Received 1 November 2014; revised 29 November 2014; accepted 16 December 2014

ABSTRACT

In this paper, we introduce the concept of a (weak) minimizer of order k for a nonsmooth vector optimization problem over cones. Generalized classes of higher-order cone-nonsmooth (F, ρ)- convex functions are introduced and sufficient optimality results are proved involving these classes. Also, a unified dual is associated with the considered primal problem, and weak and strong duality results are established.

Keywords:

Nonsmooth Vector Optimization over Cones, (Weak) Minimizers of Order k, Nonsmooth (F, ρ)-Convex Function of Order k

1. Introduction

It is well known that the notion of convexity plays a key role in optimization theory [1] [2] . In the literature, various generalizations of convexity have been considered. One such generalization is that of a -convex function introduced by Vial [3] . Hanson and Mond [4] defined the notion of an F-convex function. As an extended unification of the two concepts, Preda [5] introduced the concept of a -convex function. Antczak gave the notion of a locally Lipschitz -convex scalar function of order k [6] and a differentiable - convex vector function of order 2 [7] .

L. Cromme [8] defined the concept of a strict local minimizer of order k for a scalar optimization problem. This concept plays a fundamental role in convergence analysis of iterative numerical methods [8] and in stability results [9] . The definition of a strict local minimizer of order 2 is generalized to the vectorial case by Antczak [7] .

Recently, Bhatia and Sahay [10] introduced the concept of a higher-order strict minimizer with respect to a nonlinear function for a differentiable multiobjective optimization problem. They proved various sufficient optimality and mixed duality results involving generalized higher-order strongly invex functions.

The main purpose of this paper is to extend the concept of a higher-order minimizer to a nonsmooth vector optimization problem over cones. The paper is organized as follows. We begin in Section 2 by recalling some known concepts in the literature. We then define the notion of a (weak) minimizer of order k for a nonsmooth vector optimization problem over cones. Thereafter, we introduce various new generalized classes of cone- nonsmooth -convex functions of higher-order. In Section 3, we study several optimality conditions for higher-order minimizers via the introduced classes of functions. In Section 4, we associate a unified dual to the considered problem and establish weak and strong duality results.

2. Preliminaries and Definitions

Let be a nonempty open subset of. Let be a closed convex cone with nonempty interior and let denote the interior of K. The dual cone K^* of K is defined as

.

The strict positive dual cone of K is given by

A function is said to be locally Lipschitz at a point if for some,

, within a neighbourhood of u.

A function is said to be locally Lipschitz on S if it is locally Lipschitz at each point of S.

Definition 2.1. [11] Let be a locally Lipschitz function, then denotes the Clarke’s generalized directional derivative of at in the direction and is defined as

.

The Clarke’s generalized gradient of at u is denoted by and is defined as

.

Let be a vector valued function given by ,. Then f is said to be locally Lipschitz on S if each is locally Lipschitz on S. The generalized directional derivative of a locally Lipschitz function at in the direction is given by

.

The generalized gradient of f at u is the set

,

where is the generalized gradient of at u for.

Every element is a continuous linear operator from to and

for all.

A functional is sublinear with respect to the third variable if, for all,

(i) for all, and

(ii) for all.

(i) and (ii) together imply. (1)

We consider the following nonsmooth vector optimization problem

(NVOP) K-minimize

subject to,

where:,:, K and Q are closed convex cones with nonempty interiors in R^m and R^p respectively. We assume that for each and for each are locally Lipschitz on S.

Let denote the set of all feasible solutions of (NVOP).

The following solution concepts are well known in the literature of vector optimization theory.

Definition 2.2. A point, is said to be

(i) a weak minimizer (weakly efficient solution) of (NVOP) if for every,

(ii) a minimizer (efficient solution) of (NVOP) if for every,

With the idea of analyzing the convergence and stability of iterative numerical methods, L. Cromme [8] introduced the notion of a “strict local minimizer of order k”. As a recent advancement on this platform, Bhatia and Sahay [10] defined the concept of a higher-order strict minimizer with respect to a nonlinear function for a differentiable multiobjective optimization problem. We now generalize this concept and give the definition of a higher-order (weak) minimizer with respect to a function for a nonsmooth vector optimization problem over cones.

Definition 2.3. A point is said to be

(i) a weak minimizer of order for (NVOP) with respect to, if there exists a vector such that, for every

;

(ii) a minimizer of order for (NVOP) with respect to, if there exists a vector such that, for every

Remark 2.1. (1) If f is a scalar valued function, and, the definition of a weak minimizer of order k reduces to the definition of a strict minimizer of order k (see [8] [9] [12] [13] ).

