Abstract and Applied AnalysisVolume 2011 (2011), Article ID 103597, 14 pagesdoi:10.1155/2011/103597

Research Article

Generalized Second-Order Mixed Symmetric Duality in Nondifferentiable Mathematical Programming

Ravi P. Agarwal,¹ Izhar Ahmad,¹ S. K. Gupta,² and N. Kailey³

¹Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia²Department of Mathematics, Indian Institute of Technology Patna, Patna 800 013, India³School of Mathematics and Computer Applications, Thapar University, Patiala 147 004, India

Received 30 December 2010; Accepted 24 January 2011

Academic Editor: Elena Litsyn

Copyright © 2011 Ravi P. Agarwal et al. Ravi P. Agarwal et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper is concerned with a pair of second-order mixed symmetric dual programs involving nondifferentiable functions. Weak, strong, and converse duality theorems are proved for aforementioned pair using the notion of second-order -convexity/pseudoconvexity assumptions.

1. Introduction

Duality is a fruitful theory in mathematical programming and is useful both theoretically and practically. Duality as used in our daily life means the sort of harmony of two opposite or complementary parts through which they integrate into a whole. Symmetry is bound up with duality and, in particular, is significant in mathematics. The problem of optimizing a numerical function of one or more variables subject to constraints on the variables is called the mathematical programming, or constrained optimization, problem. When either the objective function or one or more of the constraints are nonlinear, the programming problem is called a nonlinear programming problem, a discipline playing an increasingly imperative role in such diverse fields as operations research and management science, engineering, economics, system analysis, and computer science.

Dantzig et al. [1], Mond [2], and Bazaraa and Goode [3] studied symmetric duality in nonlinear programming. Later, Chandra and Husain [4] formulated a pair of Wolfe-type nondifferentiable symmetric dual programs and proved duality results under convexity/concavity assumptions. Subsequently, Chandra et al. [5] weakened these assumptions to pseudoconvexity/pseudoconcavity. Mond and Schechter [6] presented two symmetric dual pairs involving nondifferentiable functions. Kumar and Bhatia [7] discussed multiobjective symmetric duality by using a nonlinear vector-valued function of two variables corresponding to various objectives.

Mangasarian [8] presented a dual problem associated with a primal nonlinear programming problem that involves second derivatives of the function constituting the primal problem. The study of second-order duality is significant, as it can provide a lower bound to the infimum of a primal optimization problem when it is difficult to find a feasible solution for the first-order dual. Bector and Chandra [9] achieved duality results for a pair of Mond-Weir-type second-order symmetric dual nonlinear programs. Hou and Yang [10] formulated a pair of second-order symmetric dual nondifferentiable programs and established duality theorems under second-order -pseudoconvexity assumptions.

Chandra et al. [11] and Yang et al. [12] discussed a mixed symmetric dual formulation for a nonlinear programming problem and for a class of nondifferentiable nonlinear programming problems, respectively. Later on, Ahmad [13] formulated mixed type symmetric dual in multiobjective programming problems ignoring nonnegativity restrictions of Bector et al. [14].

In this paper, a pair of second-order mixed symmetric dual programs is presented for a class of nondifferentiable nonlinear programming problems. Weak, strong, and converse duality theorems are proved using the notion of second-order -convexity/pseudoconvexity assumptions. These results generalize the known work in [6, 10–12, 15–17].

2. Preliminaries

In this section, we presented some of the basic definitions used in the paper.

Definition 2.1. Let be a compact convex set in . The support function of is defined by A support function, being convex and everywhere finite, has a subdifferential, that is, there exists such that The subdifferential of is given by For any set , the normal cone to at a point is defined by It can be easily seen that for a compact convex set , is in if and only if , or equivalently, is in .

Definition 2.2. A functional (where ) is sublinear with respect to the third variable if for all ,(i) for all , (ii), for all and for all . Let be a real-valued twice differentiable function.

Definition 2.3. is said to be second-order -convex at with respect to , if for all ,

Definition 2.4. is said to be second-order -pseudoconvex at with respect to , if for all , is second-order -concave/pseudoconcave at with respect to if is second-order -convex/pseudoconvex at with respect to .

3. Second-Order Mixed Nondifferentiable Symmetric Dual Programs

For and , let , , , and . Let denote the number of elements in . The other symbols , and are defined similarly. Let , . Then, any can be written as . Similarly, for , , can be written as . It may be noted here that if , then , , and therefore . In this case, and will be zero-dimensional, -dimensional and -dimensional Euclidean spaces, respectively. The other situations are , or .

Now we formulate the following pair of mixed nondifferentiable second-order symmetric dual programs and discuss their duality results.

Primal problem (SMNP)minimize subject to

Dual problem (SMND)maximize subject to where (i) and are differentiable functions, (ii), , and are compact convex sets in , , and , respectively, (iii), , and .

