).
Proof: Let if possible,
be not a Pareto minimum of (VOP).
Then there exists
such that
.
Since f is strictly K-pseudoconvex at
we get
.
As
, we have
.
Now proceeding as in the converse part of Theorem 3.1, we get a contradiction to Lemma 3.1. Hence
is a Pareto minimum of (VOP).
Theorem 3.5: Let f be strongly K-pseudoconvex at
and the feasible set F be convex and suppose that there exist
and
such that (7) and (8) hold. Then
is a strong minimum of (VOP).
Proof: Let if possible,
be not a strong minimum of (VOP).
Then there exists
such that
.
Since f is strongly K-pseudoconvex at
we get
.
As
, we have
.
Again proceeding as in the converse part of Theorem 3.1, we get a contradiction. Hence
is a strong minimum of (VOP).
4. Duality
With the primal problem (VOP), we associate the following Mond-Weir type dual program (MDP):
(MDP) K-maximize ![](https://www.scirp.org/html/10-1040268\6880a0cd-686e-4a52-9e54-0a7fe3de9b8d.jpg)
subject to
(12)
(13)
.
Let FD denote the set of feasible solutions of (MDP).
Definition 4.1: A point
is said to be a weak maximum of (MDP) if
.
Let
denote the set of weak maximum solutions of (MDP).
Theorem 4.1: (Weak Duality) Let
and
. Assume that f is K-pseudoconvex at y and the feasible set F is convex, then
.
Proof: Let
and
. Suppose to the contrary that
(14)
Since f is K-pseudoconvex at y, (14) implies
.
As
, we get
. (15)
Since
, therefore by Lemma 3.1,
. (16)
Adding (15) and (16), we have
which contradicts (12). Hence,
.
Theorem 4.2: (Strong Duality) Let
Assume that Slater-type cone constraint qualification holds at x*. If f is K-pseudoconvex at x* and the feasible set F is convex, then there exist
and
such that
Further, if the conditions of Weak Duality Theorem 4.1 hold for all
and
, then ![](https://www.scirp.org/html/10-1040268\3b64b9f8-14d6-4c40-9a15-5443ca5a73c5.jpg)
Proof: Since all the conditions of Theorem 3.2 hold, therefore there exist
and
such that
![](https://www.scirp.org/html/10-1040268\fe85be86-6d9b-4779-8847-1ff180400b0f.jpg)
and
.
Thus
Further if
, then there exists
such that
![](https://www.scirp.org/html/10-1040268\0484225a-a4b7-4a82-85aa-beb40ed32e65.jpg)
which contradicts Theorem 4.1.
Hence, ![](https://www.scirp.org/html/10-1040268\5115d350-a91f-4ea3-814a-71f2918bd388.jpg)
Theorem 4.3: (Converse Duality) Let
![](https://www.scirp.org/html/10-1040268\1586b2c9-6c29-493d-ad2b-b6b1e298dd74.jpg)
Assume that f is K-pseudoconvex at
and the feasible set F is convex. Then ![](https://www.scirp.org/html/10-1040268\4edc7f6a-c3e5-4af0-b5e5-d92f8e7e7cc4.jpg)
Proof: Suppose
Then there exists
such that
.
Since f is K-pseudoconvex at
we get
so that,
(17)
Also,
, so that by Lemma 3.1,
. (18)
Adding (17) and (18), we have
which contradicts (12). Hence, ![](https://www.scirp.org/html/10-1040268\02f034b9-774b-466a-b6fc-8b3477d16802.jpg)
5. Conclusion
This paper gives a new direction to the search for solution of a vector optimization problem over cones. We have shown that, with Slater-type cone constraint quailfication, convexity of the feasible set can replace the cone-convexity (or any of its generalization) of the constraint functions, and then we just need to assume the cone-convexity (or a suitable generalization) of the objective function to prove the necessity and sufficiency of the KKT optimality conditions. Moreover, a Mond-Weir type dual has been formulated in the modified situation and various duality results have been established.
6. Acknowledgements
The first author is grateful to the University Grants Commission (UGC), India for offering financial support.
NOTES