(2) If, and, the definition of a (weak) minimizer of order k becomes the definition of a vector strict global (weak) minimizer of order 2 given by Antczak [7] .

(3) If the definition of a weak minimizer of order k reduces to the definition of a strict minimizer of order k given by Bhatia and Sahay [10] .

Remark 2.2. (1) Clearly a minimizer of order k for (NVOP) with respect to is also a weak minimizer of order k for (NVOP) with respect to the same.

(2) A direct implication of the fact that is that, a (weak) minimizer of order k for (NVOP) with respect to is a (weak) minimizer for (NVOP).

(3) Note that if is a (weak) minimizer of order k for (NVOP) with respect to, then for all, it is also a (weak) minimizer of order for (NVOP) with respect to the same.

In the sequel, for a vector function and, denotes the vector.

We now define various classes of nonsmooth -convex functions of higher-order over cones.

Definition 2.4. A locally Lipschitz function is said to be K-nonsmooth -convex of order k with respect to at on S if there exist a sublinear (with respect to the third variable) functional and a vector such that, for each and all

.

If the above relation holds for every then f is said to be K-nonsmooth -convex of order k with respect to on S.

Remark 2.3. (1) If f is a scalar valued function and, the above definition reduces to the definition of a (locally Lipschitz) -convex function of order k with respect to given by Antczak [6] .

(2) If f is a differentiable function, , and the definition of a K-nonsmooth -convex function of order k with respect to becomes the definition of a vector -convex function of order 2 given in [7] .

(3) If, for some function and, K-nonsmooth - convexity of order k with respect to reduces to -invexity, where, introduced by Nahak and Mohapatra [14] .

(4) If is a differentiable function, and, , for some function, the above definition becomes the definition of a higher-order strongly invex function given by Bhatia and Sahay [10] .

Definition 2.5. A locally Lipschitz function is said to be K-nonsmooth -pseudoconvex type I of order k with respect to at on S if there exist a sublinear (with respect to the third variable) functional and a vector such that, for each and all,

.

Equivalently,

.

If f is K-nonsmooth -pseudoconvex type I of order k with respect to at every then f is said to be K-nonsmooth -pseudoconvex type I of order k with respect to on S.

Clearly, if f is K-nonsmooth -convex of order k with respect to, then f is K-nonsmooth - pseudoconvex type I of order k with respect to the same, however the converse may not be true as shown by the following example.

Example 2.1. Consider the following nonsmooth function, , and

Here and.

Define as

.

Let be given by, and.

Then, at.

,

for every and.

Hence, f is K-nonsmooth -pseudoconvex type I of order 3 with respect to at u on S.

However, for and.

,

so that f is not K-nonsmooth -convex of order 3 at u on S.

Definition 2.6. A locally Lipschitz function is said to be K-nonsmooth -pseudoconvex type II of order k with respect to at on S if there exist a sublinear (with respect to the third variable) functional and a vector such that, for each and all,

Equivalently,

.

If the above relation holds for every, then f is said to be K-nonsmooth -pseudoconvex type II of order k with respect to on S.

We now give an example to show that a K -nonsmooth -pseudoconvex type II function of order k with respect to may fail to be a K -nonsmooth -convex function of order k with respect to.

Example 2.2. Consider the following nonsmooth function, , and

,

Here and.

Let be given by.

and.

Then, at,

,

for every, and.

Therefore, f is K-nonsmooth -pseudoconvex type II of order with respect to at u on S.

However, for and, ,

.

Thus, f is not K-nonsmooth -convex of any order k with respect to at u on S.

Definition 2.7. A locally Lipschitz function is said to be K-nonsmooth -quasiconvex type I of order k with respect to at on S if there exist a sublinear (with respect to the third variable) functional and a vector such that, for each and all,

.

If the above relation holds at every, then f is said to be K-nonsmooth -quasiconvex type I of order k with respect to on S.

Definition 2.8. A locally Lipschitz function is said to be K-nonsmooth -quasiconvex type II of order k with respect to at on S if there exist a sublinear (with respect to the third variable) functional and a vector such that, for each and all,

.

If f is K-nonsmooth -quasiconvex type II of order k with respect to at every, then f is said to be K-nonsmooth -quasiconvex type II of order k with respect to on S.

