Non-differentiable multiobjective mixed symmetric duality under generalized convexity

Abstract

The objective of this paper is to obtain a mixed symmetric dual model for a class of non-differentiable multiobjective nonlinear programming problems where each of the objective functions contains a pair of support functions. Weak, strong and converse duality theorems are established for the model under some suitable assumptions of generalized convexity. Several special cases are also obtained.

MS Classification: 90C32; 90C46.

1 Introduction

Dorn [1] introduced symmetric duality in nonlinear programming by defining a program and its dual to be symmetric if the dual of the dual is the original problem. The symmetric duality for scalar programming has been studied extensively in the literature, one can refer to Dantzig et al. [2], Bazaraa and Goode [3], Devi [4], Mond and Weir [5, 6]. Mond and Schechter [7] studied non-differentiable symmetric duality for a class of optimization problems in which the objective functions consist of support functions. Following Mond and Schechter [7], Hou and Yang [8], Yang et al. [9], Mishra et al. [10] and Bector et al. [11] studied symmetric duality for such problems. Weir and Mond [6] presented two models for multiobjective symmetric duality. Several authors, such as the ones of [12–14], studied multiobjective second and higher order symmetric duality, motivated by Weir and Mond [6].

Very recently, Mishra et al. [10] presented a mixed symmetric dual formulation for a non-differentiable nonlinear programming problem. Bector et al. [11] introduced a mixed symmetric dual model for a class of nonlinear multiobjective programming problems. However, the models given by Bector et al. [11] as well as by Mishra et al. [10] do not allow the further weakening of generalized convexity assumptions on a part of the objective functions. Mishra et al [10] gave the weak and strong duality theorems for mixed dual model under the sublinearity. However, we note that they did not discuss the converse duality theorem for the mixed dual model.

In this paper, we introduce a model of mixed symmetric duality for a class of non-differentiable multiobjective programming problems with multiple arguments. We also establish weak, strong and converse duality theorems for the model and discuss several special cases of the model. The results of Mishra et al. [10] as well as that of Bector et al. [11] are particular cases of the results obtained in the present paper.

2 Preliminaries

Let Rⁿ be the n-dimensional Euclidean space and let be its non-negative orthant. The following convention will be used: if x, y ∈ Rⁿ , then ; ; ; x ≰ y is the negation of x ≰ y.

Let f(x, y) be a real valued twice differentiable function defined on Rⁿ × R^m . Let and denote the gradient vector of f with respect to x and y at . Also let denote the Hessian matrix of f (x, y) with respect to the first variable x at . The symbols , and are defined similarly. Consider the following multiobjective programming problem (VP):

where X is an open set of Rⁿ , f _i : X → R, i = 1, 2,..., p and h : X → R^m .

Definition 2.1 A feasible solution is said to be an efficient solution for (VP) if there exists no other x ∈ X such that .

Let C be a compact convex set in Rⁿ . The support function of C is defined by

A support function, being convex and everywhere finite, has a subdifferential [7], that is, there exists z ∈ Rⁿ such that

The subdifferential of s(x|C) is given by

For any set D ⊂ Rⁿ , the normal cone to D at a point x ∈ D is defined by

It is obvious that for a compact convex set C, y ∈ N _C (x) if and only if s(y|C) = x^Ty, or equivalently, x ∈ ∂s(y|C).

Let us consider a function F : X × X × Rⁿ → R (where X ⊂ Rⁿ ) with the properties that for all (x, y) ∈ X × X, we have

(i)F(x, y; ·) is a convex function, (ii)F(x, y; 0) ≧ 0.

If F satisfies (i) and (ii), we obviously have F(x, y; -a) ≧ - F(x, y; a) for any a ∈ Rⁿ .

For example, F(x, y; a) = M₁||a|| + M₂||a||2, where a depends on x and y, M₁, M₂ are positive constants. This function satisfies (i) and (ii), but it is neither subadditive, nor positive homogeneous, that is, the relations

(i')F(x, y; a + b) ≦ F(x, y; a) + F(x, y; b), (ii')F(x, y; ra) = rF(x, y; a) are not fulfilled for any a, b ∈ Rⁿ and r ∈ R₊. We may conclude that the class of functions that verify (i) and (ii) is more general than the class of sublinear functions with respect the third argument, i.e. those which satisfy (I') and (ii'). We notice that till now, most results in optimization theory were stated under generalized convexity assumptions involving the functions F which are sublinear. The results of this paper are obtained by using weaker assumptions with respect to the above function F.

