1. Introduction
Multiobjective optimization plays an important role in management science, operations research, and economics. The reader is referred to the recently published book [
1] for more details on vector optimization theory and applications. The classical concept of efficient solution in multiobjective optimization problems was introduced by Pareto [
2] under specific preferences. Koopmans [
3] proposed the concept of a Pareto efficient solution. After that, many scholars studied Pareto efficiency and obtained a lot of results (see the book [
4] and the reference therein). However, the set of all Pareto efficient solutions is large, and part of it cannot be characterized by a scalar minimization problem. To eliminate these abnormal solutions, various kinds of proper efficient solutions have been introduced (see Chapter 4 in [
4]), one of which was introduced by Borwein [
5] and was called the Borwein proper efficient solution by some later researchers. Since Borwein proper efficiency highlights the geometric property and abandons noneffective decisions in decision making, it has become a standard concept in vector optimization literature (see [
6,
7,
8]).
In this paper, we consider the following multiobjective mathematical programming with vanishing constraints (MMPVC for short):
where
are locally Lipschitz functions with
, and
.
The MMPVC is a complicated programming problem since it involves product function in its constraints with . This complicatedness brings us two difficulties. One is that the feasible set usually is not a convex set; the other is that the constrained property of vanishes in the case .
In the special case
, MMPVC reduces to the mathematical programming with vanishing constraints (MPVC for short) which was introduced by Achtziger and Kanzow [
9]. MPVCs not only play an important role in topology optimization which is a powerful tool in mechanical structures design, but also extend another group of programming problems called mathematical programming with equilibrium constraints (see [
9]). For these two reasons, the MPVCs have attracted some researchers’ interest. Some stationary conditions of Karush–Kuhn–Tucker-type optimality conditions are given under various qualification condition by the classical subdifferential and normal cones, such as Clarke subdifferential and Clarke normal cone. Readers are referred to the reference [
9,
10,
11,
12] for smooth MPVCs and [
13,
14,
15] for nonsmooth MPVCs.
All MPVCs mentioned above concerned a single-objective function. To the best of our knowledge, Mishra, et al. [
16] studied MMPVCs involving continuously differentiable functions for the first time. They modified some constraint qualifications such as Cottle constraint qualification, Slater constraint qualification, etc. Then, they established relationships among them and obtained the Karush–Kuhn–Tucker-type necessary optimality conditions for Pareto efficiency solutions. In [
17], the MMPVC with its objective functions being continuously differentiable and its constrained function being convex were considered. Two Abadie-type constraint qualifications were introduced and some necessary conditions for Geoffrion properly efficient solutions were given by convex subdifferentials. Recently in [
18], for the nonsmooth MMPVCs, some data qualifications characterized by Clarke subdifferential were introduced, and the relationship among them was discussed. Some stationary conditions as necessary or sufficient conditions of weakly efficient and Pareto efficient solutions were also given. Motivated by [
18], it is natural for us to consider the stationary condition for Borwein proper efficient solutions of the MMPVC.
The rest of this paper is organized as follows. In
Section 2, we introduce some notions and preliminary results which will be needed later. In
Section 3, we present our main results. Unlike [
18,
19] concerning weak efficiency and Pareto efficiency, we consider Borwein proper efficiency. We introduce
proper Abadie data qualification condition (in short,
-PADQ) for a given
. Using
-PADQ condition, we obtain a strictly strong stationary condition as a necessary condition for the Borwein proper efficient solution of problem MMPVC. Under the assumption of the convexity of objective functions and the
∂-quasi-convexity of constrained functions, we establish a strictly strong stationary condition as a sufficient condition for the Borwein proper efficient solution of problem MMPVC.
2. Preliminaries
Throughout this paper, unless stated otherwise, we always assume that X is a real Banach space, is the dual space of X, is a nonempty subset of X, is the closed unit ball of X, and , . The interior, convex hull, and closure of are denoted by ), , and cl(), respectively.