Remark 2.4. When f is a differentiable function, and, for some function, Definition 2.4 - 2.7 take the form of the corresponding definitions given by Bhatia and Sahay [10] .

3. Optimality

In this section, we obtain various nonsmooth Fritz John type and Karush-Kuhn-Tucker (KKT) type necessary and sufficient optimality conditions for a feasible solution to be a (weak) minimizer of order k for (NVOP).

On the lines of Craven [15] we define Slater-type cone constraint qualification as follows:

Definition 3.1. The problem (NVOP) is said to satisfy Slater-type cone constraint qualification at if, for all, there exists a vector such that.

Remark 3.1. The following inclusion relation is worth noticing.

For and,

Thus,

. (2)

Since a weak minimizer of order for (NVOP) is a weak minimizer for (NVOP), the following nonsmooth Fritz John type necessary optimality conditions can be easily obtained from Craven [15] .

Theorem 3.1. If a vector is a weak minimizer of order k with respect to for (NVOP) with, then there exist Lagrange multipliers and not both zero, such that

.

The necessary nonsmooth KKT type optimality conditions for (NVOP) can be given in the following form.

Theorem 3.2. If a vector is a weak minimizer of order k with respect to for (NVOP) with and if Slater-type cone constraint qualification holds at, then there exist Lagrange multipliers and, such that

(3)

. (4)

Proof. Assume that is a weak minimizer of order k with respect to for (NVOP), then by Theorem 3.1 there exist and, not both zero, such that (3) and (4) hold.

If possible, suppose. Then, and (3) reduces to

.

So there exists such that

. (5)

Now, since Slater-type cone constraint qualification holds at, we have for all, there exists a vector such that. Since, we get. In particular,. On the contrary (5) implies. This contradiction justifies.

Now, we give sufficient optimality conditions for a feasible solution to be a higher-order (weak) minimizer for (NVOP).

Theorem 3.3. Let be a feasible solution for (NVOP) and suppose there exist vectors, and, such that

(6)

. (7)

Further, assume that f is K-nonsmooth -convex of order k with respect to at on and g is Q-nonsmooth -convex of order k with respect to the same at on. If and, then is a weak minimizer of order k with respect to for (NVOP).

Proof. Assume on the contrary that is not a weak minimizer of order k with respect to for (NVOP). Then, for any, there exists a vector such that,

.

As, the above relation holds in particular for, so that we have

. (8)

As (6) holds, there exist and such that

. (9)

Since f is K-nonsmooth -convex of order k with respect to at on, we have

. (10)

Adding (8) and (10), we get

.

As, we obtain

. (11)

Also, since g is Q-nonsmooth convex of order k with respect to at on and, we have

.

However, , and (7) together give

. (12)

Adding (11) and (12), we get

,

which implies that

.

Using sublinearity of F under the assumption and, we obtain

,

which on using (9) and (1), gives

.

This is impossible as and, so that, and norm is a non-negative function. Hence is a weak minimizer of order with respect to for (NVOP).

Theorem 3.4. Suppose there exists a feasible solution for (NVOP) and vectors and such that (6) and (7) hold. Moreover, assume that f is K-nonsmooth -pseudoconvex type I of order k with respect to at on and is -nonsmooth -quasiconvex type I of order k with respect to the same at on. If and, then is a weak minimizer of order k with respect to for (NVOP).

Proof: Let if possible, be not a weak minimizer of order k with respect to ω for (NVOP). Then, for any, there exists such that,

.

Since taking, in particular, in the above relation, we obtain

. (13)

As (6) holds, there exist and such that (9) holds.

Since f is K-nonsmooth -pseudoconvex type I of order k with respect to ω at on, (13) implies

.

As, we have

. (14)

Now, means, so that. This along with (7) gives

. (15)

If, then (15) implies.

Since g is Q-nonsmooth -quasiconvex type I of order k with respect to at on, therefore

,

so that

. (16)

If, then also (16) holds.

Now, proceeding as in Theorem 3.3, we get a contradiction. Hence, is a weak minimizer of order k with respect to for (NVOP).

Theorem 3.5. Assume that all the conditions of Theorem 3.3 (Theorem 3.4) hold with. Then is a minimizer of order k with respect to for (NVOP).

Proof: Let if possible, be not a minimizer of order k with respect to for (NVOP), then for any there exists such that

. (17)

Proceeding on similar lines as in proof of Theorem 3.3 (Theorem3.4) and using (17) we have

.

As, we get

.