Theorem 3.1 (Weak duality). Let be feasible for (SMNP) and be feasible for (SMND). Let the sublinear functionals , , and satisfy the following conditions: Suppose that (i) is second-order -convex at , and is second-order -concave at , (ii) is second-order -pseudoconvex at , and is second-order -pseudoconcave at . Then,

Proof. By the second-order -convexity of at and the second-order -concavity of at , we have Since is feasible for primal problem (SMNP) and is feasible for dual problem (SMND), by the dual constraint (3.7), the vector , and so from the hypothesis (A), we obtain Similarly, for the vector .Using (3.14) in (3.12) and (3.15) in (3.13), we have Adding the above two inequalities, we obtain Substituting the values of and in (3.17), we get Using and , we have By hypothesis (C) and the dual constraint (3.8), we obtain which on using the dual constraint (3.9) yields Since is second-order -pseudoconvex at , we have Similarly, from (3.3) and (3.4) and hypothesis (D) along with second-order -pseudoconcavity of at , we get Adding (3.22) and (3.23), we obtain Using and , we have Inequalities (3.19) and (3.25) together yield that is,.

Theorem 3.2 (Strong duality). Let and be differentiable functions, and let be a local optimal solution of (SMNP). Suppose that (i)the matrix is non singular, (ii) is positive definite, or is negative definite, and , (iii), (iv)one of the matrices ,  , is positive or negative definite. Then, there exist and such that is feasible for (SMND), and the objective function values of (SMNP) and (SMND) are equal. Furthermore, if the assumptions of weak duality (Theorem 3.1) are satisfied for all feasible solutions of (SMNP) and (SMND), then and are global optimal solutions for (SMNP) and (SMND), respectively.

Proof. Since is a local optimal solution of (SMNP), there exist , , , , , and such that the following by Fritz John optimality conditions [18] are satisfied at By hypothesis (i), (3.31) gives Since is positive or negative definite, (3.32) yields Suppose that , then (3.42) implies Using (3.42) in (3.30), we get which on using hypothesis (iii) and yields As , therefore the equations and give and , respectively. Further, (3.41) implies . Consequently, , contradicting (3.40). Hence, we have Subtracting (3.35) from (3.34) yields Using (3.42) and (3.46) in the above equation, we get which contradicts hypothesis (ii) unless Equation (3.42) yields Using (3.49) and (3.50) in (3.30), we obtain which on using hypothesis (iii) and (3.49) gives Since , therefore Now, using (3.41) and (3.46) in (3.29), we get which by hypothesis (iv) implies By (3.41) and (3.55), we have Using (3.46), (3.55), and (3.56) in (3.27), we get Equations (3.28), (3.46), (3.49), and (3.50) give and hence, we also have Thus, (, , , ) satisfies the dual constraints from (3.7) to (3.10), and so it is a feasible solution for the dual problem (SMND). Further, using (3.46), (3.55), and (3.56) in (3.33), we obtain Moreover, since and , (3.36) implies so that From (3.37), (3.50), (3.52), and (3.53), we get Since is a compact convex set in , Therefore, using (3.38), (3.39), (3.49), (3.55), (3.57), and (3.60)–(3.63), we obtain that is, the two objective function values are equal.Finally, from Theorem 3.1, we get that and are global optimal solutions for (SMNP) and (SMND), respectively.

Theorem 3.3 (Converse duality). Let and be differentiable functions, and let be a local optimal solution of (SMND). Suppose that (i)the matrix is non singular, (ii) is positive definite and or is negative definite and , (iii), (iv)one of the matrices , , is positive or negative definite. Then, there exist and such that is feasible for (SMNP) and the objective function values of (SMNP), and (SMND) are equal. Furthermore, if the assumptions of weak duality (Theorem 3.1) are satisfied for all feasible solutions of (SMNP) and (SMND), then and are global optimal solutions for (SMND) and (SMNP), respectively.

Proof. It follows on the lines of Theorem 3.2.

4. Special Cases

In this section, we consider some of the special cases of the problems studied in Section 3. (i)If and , then our problems (SMNP) and (SMND) reduce to the programs (PP) and (DP) studied in Gulati and Gupta [17]. (ii)If , , and , then (SMNP) and (SMND) are reduced to the programs (SP) and (SD) studied in Gulati et al. [16] with the omission of nonnegativity constraints from (SP) and (SD). (iii)If , , , and , then (SMNP) and (SMND) become a pair of symmetric nondifferentiable dual programs considered in Mond and Schechter [6] with the omission of nonnegativity constraints from the programs (P) and (D) studied in Mond and Schechter. (iv)If , , , , , and , then the programs (WP) and (WD) of [15] are obtained with the omission of nonnegativity constraints from (WP) and (WD). (v)If and in (SMNP) and (SMND), then the programs studied in [10] are obtained. (vi)If , , , and in (SMNP) and (SMND), then the programs (SP1) and (SD1) of [16] are obtained with the omission of nonnegativity constraints from (SP1) and (SD1). (vii)If , , , and , then (SMNP) and (SMND) become a pair of symmetric nondifferentiable dual programs considered in [6] with the omission of nonnegativity constraints from the programs (P1) and (D1) studied in Mond and Schechter.(viii)If , , , , , and , then (SMNP) and (SMND) become a pair of single objective symmetric differentiable dual programs considered in [15] with the omission of nonnegativity constraints from (MP) and (MD). (ix)By eliminating the second-order and nondifferentiable terms, our problems (SMNP) and (SMND) reduce to the mixed symmetric dual programs studied by Chandra et al. [11] with the omission of , , , and from the programs studied in Chandra et al. [11]. (x)By eliminating the second-order terms, our problems are reduced to the programs (MP) and (MD) studied in [12] with the omission of nonnegativity constraints from (MP) and (MD).