The set
is called a cone if
for all
and
. Clearly, a cone
is convex if and only if
. The cone generated by
is defined as
The negative and strict negative polar cone of
are, respectively, defined as
For
, the contingent cone of
at
is the set
Let
, and
. The function
is said to be upper directionally differentiable at
in the direction
u if
exists, where
means that
and
t converges to 0. The function
is said to be directionally differentiable at
in the direction
u, if
exists. Clearly, if
is directionally differentiable at
in the direction
u, then
. If for all
exists and
is a continuous linear mapping, then
is said to be G
teaux differentiable at
.
Let
Y be a Banach space,
, a mapping
is said to be locally Lipschitz at
z, if there exist
and
such that
If is locally Lipschitz at each point of , then is said to be locally Lipschitz on .
Let
,
and
be locally Lipschitz at
. The Clarke generalized directional derivative of
at
in the direction
u is defined as
The set
is called the Clarke subdifferential of
at
.
Lemma 1 ([
20])
. Let be a function, , , and ψ be locally Lipschitz at . Then:- (i)
is a nonempty compact convex set;
- (ii)
There exists such that ;
- (iii)
Definition 1 ([
21]).
Let be a function, and ψ be locally Lipschitz at . The function ψ is called ∂-quasi convex at , if for all , Definition 2 ([
5]).
Let A be a nonempty subset of X and Θ be a pointed convex cone of X. A point is called a Borwein proper efficient point of A, ifThe set of all Borwein proper efficient points of A is denoted by .
The following lemma is a standard separation theorem for two convex sets.
Lemma 2 ([
4]).
Let A be a nonempty compact convex subset of X and B be a nonempty closed convex subset of X. Then, if and only if there exist and such that For convenience of the readers, we give the important notations mentioned above in
Table 1.
3. Main Results
In this section, we establish necessary and sufficient optimality conditions for the Borwein proper efficient solution of problem MMPVC. The feasible set of problem MMPVC is denoted as follows:
We always assume that
, and
will be fixed in the remainder of this paper. Following [
9,
13,
18], we define the index sets as follows:
Let and . Obviously, .
For each
, set
, and define
Let . Now, using Definition 2, we can define a Borwein proper efficient solution of problem MMPVC.
Definition 3. A point is said to be a Borwein proper efficient solution of problem MMPVC, if , that is,The set of all Borwein proper efficient solutions is denoted by . Ifthen is called a Pareto efficient solution of problem MMPVC. The set of all Pareto efficient solutions is denoted by . Lemma 3. Suppose that , and is locally Lipschitz at for all . Then, Proof. We divide p into two cases: and .
Case 1:
. In this case, (
1) equals
Suppose to the contrary that there exists some
, then there exist
with
,
with
such that
for all
n. Since
is locally Lipschitz at
and
, it follows from Lemma 1 that there exists
such that
Since
is locally Lipschitz at
, there exist
and
such that for all
,
Since
,
, there exists a positive integer number
N such that for all
,
By (iii) of Lemma 1, we have
As
we obtain
which contradicts
since
.
Case 2:
. To verify (
1), it suffices to prove that
Without loss of generality, we only need to show that
Suppose to the contrary that there exists
. Since
, there exist
with
,
with
such that
. Using the same proof of case
, we obtain
By the definition of
, we have
Since
is locally Lipschitz at
, we have
Therefore,
which contradicts
since
. Therefore,
In conclusion, Equation (
1) is verified. □
Remark 1. In [19] (Lemma 5.1) (also see [18] (Lemma 2)), Li proved that if , thenwhereIt is known that a Borwein proper efficient solution is a Pareto efficient solution, but the converse is not true. To illustrate that Lemma 3 sharpens Li’s result, it suffices to give an example that is a strict subset of . See the following example. Example 1. In problem MMPVC, we take , and let be defined by Clearly, is a Borwein proper efficient solution. We calculate thatTherefore, is a strict subset of . Under some mild conditions, is a subset of .
