Minimax fractional programming problem involving nonsmooth g eneralized α-univex functions

a Department of Applied Mathematics, Indian School of Mines, Dhanbad-826004, Jharkhand, India Email: anurag_jais123@yahoo.com b Department of Applied Mathematics, Birla Institute of Technology, Patna Campus, Patna-800014, Bihar, India Email: rajnish_kumar1012@yahoo.com c Department of Applied Mathematics, Birla Institute of Technology, Mesra, Ranchi-835215, Jharkhand, India Email: dilipopsranchi@gmail.com


Introduction
Fractional programming models have become a subject of wide interest since they provide a universal apparatus for a wide class of models.For example, it can be used in engineering, corporate planning, agricultural planning, public policy decision making, financial analysis of a firm, health care, and educational planning.In these sorts of problems the objective function is usually given as a ratio of functions in fractional programming form (see Stancu Minasion [20]).The problems, in which both a minimization and a maximization process of fractional objectives are performed, are usually called in decision science as generalized minimax fractional programming problems.These problems have arisen in game theory [3], goal programming [4], minimum risk problems [21], economics [22] and multiobjective programming [23].
Nonlinear programming problems containing square roots of positive semidefinite quadratic forms have arisen in stochastic programming, in multifacility location problems, and in portfolio selection problems, among others.A fairly extensive list of references pertaining to various aspects of these problems is given in Zalmai [26].Generalizations of convexity related to optimality conditions and duality for minimax fractional programming problems have been of much interest in the recent past and many contributions have been made to this development.For example, see [1,5,[8][9][10][11][12][13][14][15][16][17][18][19][20]24] and the references cited therein.Yadav and Mukherjee [24] formulated two dual models for minimax fractional programming problem and established some duality results.In view of some omissions and inconsistencies in Yadav and Mukherjee [24], Chandra and Kumar [5] constructed two dual models, and proved various duality theorems under convexity assumptions.
The necessary and sufficient conditions for generalized minimax programming were first developed by Schmitendorf [19].Bector and Bhatia [1] relaxed the convexity assumptions in the sufficient optimality condition in [19] and also employed the optimality conditions to construct several dual models which involve pseudo-convex and quasi-convex functions, and derived weak and strong duality theorems.
Liu [13,14] obtained the necessary and sufficient optimality conditions and derived duality theorems for a class of nonsmooth multiobjective fractional programming problems involving    , F -convex and pseudoinvex functions.Lai and Lee [12] focus his study on nondifferentiable minimax fractional programming problems and its two parameterfree dual models.They also established weak, strong and strict converse duality theorems under the assumptions of pseudo/quasi-convex functions.In the formulation of the dual models in [12] optimality conditions given in [11] are used.Zheng and Cheng [25]  To relax the definition of invex function recently Noor [18] introduced the concept of invex functions.Mishra and Rautela [17] study a nondifferentiable minimax fractional programming problem under the assumption of generalized α-type I invex which has been defined in the setting of Clarke's derivative and established sufficient optimality conditions and duality theorems for the three different type of dual problems.
Bector et al. [2] established optimality and duality results for a nonlinear multiobjective programming problem involving univex functions which have been defined by relaxing the definition of an invex function by Bector et al. [2] itself.
In this paper, firstly we introduce the concept of nonsmooth  -univex functions and a counter example is given to show that there exists a function which is nonsmooth  -univex but not  -type I invex given in [17].Then we establish sufficient optimality conditions for nondifferentiable minimax fractional programming problems involving the aforesaid functions.Finally, weak, strong and strict converse duality theorems are discussed in order to relate the efficient solutions of primal problem and its three different types of dual models.
This paper is organized as follows.Section 2 is devoted to some definitions and notations.In Section 3, we derive the sufficient optimality conditions for nondifferentiable minimax fractional programming problems under the assumption of generalized  -univex functions.
Duality results are presented in Sections 4-6.This work extends the works of Mishra and Rautela [17] and partially the results of Jayswal [10] to the nonsmooth case.