Throughout the paper, we always assume that F, G : X × X × Rⁿ → R satisfy (i) and (ii).

Definition 2.2 Let X ⊂ Rⁿ , Y ⊂ R^m . f(·, y) is said to be F-convex at , for fixed y ∈ Y, if

Definition 2.3 Let X ⊂ Rⁿ , Y ⊂ R^m . f(x,·) is said to be F-concave at , for fixed x ∈ X, if

Definition 2.4 Let X ⊂ Rⁿ , Y ⊂ R^m . f(·, y) is said to be F-pseudoconvex at , for fixed y ∈ Y, if

Definition 2.5 Let X ⊂ Rⁿ , Y ⊂ R^m . f(x,·) is said to be F-pseudoconcave at , for fixed x ∈ X, if

3 Mixed type multiobjective symmetric duality

For N = {1, 2,..., n} and M = {1, 2,..., m}, let J₁ ⊂ N, K₁ ⊂ M and J₂ = N\J₁ and K₂ = M\K₁. Let |J₁| denote the number of elements in the set J₁. The other numbers |J₂|, |K₁| and |K₂| are defined similarly. Notice that if J₁ = ∅, then J₂ = N, that is, |J₁| = 0 and |J₂| = n. Hence, is zero-dimensional Euclidean space and is n-dimensional Euclidean space. It is clear that any x ∈ Rⁿ can be written as x = (x¹, x²), , . Similarly, any y ∈ R^m can be written as y = (y¹, y²), , . Let and be twice continuously differentiable functions and e = (1, 1,..., 1) ∈ R^l.

Now we can introduce the following pair of non-differentiable multiobjective programs and discuss their duality theorems under some mild assumptions of generalized convexity.

Primal problem (MP):

(1)

(2)

(3)

(4)

(5)

(6)

(7)

Dual problem (MD):

(8)

(9)

(10)

(11)

(12)

(13)

(14)

where

and is a compact and convex subset of for i = i = 1, 2,..., l and is a compact and convex subset of for i = 1, 2,..., l. Similarly, is a compact and convex subset of for i = 1, 2,..., l and is a compact and convex subset of for i = 1, 2,..., l.

Theorem 3.1(Weak duality). Let (x¹, x², y¹, y², z¹, z², λ) be feasible for (MP) and (u¹, u², v¹, v², w¹, w², λ) be feasible for (MD). Suppose that for i = 1, 2,..., l, is F₁-convex for fixed v¹, is F₂-concave for fixed x₁, is G₁-convex for fixed v² and is G₂-concave for fixed x², and the following conditions are satisfied:

• (I) F ₁(x ¹, u ¹; a) + (u ¹) ^Ta ≧ 0 if a ≧ 0;

• (II) G ₁(x ², u ²; b) + (u ²) ^Tb ≧ 0 if b ≧ 0;

• (III) F ₂(v ¹, y ¹; c) + (y ¹) ^Tc ≧ 0 if c ≧ 0; and

• (IV) G ₂(v ², y ²; d) + (y ²) ^Td ≧ 0 if d ≧ 0.

Then H(x¹, x², y¹, y², z¹, z², λ) ≰ G(u¹, u², v¹, v², w¹, w², λ).

Proof. Assume that the result is not true, that is H(x¹, x², y¹, y², z¹, z², λ) ≤ G(u¹, u², v¹, v², w¹, w², λ). Then, since λ > 0, we have

(15)

By the F 1-convexity of , we have

, for i = 1,2,..., l.

From (7), (14) and F₁ satisfying (i) and (ii), the above inequality yields

(16)

By the duality constraint (8) and conditions (I), we get

From (10), (16) and the above inequality, we obtain

(17)

By the F₂-concavity of , we have, for i = 1, 2,..., l,

From (7), (14) and F₂ satisfying (i) and (ii), the above inequality yields

(18)

By the primal constraint (1) and conditions (III), we get

From (3), (18) and the above inequality, we obtain

(19)

Using and for i = 1, 2,..., l, it follows from (17) and (19), that

(20)

Similarly, by the G₁-convexity of and G₂-concavity of , for i = 1, 2,..., l, and condition (II) and (IV), we get

(21)

From (20) and (21), we have

which is a contradiction to (15). Hence H(x¹, x², y¹, y², z¹, z², λ) ≰ G(u¹, u², v¹, v², w¹, w², λ).

