Nonconvex composite multiobjective nonsmooth fractional programming
© Kim and Kim; licensee Springer. 2013
Received: 4 April 2013
Accepted: 25 September 2013
Published: 8 November 2013
We consider nonsmooth multiobjective programs where the objective function is a fractional composition of invex functions and locally Lipschitz and Gâteaux differentiable functions. Kuhn-Tucker necessary and sufficient optimality conditions for weakly efficient solutions are presented. We formulate dual problems and establish weak, strong and converse duality theorems for a weakly efficient solution.
MSC:90C46, 90C29, 90C32.
Keywordscomposite functions fractional programming nonsmooth programming multiobjective problems necessary and sufficient conditions duality
Recently there has been an increasing interest in developing optimality conditions and duality relations for nonsmooth multiobjective programming problems involving locally Lipschitz functions. Many authors have studied under kinds of generalized convexity, and some results have been obtained. Schaible  and Bector et al.  derived some Kuhn-Tucker necessary and sufficient optimality conditions for the multiobjective fractional programming. By using ρ-invexity of a fractional function, Kim  obtained necessary and sufficient optimality conditions and duality theorems for nonsmooth multiobjective fractional programming problems. Lai and Ho  established sufficient optimality conditions for multiobjective fractional programming problems involving exponential V-r-invex Lipschitz functions. In , Kim and Schaible considered nonsmooth multiobjective programming problems with inequality and equality constraints involving locally Lipschitz functions and presented several sufficient optimality conditions under various invexity assumptions and regularity conditions. Soghra Nobakhtian  obtained optimality conditions and a mixed dual model for nonsmooth fractional multiobjective programming problems. Jeyakumar and Yang  considered nonsmooth constrained multiobjective optimization problems where the objective function and the constraints are compositions of convex functions and locally Lipschitz and Gâteaux differentiable functions. Lagrangian necessary conditions and new sufficient optimality conditions for efficient and properly efficient solutions were presented. Mishra and Mukherjee  extended the work of Jeyakumar and Yang  and the constraints are compositions of V-invex functions.
The present article begins with an extension of the results in [7, 8] from the nonfractional to the fractional case. We consider nonsmooth multiobjective programs where the objective functions are fractional compositions of invex functions and locally Lipschitz and Gâteaux differentiable functions. Kuhn-Tucker necessary conditions and sufficient optimality conditions for weakly efficient solutions are presented. We formulate dual problems and establish weak, strong and converse duality theorems for a weakly efficient solution.
Proposition 2.1 
If, in addition, , and if f and −h are regular at x, then equality holds and is regular at x.
C is an open convex subset of a Banach space X,
, , , and , , are real-valued locally Lipschitz functions on , and and are locally Lipschitz and Gâteaux differentiable functions from X into with Gâteaux derivatives and , respectively, but are not necessarily continuously Fréchet differentiable or strictly differentiable ,
, , ,
and are regular.
Definition 2.2 
Definition 2.3 
The following lemma is needed in necessary optimality conditions, weak duality and converse duality.
Lemma 2.1 
If , , and are invex at u with respect to , and and are regular at u, then is V-invex at u with respect to , where .
3 Optimality conditions
Recall that is the upper Dini-directional derivative of at x in the direction of h, and is the Clarke subdifferential of f at . The function in (3.1) is called upper convex approximation of at x, see [11, 12].
Note that for a set C, int C denotes the interior of C, and , denotes the dual cone of C, where is the topological dual space of X. It is also worth noting that for a convex set C, the closure of the cone generated by the set C at a point a, cl , is the tangent cone of C at a, and the dual is the normal cone of C at a, see [9, 13].
Theorem 3.1 (Necessary optimality conditions)
Proof Let , , .
has a solution. So, there exists such that , , , whenever . Since for and is continuous in a neighbourhood of a, there exists such that , whenever , . Let . Then is a feasible solution for (P) and , for sufficiently small α such that .
This contradicts the fact that a is a weakly efficient solution for (P). Hence (3.2) has no solution.
Hence, the conclusion holds. □
We present new conditions under which the optimality conditions (KT) become sufficient for weakly efficient solutions.
The following null space condition is as in :
where , and , , are real positive constants. Let us denote the null space of a function H by .
Recall, from the generalized Farkas lemma , that if and only if . This observation prompts us to define the following general null space condition:
Equivalently, the null space condition means that for each , there exist real constants , , and , , and such that and . For our problem (P), we assume the following generalized null space condition for invex function (GNCI):
For each , there exist real constants , , and , , and such that and .
Note that when and and , the generalized null space condition for invex function (GNCI) reduces to (NC).
Theorem 3.2 (Sufficient optimality conditions)
For the problem (P), assume that , and are invex functions and and are locally Lipschitz and Gâteaux differentiable functions. Let u be feasible for (P). Suppose that the optimality conditions (KT) hold at u. If (GNCI) holds at each feasible point x of (P), then u is a weakly efficient solution of (P).
This is a contradiction and hence u is a weakly efficient solution for (P). □
4 Duality theorems
Theorem 4.1 (Weak duality)
This is a contradiction. The proof is completed by noting that when the conclusion trivially holds. □
Theorem 4.2 (Strong duality)
For the problem (P), assume that the generalized Slater constraint qualification holds. If u is a weakly efficient solution for (P), then there exist , , , such that is a weakly efficient solution for (D).
Since is a feasible solution for (D), is a weakly efficient solution for (D). Hence the result holds. □
Theorem 4.3 (Converse duality)
Let be a weakly efficient solution of (D), and let a be a feasible solution of (P). Assume that , and are invex functions and and are locally Lipschitz and Gâteaux differentiable functions. Moreover, (GNCI) holds with . Then u is a weakly efficient solution of (P).
This is a contradiction. □
This work was supported by a research grant of Pukyong National University (2013). The authors wish to thank the anonymous referees for their suggestions and comments.
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