This leads to a contradiction as in Theorem 3.3 (Theorem 3.4). Hence, is a minimizer of order k with respect to for (NVOP).

4. Unified Duality

On the lines of Cambini and Carosi [16] , we associate with our primal problem (NVOP), the following unified dual problem (NVUD).

(NVUD) K-maximize

subject to (18)

(19)

where, , , and is a 0 - 1 parameter.

Note that Wolfe dual and Mond-Weir dual can be obtained from (NVUD) on taking and respectively.

Definition 4.1. Given the problem (NVOP) and given a vector we define the following Lagrange function:

.

Theorem 4.1. (Weak Duality) Let x be feasible for (NVOP) and be feasible for (NVUD). If f is K-nonsmooth -convex of order k with respect to at y on and g is Q-nonsmooth -convex of order k with respect to the same at y on, with and

, (20)

then,

.

Proof: Assume on the contrary that

. (21)

Since is feasible for (NVUD), therefore by (2), there exist and such that

. (22)

Since f is K-nonsmooth -convex of order k with respect to at y on, we have

(23)

Adding (21) and (23), we obtain

.

As, we get

. (24)

Also, since g is Q-nonsmooth -convex of order k with respect to at y on and, we have

. (25)

Adding (24) and (25), we get

or,

Using sublinearity of F under the assumption that and, together with (22), (1) and (20), we obtain

.

As and, so that and we have.

This contradicts the feasibility of, hence the result.

Theorem 4.2. (Weak Duality) Let x be feasible for (NVOP) and be feasible for (NVUD) with and. Suppose the following conditions hold:

(i) If is K-nonsmooth -pseudoconvex type II of order k with respect to at y on, and

(ii) If, f is K-nonsmooth -pseudoconvex type II of order k with respect to at y on and g is Q-nonsmooth -quasiconvex type I of order k with respect to at y on.

Then, we have

.

Proof: Case (i): Let and on the contrary assume that,

. (26)

Since x is feasible for (NVOP) and, therefore. Further, so that

. (27)

Adding (26) and (27), we get

.

That is,

.

As is K-nonsmooth -pseudoconvex type II of order k with respect to, we have for all

.

Since, , we get

,

or

,

so that

. (28)

Now, since is feasible for (NVUD),

Therefore, there exists such that. Substituting in (28) and then using (1), we get

,

which is a contradiction, as and norm is a non-negative function.

Case (ii): Let, then we have to prove that

.

Let if possible,

.

Since f is K-nonsmooth -pseudoconvex type II of order k with respect to at y on, we have

.

As, we get

. (29)

Since x is feasible for (NVOP) and is feasible for (NVUD), we have

. (30)

If, (30) implies.

As g is Q-nonsmooth -quasiconvex type I of order k with respect to at y on, we get

.

Since, we have

. (31)

If, then also (31) holds.

Since is feasible for (NVUD), by Remark 3.1, there exist and such that (22) holds.

Adding (29) and (31), we get

,

or

.

Using sublinearity of F with the fact that and and then using (22) and (1), we obtain

.

This contradicts the assumption that, hence the result.

Theorem 4.3. (Strong Duality) Let be a weak minimizer of order k with respect to for (NVOP) with, at which Slater-type cone constraint qualification holds. Then there exist such that is feasible for (NVUD). Further, if the conditions of Weak Duality Theorem 4.1 (Theorem 4.2) hold for all feasible x for (NVOP) and all feasible for (NVUD), then is a weak maximizer of order k with respect to for (NVUD).

Proof: As is a weak minimizer of order k with respect to for (NVOP), by Theorem 3.2 there exist such that

, (32)

. (33)

Since, Equations (32) and (33) can be written as

,

.

Thus, is a feasible solution for (NVUD). Further, if is not a weak maximizer of order k with respect to for (NVUD), then for any, there exists a feasible solution of (NVUD) such that

or,

Since, , so that we have

which contradicts Theorem 4.1 (Theorem 4.2). Hence is a weak maximizer of order k with respect to for (NVUD).

5. Conclusion

In this paper, we introduced the concept of a higher-order (weak) minimizer for a nonsmooth vector optimization problem over cones. Furthermore, to study the new solution concept, we defined new generalized classes of cone-nonsmooth (F, ρ)-convex functions and established several sufficient optimality and duality results using these classes. The results obtained in this paper will be helpful in studying the stability and convergence analysis of iterative procedures for various optimization problems.

References

NOTES

^*Corresponding author.