Remark 3.1. Theorem 3.1 can be established for more general classes of functions such as F₁-pseudoconvexity and F₂-pseudoconcavity, and G₁-pseudoconvexity and G₂-pseudoconcavity on the functions involved in the above theorem. The proofs will follow the same lines as that of Theorem 3.1.

Strong duality theorem for the given model can be established on the lines of the proof of Theorem 2 of Yang et al. [9].

Theorem 3.2(Strong duality). Let be an efficient solution for (MP), fix in (MD), and suppose that

(A1) either the matrices and are positive definite; or and are negative definite; and

(A2) the sets and are linearly independent.

Then is feasible for (MD) and the corresponding objective function values are equal. If in addition the hypotheses of Theorem 3.1 hold, then there exist , such that is an efficient solution for (MD).

Mishra et al. [10] gave weak and strong duality theorems for the mixed model. However, we note that they did not discuss the converse duality theorem for the mixed dual model. Here, we will give a converse duality theorem for the model under some weaker assumptions.

Theorem 3.3(Converse duality). Let be an efficient solution for (MD), in (MP), and suppose that

(B1) either the matrices and are positive definite; or and are negative definite; and

(B2) the sets and are linearly independent.

Then is feasible for (MP) and the corresponding objective function values are equal. If in addition the hypotheses of Theorem 3.1 hold, then there exist , such that is an efficient solution for (MP).

Proof. Since be an efficient solution for (MD), by the modifying Fritz-John conditions [7], there exist α ∈ R^l , , , β₁ ∈ R, β₂ ∈ R, , , δ ∈ R^l such that

(22)

(23)

(24)

(25)

(26)

(27)

(28)

(29)

(30)

(31)

(32)

(33)

(34)

(35)

(36)

(37)

(38)

From (22) and (23), we get

(39)

From (31)-(34), we have

(40)

Substituting (40) into (39), we obtain

Since λ > 0, it follows from (37), that δ = 0. From δ = 0 and (30), the above equation yields

(41)

From (A1) and (41), we obtain

(42)

From (22), (23), (42) and (A2), we get

(43)

If β₁ = 0, then from (43) and (42), β₂ = 0, α = 0, α₁ = 0, α₂ = 0, and from (24) and (26), μ₁ = 0, μ₂ = 0. This contradicts (38). Hence β₁ = β₂ > 0 and α > 0.

From (38) and (42), we have

(44)

By (24), (38) and (43), we have

(45)

By (26), (38) and (43), we have

(46)

From (24), (35), (42) and (43), we have

(47)

From (26), (36), (42) and (43), we have

(48)

Hence from (12)-(14) and (44)-(48), is feasible for (MP). Now from (28), (42) and α > 0, we have , i = 1, 2,..., l, that is

(49)

From (29), (42) and α > 0, we have

(50)

Finally, from (25), (27), (49) and (50), for all i = 1, 2,..., l, we give,

(51)

Thus . By the weak duality and (51), is an efficient solution for (MD).

4 Special cases

In this section, we consider some special cases of problems (MP) and (MD) by choosing particular forms of compact convex sets, and the number of objective and constraint functions:

• (i) If F(x, y; ·) is sublinear, then (MP) and (MD) reduce to the pair of problems (MP2) and (MD2) studied in Mishra et al. [10].

• (ii) If F(x, y; ·) is sublinear, |J ₂| = 0, |K ₂| = 0 and l = 1, then (MP) and (MD) reduce to the pair of problems (P1) and (D1) of Mond and Schechter [7]. Thus (MP) and (MD) become multiobjective extension of the pair of problems (P1) and (D1) in [7].

• (iii) If F(x, y; ·) is sublinear and l = 1, then (MP) and (MD) are an extension of the pair of problems studied in Yang et al. [9].

• (iv) From the symmetry of primal and dual problems (MP) and (MD), we can construct other new symmetric dual pairs. For example, if we take and , where , , i = 1,2,,..., l, are positive semi definite matrices, then it can be easily verified that , and , i = 1, 2,..., l. Thus, a number of new symmetric dual pairs and duality results can be established.