Proposition 1. Suppose that ; is locally Lipschitz at and directionally differentiable at in any direction for all . Then, Proof. To verify Equation (
2), it suffices to show that
Without loss of generality, we only need to show that
We divide p into two cases: and .
Case 1:
. In this case,
, and hence
Equation (
3) is verified.
Case 2:
. Let
; then, there exist
with
,
with
such that
. By the definition of
, we obtain
and
and hence
Since
is locally Lipschitz at
and directionally differentiable at
in any direction, we obtain
This implies that
, and so Equation (
3) is verified. □
Before we give necessary conditions for the Borwein proper efficient solution of problem MMPVC, we introduce the following two definitions.
Definition 4. Let , for each , be a nonempty convex set of , , and . The setis called the ε-core of D. Definition 5. Let . We say that problem MMPVC satisfies ε proper Abadie data qualification (in short, ε-PADQ) at , ifwhere Remark 2. Assume that . Sincewe deduce that if problem MMPVC satisfies -PADQ, then it satisfies -PADQ. If we take , replace and “♯” with and “−”, respectively, in Definition 5, then ε-PADQ reduces to EADQ introduced in [18] (Definition 2). The meaning of was introduced in [13]. Here, -PADQ reveals the relationship between the subdifferentials of the objective functions and the constrained functions and the feasible set of problem MMPVC. It is somewhat abstract, which leads to the difficulty of verifying it for a general problem MMPVC. However, under some mild conditions, -PADQ is easy to verify.
Proposition 2. Let , with . Suppose that , , and are locally Lipschitz at . If one of the following conditions holds:
- (i)
;
- (ii)
;
- (iii)
;
- (iv)
is Gteaux differentiable at and
then problem MMPVC satisfies ε-PADQ at . Proof. Let
To verify problem MMPVC satisfying
-PADQ at
, it suffices to show that
Assume that (i) holds. Then, we have
. By the definition of negative polar cone, we have
. By the definition of
, we obtain
. To verify (
4), it suffices to prove that
Obviously, (
5) holds true if
. Now, let
. Then, we have
and so,
Therefore,
, which verifies Equation (
5).
Assume that (ii) holds. Then, (i) also holds since . Therefore, problem MMPVC satisfies -PADQ at .
Assume that (iii) holds. Then,
. By the definition of strict negative polar cone, we have
, and so Equation (
4) is verified. Therefore, we deduce that problem MMPVC satisfies
-PADQ at
.
Finally, assume that (iv) holds. Since
is G
teaux differentiable at
, we have
Here, (iv) implies that
. By (iii), problem MMPVC satisfies
-PADQ at
. □
Now, we give stationary conditions for the Borwein proper efficient solution of problem MMPVC.
Theorem 1. Let with . Suppose that , , , and are locally Lipschitz at , is a closed set, and problem MMPVC satisfies ε-PADQ at and
: for all with , ⇒
Then, there exist , , such thatand Proof. Since , , and are locally Lipschitz at , with Lemma 1, , , and are nonempty compact convex sets. Hence, is a compact convex set.
Firstly, we will prove that there exists
such that
Suppose to the contrary that for all positive integer numbers
n with
,
Since
is a closed convex set, applying Lemma 2 to the above equation, there exists
with
such that
Since problem MMPVC satisfies
-PADQ at
, with (
11), (
12), and Remark 2, we obtain
whenever
. Since
is bounded, we may assume that
. Since
is a closed set, we obtain
The condition (
) and Equation (
13) imply that
From Equations (
14) and (
15), we obtain
which contradicts Lemma 3. Therefore, (
10) is justified and
This implies that there exist nonnegative real numbers
with
such that
Using the same approach of Theorem 4.1 in [
13], we let
and the conclusion is proved. □
Remark 3. In Theorem 4 of [18], Sadeghieh et al. established the following result. Suppose that , andIf , , and are polyhedron, then Equations (6)–(9) hold. In [18], Equations (6)–(9) are called “strong strongly stationary conditions" and by “strong " in short. In this paper, we call the conditions (6)–(9) “strictly strong stationary conditions" only from grammar angles. Remark 4. In Theorem 1, if there exists a such thatthen the condition holds true. In fact, for all with and , we have . Let be arbitrarily given. Take . Then, we haveandLetting in (17), we have . Combining and , we obtain . This inequality implies that , and so the condition holds. In problem MMPVC, condition does not always hold true. See the following Example 2.
