Abstract

Three approaches of second order mixed type duality are introduced for a nondifferentiable multiobjective fractional programming problem in which the numerator and denominator of objective function contain square root of positive semidefinite quadratic form. Also, the necessary and sufficient conditions of efficient solution for fractional programming are established and a parameterization technique is used to establish duality results under generalized second order -univexity assumption.

1. Introduction

A fractional programming problem arises in many types of optimization problems such as portfolio selection, production, information theory, and numerous decision making problems in management science. More specifically, it can be used in engineering and economics to minimize a ratio of physical or economical function or both, such as cost/time, cost/volume, and cost/benefit, in order to measure the efficiency or productivity of the system. Many economic, noneconomic, and indirect applications of fractional programming problem have also been given by Bector [1], Bector and Chandra [2], Craven [3], Mond and Weir [4], Stancu-Minasian [5], Schaible and Ibaraki [6], Ahmad et al. [7], Ahmad and Sharma [8], and Gulati et al. [9].

The central concept in optimization is known as the duality theory which asserts that, given a (primal) minimization problem, the infimum value of the primal problem cannot be smaller than the supermom value of the associated (dual) maximization problem and the optimal values of primal and dual problems are equal. Duality in fractional programming is an important class of duality theory and several contributions have been made in the past [1, 5, 8, 10–14]. Second order duality provides a tighter bound for the value of the objective function when approximations are used. For more details, one can consult [15, page 93]. Another advantage of second order duality when applicable is that if a feasible point in the primal is given and first order duality does not apply, then we can use second order duality to provide a lower bound of the value of the primal problem (see [4]).

Multiobjective fractional programming duality has been of much interest in the recent past. Schaible [16] and Bector et al. [11] derived Fritz John and Karush-Kuhn Tucker necessary and sufficient optimality condition for a class of nondifferentiable convex multiobjective fractional programming problems and established some duality theorems. Liang et al. [17, 18] discussed the optimality condition and duality for nonlinear fractional programming. Santos et al. [19] discussed the optimality and duality for nonsmooth multiobjective fractional programming with generalized convexity. Bector et al. [20] and Xu [21] gave a mixed type duality for fractional programming, established some sufficient conditions, and obtained various duality results between the mixed dual and primal problem. Zhou and Wang [22] introduced a class of mixed type dual for nonsmooth multiobjective fractional programming and established the duality results under (, ) invexity assumption.

Duality for various forms of mathematical problems involving square roots of positive semidefinite quadratic forms has been discussed by many authors [10, 23–25]. Mond [25] considered a nonlinear fractional programming problem involving square roots of positive semidefinite quadratic form in the numerator and denominator and proved the necessary and sufficient condition for optimality. Kim et al. [26, 27] formulated a nondifferentiable multiobjective fractional problem in which numerators contain support function. One of the most known approaches used for solving nonlinear fractional programming problem is called parametric approach. Dinklebaeh [28] and Jagannathan [12] introduced this approach that was used later by Osuna-Gmez et al. [13] to characterize solution of a multiobjective fractional problem under generalized convexity. Tripathy [14] introduced three approaches given by Dinklebaeh [28] and Jagannathan [12] for both primal and mixed type dual of a nondifferentiable multiobjective fractional programming and established the duality results under generalized -univexity.

To relax convexity assumption imposed on the function in theorems on optimality and duality, various generalized convexity concepts have been proposed. Hanson [29] introduced the class of invex functions. Bector et al. [30] introduced univex function. Mishra [31] derived the optimality condition for multiobjective programming with generalized univexity. Jayswal [32] presented minimax fractional programming under generalized -univexity assumption.

Motivated by the earlier authors in this paper, we have introduced three approaches given by Dinklebaeh [28] and Jagannathan [12] for both primal and second order mixed type dual of a nondifferentiable multiobjective fractional programming problem in which the numerator and denominator of objective function contain square root of positive semidefinite quadratic form. Also we have established the necessary and sufficient optimality condition and used a parameterization technique to establish duality results under generalized -univexity assumption.