Preliminaries
Throughout this paper, let n R be the ndimensional Euclidean space and n R  be its nonnegative orthant.Let X be a nonempty subset of n R .First, we recall the following definitions.
It is well known that the  -invex set need not be a convex set, see Noor [18].Definition 2.5 [18] The function f on the invex set is said to be  -preinvex with respect to , if Note that every convex function is a preinvex function, but the converse is not true.For example, the function The following example shows that  -preinvex , which indicates that F is  -preinvex with respect to  and  on X .
From now onwards, unless otherwise is specified, we assume that X is a nonemptyinvex set with respect to  and  .Consider the following nondifferentiable minimax fractional programming problem:  We shall make use of the following generalized Schwartz inequality: for some the equality holds when In order to relax the convexity assumption in the above problem, we impose the following definitions.Let R X f  : be a locally Lipschitz function.

Definition 2.6
The function f is said to be and  , if there exist , : given in Jayswal [10].
, then we get the definition of  -type I invex given in Mishra and Rautela [17].
It is noted that, not every  -univex function is  -type I invex function [17].We have the following counter-example, which shows that the On the other hand, if we take 0  x , we have which shows that f is not  -type I invex at a with respect to same and  .

Definition 2.7
The function f is said to be pseudo The following example shows that there exists function which is pseudo  -univex but neither  -type I invex nor pseudo -type I invex.
and . But f is neither  -type I invex nor pseudo -type I invex with respect to same and as can be seen by taking a x  .

Definition 2.8
The function f is said to be strict pseudo -univex at and  .

Definition 2.9
The function f is said to be quasi The following example shows that quasi univex function exists.
and  be same as in Example 2.3.However, if we define The following example shows that there exists function which is quasi  -univex but not pseudo  -type I univex not pseudo -type I invex and not  -type I invex.

Example 2.6 The function
and  nor pseudo  -type I invex with respect to  and  on R .Also it can be easily seen that for a x  , f is not  -type I invex with respect to  and  on R .
The following result from [12] is needed in the sequel.

Lemma 2.1 Let 0
x be an optimal solution for (P) satisfying 0, , It should be noted that both the matrices A and B are positive definite at the solution 0 x in the above Lemma.If one of 0 0 , x Ax and 0 0 , x Bx is zero, or both A and B are singular are nonnegative functions defined on

Sufficient Optimality Condition
We now establish sufficient optimality conditions for (P) under the assumptions of generalized univexity discussed in previous section.Theorem 3.1 Suppose that P x   0 be a feasible solution for (P).Suppose that there exist (7).Assume that one of the following conditions holds: x is an optimal solution of (P).
If hypothesis (b) holds, by the positivity of 0 , b and from the inequality (9), we get . By the pseudo  -univexity of  , the above inequality give . (10) From ( 10) and (3), we get By the condition and the positivity of 1 b , (12) gives and the above inequality, we get (11).
For hypothesis (c) the proof is similar to the proof of case (b).This completes the proof.□

First Duality Model
In this section, we consider the following dual to (P): (DI) where 13) -( 16) , , 1 is empty, then we define the supremum over it to be -∞.In this section we denote Proof.Suppose contrary to the result, that is Therefore we get the following relation 2), ( 14), (16) and the above inequality, we get If hypothesis (a) holds, then (by the feasibility of x for (P) and ( 15)).