Example 2. Let . Consider the following problem MMPVC1:whereWe can verify that is a Borwein proper efficient solution of problem MMPVC1. Now, we prove that condition does not hold true. We calculate that Let . Then, , , and . However, .
In the following, we give an example to illustrate Theorem 1.
Example 3. Let . Consider the following problem MMPVC2:whereWe can verify that is a Borwein proper efficient solution of problem MMPVC2. We calculate that Clearly, is a closed set. For , we calculate thatHence, problem MMPVC2 satisfies -PADQ condition at . In the following, we will verify that condition (Υ) holds. Assume thatThen, we haveBy sending in (18), we have Taking in (19), we obtain . Combining , we obtain , and soHence, the condition (Υ) is verified. All conditions of Theorem 1 are satisfied. By Theorem 1, Equations (6)–(9) hold. In fact, take then However, since is not a polyhedron, Theorem 4 of [18] cannot be applied to MMPVC2. For
, we suppose that problem MMPVC satisfies the strictly strong stationary condition at
, that is, satisfies (
6)–(
9). Motivated by [
18,
22], we define the index sets as follows:
Now, we give sufficient optimality conditions for the Borwein proper efficient solution of problem MMPVC in terms of the strictly strong stationary condition.
Theorem 2. Suppose that problem MMPVC satisfies (6)–(9) at , , , and are locally Lipschitz at , is a convex function, and , , and are ∂-quasi-convex at . - (i)
Then, is a local Borwein proper efficient solution of problem MMPVC,
- (ii)
If , then is a Borwein proper efficient solution of problem MMPVC.
Proof. (i) Since
, we have
Since
, we have
Since
and
are continuous functions with (
22) and (
23), there exists a neighborhood
U of
such that
Since problem MMPVC satisfies (
6)–(
9), there exist
,
,
such that
Suppose to the contrary that
is not a local Borwein proper efficient solution. Then, there exist
and a neighborhood
V of
such that
Thus, there exist
with
and
such that
and at least one
; without loss of generality we may assume that
. Since
is a continuous convex function, we have
resulting in
and so
By (
26), we have
implying
Since
and
are
∂-quasiconvex at
, it follows from (
22)–(
25) that
On the other hand, by the definition of index sets, combining
, we have
Since
, the
∂-quasi convexity of
and (
28) imply that
The above inequality, Equations (
20) and (
21) imply that
Adding (
29)–(
32) and noting
, we have
which contradicts (
27). Therefore,
is a local Borwein proper efficient solution of problem MMPVC.
(ii). Now, assume that
. We begin our proof from “since problem MMPVC satisfies (
6)–(
9)” in the proof (i) and remove the neighborhoods
U and
V and
from it. We immediately obtain that
is a global Borwein proper efficient solution of problem MMPVC. □
Remark 5. In Theorem 10 of [18], Sadeghieh et al. established the following result. Suppose that is ∂-pseudoconvex at , and other conditions are the same as Theorem 2. Then, Theorem 2 holds for Pareto efficient solutions. In the following, we give an example to illustrate Theorem 2.
Example 4. Let . Consider the following problem MMPVC3:where Let . We can calculate that Let . Then, we can calculate thatandand , which imply that (6)–(9) hold. Obviously, is ∂-quasi-convex at . Since , by Theorem 2, a global Borwein proper efficient solution of problem MMPVC3.