2. Notations and Preliminaries

Let be the -dimensional Euclidean space and its nonnegative orthant. The following conventions for inequality will be used throughout this paper. For any , , we denote the following.(i), for all .(ii), for all .Throughout the paper, let be a nonempty open subset of .

Consider the following nondifferentiable multiobjective fractional programming problem.

2.1. Multiobjective Fractional Primal Problem

(i) MFP0. Minimize  where (ii) MFP1. Minimize  where   are fixed parameters.(iii). Minimize ; is -dimensional strictly positive vector, all subject to same constraint where , , and ; , , are differentiable functions, and , are positive semidefinite matrices of order . In the sequel, we assume that and on for .

Let for all feasible solutions of MFP0, MFP1, and and denote , , , and . It is obvious that .

Throughout the paper, consider , , , .

Assume that satisfying or and , . For we can write .

Definition 1. The real differentiable function is said to be second order -univex at with respect to , , and , if

Definition 2. The real differentiable function is said to be second order -pseudounivex at with respect to , , and , if

Definition 3. The real differentiable function is said to be second order -quasiunivex at with respect to , , and , if

Remark 4. If , the above definitions reduce to the definitions of -univex, -pseudounivex, and -quasiunivex as introduced in [14].

Definition 5. A feasible point is said to be efficient for MFP1, if there exists no other feasible point in MFP1 such that , , and for some .

Definition 6 (see [33]). A feasible point is said to be properly efficient for MFP1, if it is efficient and there exist such that, for each and for all feasible point in MFP1 satisfying , we have for some such that .

We assume that , , for all .

Definition 7 (generalized Schwarz Inequality). Let be a positive semidefinite matrix of order . Then, for all , .

The equality holds if for some .

Let and .

Then define the set satisfying any one of the following conditions:(a), , , ;(b), , , ;(c), , , ;(d), , , .

Lemma 8 (see [33]). If is an optimal solution of , then is properly efficient for MFP1.

Lemma 9 (see [12]). is an efficient solution for MFP0 if and only if it is an efficient solution of MFP1 with .

Lemma 10 (see [10] necessary optimality condition). If is an optimal solution of () such that , then there exist , , and such that

Theorem 11 (sufficient optimality condition). Let be a feasible solution of MFP1 and there exist ; , , and satisfying the condition in Lemma 10 at . Furthermore suppose that the following conditions hold.(i) is second order -pseudounivex with respect to , and at , with , where , , , ; , , , , and satisfying .(ii).
Then is an efficient solution of MFP1.

Proof. Suppose that the hypothesis holds.
Since the conditions of Lemma 10 are satisfied, from (9) and (13), we have Also from hypothesis (i), we have
So we can write .
Now for , we can write .
For , we have .
Since is second order -pseudounivex with respect to , and at , we have , and using the properties of , it gives Since , the above inequality implies Suppose that is not efficient solution of MFP1; then there exist such that and , for some .
The above relation, together with the relation , implies that From the relations (5), (11), and (14), we get Consequently (20) and (21) yield This contradicts (18). Hence is an efficient solution for MFP1.

Theorem 12 (sufficient optimality condition). Let be a feasible solution of MFP1 and there exist ; , , and satisfying the condition in Lemma 10 at . Furthermore suppose that the following conditions hold.(i) is second order -pseudounivex with respect to , and at and is second order -quasiunivex with respect to , and at with and where , , , ; , , , and satisfying and .(ii). Then is an efficient solution of .

Proof. Suppose hypothesis holds.
From the relations (5), (11), and (14), we get Also from hypothesis (i), we get .
So we have the following: Hence, the -quasiunivexity of with respect to , and implies the following: From (9), we get Using (25) in (26), we get Since , we get So, we have Since is -pseudounivex with respect to , and , we obtained Using the property of and , we get If were not an efficient solution to MFP1, then, from (20), we have This contradicts (31).
Therefore, is an efficient solution for MFP1.