If hypothesis (b) holds, by the positivity of
and from the inequality (17), we get By the pseudo  -univexity of , the above inequality gives From ( 18) and ( 13), we get Since , , and the positivity of 1 b , the above inequality yield and from the above inequality, we get By the positivity of 1 , which contradicts (19).
For hypothesis (c) the proof is similar to that of the proof given above for case (b).□

Theorem 4.2 (Strong duality). Assume that *
x is an optimal solution for (P) and * x satisfies a constraints qualification for (P).
and the positivity of 0 b , the above inequality yield . Now from (13) and the above inequality, we get Since , , and the positivity of 1 b , the above inequality yield and from the above inequality, we get .
By the positivity of 1 which contradicts to (20).Hence, we get The above inequality contradicts the fact that For hypothesis (b) the proof is similar to that of the proof given above for case (a).□

Second Duality Model
In this section, we formulate the Wolfe-type dual model to problem (P) as follows: where is empty, then we define the supremum over it to be -∞.In this section, we denote Proof.Suppose contrary to the result that for each Following as in [12], we get Now if condition (a) holds, then (21))  (by the feasibility of x for (P) and ( 22)).
Since   If hypothesis (b) holds, by the positivity of 0 , b and from the inequality (26), we get By the pseudo  -univexity of 1  , the above inequality gives From ( 21) and ( 27), we get Since , , By the condition and from the above inequality, we get By the positivity of 1 , which contradicts (28).
The proof is similar when hypothesis (c) holds.This completes the proof.□ Theorem 5.2 (Strong duality).Assume that * x is an optimal solution for (P) and * x satisfies a constraints qualification for (P).
is strictly  -univex with respect to ; that is, z is an optimal solution for (P).Following as in [12], we get By the positivity of 0 b , and from the inequality (31), we get Now from (32) and ( 21), we get By the condition and the positivity of 0 b , from (31), we get

Third Duality Model
In this section we take the following form of Lemma 2.1: Lemma 6.1 Let * x be an optimal solution for (P).

Assume that
Now we consider the following parameter free dual problem for (P):    in the above Theorem 6.1, 6.2 and 6.3 we get Theorem 6.1, 6.2 and 6.3 given in [17].

Conclusion and Further Developments
In this paper, we have introduced the classes of  -univex and generalized  -univex functions where the involved functions are locally Lipschitz, and have used these different classes of functions to derive sufficient optimality conditions and three types of duality results for nondifferentiable minimax fractional programming problems.The results developed in this paper improve and generalize a number of existing results in the literature.In fact, some researchers have paid much attention on extending some known results for univex functions.Hence, for this purpose, we may conclude that this paper enriched optimization theory in the view of mathematics.
Furthermore, the results developed in this paper can be generalized to the following nondifferentiable multiobjective programming problem: semi-definite matrices and Y , an  -invex set, is a compact subset of m R .Let P  be the set of all feasible solutions of (P).For each   , ,

2. 1 ,
then the result of Lemma 2.1 still holds.Throughout the paper, we assume that 0 b and 1 b If the functions g f , and h are continuous differentiable, then Theorem 3.1 above reduces to Theorem 3.1 given in[10].(ii) Evidently, if we choose Theorem 3.1, we get Theorem 3.1 given in[17].

1 
holds, then by the strict and from the above inequality, we get

1 
hypothesis (b) holds, then by the strict pseudo  -univexity of and of the proof is similar to the case of case (a).This completes the proof.□ Remark 5.1 (i) If the functions g f , and h are continuous differentiable, then the above Theorem 5.1, 5.2 and 5.3 reduces to Theorem 5.1, 5.2 and 5.3 given in [10].


, then we define the supremum over it to be -∞.Throughout this section for the sake of simplicity, we denote by Now we shall state weak, strong and strict converse duality theorems without proof as they can be proved in thelight of Theorem 5.1 to Theorem 5.3, proved in the previous section.

Theorem 6 . 1 (
Weak duality) Let P x   be a feasible solution for (P) and let   1 (Weak duality).Let is, z is an optimal solution for [12]lowing as in[12], we get [10] the functions g f , and h are continuous differentiable, then the above Theorem 6.1, 6.2 and 6.3 reduces to Theorem 6.1, 6.2 and 6.3 given in[10].
positive semidefinite symmetric matrix.This will orient the future research of the authors.