Open Access

Non-differentiable multiobjective mixed symmetric duality under generalized convexity

Journal of Inequalities and Applications20112011:23

https://doi.org/10.1186/1029-242X-2011-23

Received: 21 January 2011

Accepted: 21 July 2011

Published: 21 July 2011

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.

Keywords

symmetric dualitynon-differentiable nonlinear programminggeneralized convexitysupport function

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 [1214], 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 R n be the n-dimensional Euclidean space and let be its non-negative orthant. The following convention will be used: if x, y R n , then ; ; ; x y is the negation of x y.

Let f(x, y) be a real valued twice differentiable function defined on R n × 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 R n , f i : XR, i = 1, 2,..., p and h : XR 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 R n . 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 R n such that
The subdifferential of s(x|C) is given by
For any set D R n , 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 T y, or equivalently, x ∂s(y|C).

Let us consider a function F : X × X × R n R (where X R n ) 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 R n .

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 R n 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 × R n R satisfy (i) and (ii).

Definition 2.2 Let X R n , Y R m . f(·, y) is said to be F-convex at , for fixed y Y, if
Definition 2.3 Let X R n , Y R m . f(x,·) is said to be F-concave at , for fixed x X, if
Definition 2.4 Let X R n , Y R m . f(·, y) is said to be F-pseudoconvex at , for fixed y Y, if
Definition 2.5 Let X R n , 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 R n 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:
  1. (I)

    F 1(x 1, u 1; a) + (u 1) T a 0 if a 0;

     
  2. (II)

    G 1(x 2, u 2; b) + (u 2) T b 0 if b 0;

     
  3. (III)

    F 2(v 1, y 1; c) + (y 1) T c 0 if c 0; and

     
  4. (IV)

    G 2(v 2, y 2; d) + (y 2) T d 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:
  1. (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].

     
  2. (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].

     
  3. (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].

     
  4. (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.

     

Declarations

Acknowledgements

This study was supported by the Education Committee Project Research Foundation of Chongqing (No.KJ110624), the Doctoral Foundation of Chongqing Normal University (No.10XLB015) and Chongqing Key Lab of Operations Research and System Engineering.

Authors’ Affiliations

(1)
Department of Mathematics, Chongqing Normal University

References

  1. Dorn WS: A symmetric dual theorem for quadratic programming. J Oper Res Soc Jpn 1960, 2: 93–97.Google Scholar
  2. Dantzig GB, Eisenberg E, Cottle RW: Symmetric dual nonlinear programs. Pacific J Math 1965, 15: 809–812.MATHMathSciNetView ArticleGoogle Scholar
  3. Bazaraa MS, Goode JJ: On symmetric duality in nonlinear programming. Oper Res 1973, 21: 1–9. 10.1287/opre.21.1.1MATHMathSciNetView ArticleGoogle Scholar
  4. Devi G: Symmetric duality for nonlinear programming problem involving g-convex functions. Eur J Oper Res 1998, 104: 615–621. 10.1016/S0377-2217(97)00020-9MATHView ArticleGoogle Scholar
  5. Mond B, Weir T: Symmetric duality for nonlinear multiobjective programming. In Recent Developments in Mathematical Programming. Edited by: Kumar S. Gordon and Breach, London; 1991.Google Scholar
  6. Weir T, Mond B: Symmetric and self duality in multiobjective programming. Asia Pacific J Oper Res 1991, 5: 75–87.MathSciNetGoogle Scholar
  7. Mond B, Schechter M: Nondifferentiable symmetric duality. Bull Aust Math Soc 1996, 5: 177–188.MathSciNetView ArticleGoogle Scholar
  8. Hou SH, Yang XM: On second order symmetric duality in nondifferentiable programming. J Math Anal Appl 2001, 255: 491–498. 10.1006/jmaa.2000.7242MATHMathSciNetView ArticleGoogle Scholar
  9. Yang XM, Teo KL, Yang XQ: Mixed symmetric duality in nondifferentiable mathematical programming. Indian J Pure Appl Math 2003, 34: 805–815.MATHMathSciNetGoogle Scholar
  10. Mishra SK, Wang SY, Lai KK, Yang FM: Mixed symmetric duality in nondifferentiable multiobjective mathematical programming. Eur J Oper Res 2007, 181: 1–9. 10.1016/j.ejor.2006.04.041MATHMathSciNetView ArticleGoogle Scholar
  11. Bector CR, Chandra S: Abha: On mixed symmetric duality in multiobjective programming. Opsearch 1999, 36: 399–407.MATHMathSciNetGoogle Scholar
  12. Yang XM, Yang XQ, Teo KL, Hou SH: Second order symmetric duality in non-differentiable multiobjective programming with F-convexity. Eur J Oper Res 2005, 164: 406–416. 10.1016/j.ejor.2003.04.007MATHMathSciNetView ArticleGoogle Scholar
  13. Yang XM, Yang XQ, Teo KL, Hou SH: Multiobjective second order symmetric duality with F-convexity. Eur J Oper Res 2005, 165: 585–591. 10.1016/j.ejor.2004.01.028MATHMathSciNetView ArticleGoogle Scholar
  14. Chen X: Higher-order symmetric duality in nondifferentiable multiobjective programming problems. J Math Anal Appl 2004, 290: 423–435. 10.1016/j.jmaa.2003.10.004MATHMathSciNetView ArticleGoogle Scholar

Copyright

© Li and Gao; licensee Springer. 2011

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.