5. Concluding Remarks

It is to be noted that previously known results [6, 10–12, 15–17] are special cases of our study. It is not clear whether the second-order mixed symmetric duality in mathematical programming can be further extended to higher-order multiobjective symmetric dual programs formulated in [19].

Acknowledgments

Ravi P. Agarwal and Izhar Ahmad are thankful to the King Fahd University of Petroleum and Minerals, Saudi Arabia for providing financial support to carry out this research.

References

G. B. Dantzig, E. Eisenberg, and R. W. Cottle, “Symmetric dual nonlinear programs,” Pacific Journal of Mathematics, vol. 15, pp. 809–812, 1965.

B. Mond, “A symmetric dual theorem for non-linear programs,” Quarterly of Applied Mathematics, vol. 23, pp. 265–269, 1965.

M. S. Bazaraa and J. J. Goode, “On symmetric duality in nonlinear programming,” Operations Research, vol. 21, pp. 1–9, 1973.

S. Chandra and I. Husain, “Symmetric dual nondifferentiable programs,” Bulletin of the Australian Mathematical Society, vol. 24, no. 2, pp. 295–307, 1981.

S. Chandra, B. D. Craven, and B. Mond, “Generalized concavity and duality with a square root term,” Optimization, vol. 16, no. 5, pp. 653–662, 1985.

B. Mond and M. Schechter, “Nondifferentiable symmetric duality,” Bulletin of the Australian Mathematical Society, vol. 53, no. 2, pp. 177–188, 1996.

P. Kumar and D. Bhatia, “A note on symmetric duality for multiobjective non-linear program,” Opsearch, vol. 32, no. 3, pp. 172–183, 1995.

O. L. Mangasarian, “Second- and higher-order duality in nonlinear programming,” Journal of Mathematical Analysis and Applications, vol. 51, no. 3, pp. 607–620, 1975.

C. R. Bector and S. Chandra, “Second order symmetric and self-dual programs,” Opsearch, vol. 23, no. 2, pp. 89–95, 1986.

S. H. Hou and X. M. Yang, “On second-order symmetric duality in nondifferentiable programming,” Journal of Mathematical Analysis and Applications, vol. 255, no. 2, pp. 491–498, 2001.

S. Chandra, I. Husain, and Abha, “On mixed symmetric duality in mathematical programming,” Opsearch, vol. 36, no. 2, pp. 165–171, 1999.

X. M. Yang, K. L. Teo, and X. Q. Yang, “Mixed symmetric duality in nondifferentiable mathematical programming,” Indian Journal of Pure and Applied Mathematics, vol. 34, no. 5, pp. 805–815, 2003.

I. Ahmad, “Multiobjective mixed symmetric duality with invexity,” New Zealand Journal of Mathematics, vol. 34, no. 1, pp. 1–9, 2005.

C. R. Bector, S. Chandra, and Abha, “On mixed symmetric duality in multiobjective programming,” Opsearch, vol. 36, no. 4, pp. 399–407, 1999.

S. Chandra, A. Goyal, and I. Husain, “On symmetric duality in mathematical programming with -convexity,” Optimization, vol. 43, no. 1, pp. 1–18, 1998.

T. R. Gulati, I. Ahmad, and I. Husain, “Second order symmetric duality with generalized convexity,” Opsearch, vol. 38, no. 2, pp. 210–222, 2001.

T. R. Gulati and S. K. Gupta, “Wolfe type second-order symmetric duality in nondifferentiable programming,” Journal of Mathematical Analysis and Applications, vol. 310, no. 1, pp. 247–253, 2005.

M. Schechter, “More on subgradient duality,” Journal of Mathematical Analysis and Applications, vol. 71, no. 1, pp. 251–262, 1979.

R. P. Agarwal, I. Ahmad, and A. Jayswal, “Higher order symmetric duality in nondifferentiable multi-objective programming problems involving generalized cone convex functions,” Mathematical and Computer Modelling, vol. 52, no. 9-10, pp. 1644–1650, 2010.