1 INTRODUCTION

The importance of minimax problems is well known in optimization theory as they occur in enormous numbers of applications in economics and engineers. Over the last decade much research has been conducted on sufficiency and duality for minimax fractional programming problems, which are not necessarily smooth. The interested reader is referred to^{1}^{), (}^{2}^{), (}^{3}^{), (}^{4}^{), (}^{11}^{), (}^{13}^{), (}^{16}^{), (}^{23}^{), (}^{24}^{), (}^{27} for more information of sufficiency and duality for minimax fractional programming problems and to^{5}^{), (}^{9}^{), (}^{10}^{), (}^{12}^{), (}^{17} for some of its applications in practice.

There exists a generalization of convexity to locally Lipschitz functions, with derivative replaced by the Clarke generalized gradient (see e.g. ^{3}^{), (}^{4}^{), (}^{15}^{), (}^{16}^{), (}^{27}). Antczak and Stasiak^{4} introduced a new class of nonconvex nondifferentiable functions, called locally Lipschitz (Φ, ρ)-invex functions as a generalization of (Φ, ρ)-invexity notion introduced by Caristi et al.^{6}, with the tool Clarke generalized subgradient. Later, Antczak^{3} established parametric and nonparametric optimality conditions and several duality results in the sense of Mond-Weir and Wolfe for a new class of nonconvex nonsmooth minimax programming problems involving nondifferentiable (Φ, ρ)-invex functions. However, the results cannot be applied to generalized fractional programming involving equality constraints.

During the last two decades there has been an extremely rapid development in subdifferential calculus of nonsmooth analysis and which is well recognized for its many applications to optimization theory. The Mordukhovich subdifferential is a highly important notion in nonsmooth analysis and closely related to optimality conditions of locally Lipschitzian functions of optimization theory (see^{19}^{), (}^{22}). The Mordukhovich subdifferential is a closed subset of the Clarke subdifferential and this subdifferentials are in general nonconvex sets, unlike the well-known Clarke subdifferentials. Therefore, from the point of view of optimization and its applications, the descriptions of the optimality conditions and calculus rules in terms of Mordukhovich subdifferentials provide sharp results than those given in terms of the Clarke generalized gradient (see e.g.^{7}^{), (}^{8}^{), (}^{18}).

Sach et al.^{21} observed that the usual notion of invexity is suitable for optimization problem with inequality constraints, but it is not suitable for optimization problem with equality constraints. Therefore, Sach et al.^{21} defined the notion of infine nonsmooth functions for locally Lipschitz functions, which is a generalization of invexity^{14} and studied several characterizations of infineness property. Very recently, Chuong^{7} introduced the concept of *L*-invex-infine functions by employing the limiting/Mordukhovich subdifferential instead of the Clarke subdifferential one which has been used before in the definitions of invex-infine functions^{20}^{), (}^{21}.

Consequently, in the present paper, we concentrate on studying nonsmooth minimax fractional programming problem with inequality and equality constraints to derive optimality conditions and duality results by means of employing *L*-invex-infine functions. Although many efforts have been made on this topic, it still remains a very attractive and challenging area of research. There are several approaches developed in the literature, see^{1}^{), (}^{2}^{), (}^{3}^{), (}^{4}^{), (}^{8}^{), (}^{15}^{), (}^{16}^{), (}^{20}^{), (}^{21}^{), (}^{27} and the references therein.

The summary of the paper is as follows. Section 2 contains basic definitions and a few basic auxiliary results, which will be needed later in the sequel. Section 3 is devoted to the optimality conditions, and in Section 4 we turn to an investigation of the notion of duality for the nonsmooth minimax fractional programming problem. Here we propose a parametric type dual problem and prove weak, strong and strict converse duality theorems. The final Section 5 contains the concluding remarks and further developments.

2 PRELIMINARIES

In this section, we gather for convenience of reference, a number of basic definitions which will be used often throughout the sequel, and recall some auxiliary results.

Let *R ^{n}
* be the

*n*-dimensional Euclidean space and

*R*

^{n}_{+}be its non-negative orthant. Unless otherwise stated, all the spaces in this paper are Banach whose norms are always denoted by ||.||. Given a space

*X*, it's dual is denoted by

*X*

^{*}and the canonical pairing between

*X*and

*X*

^{*}is denoted by 〈...〉. The polar cone of a set

*S*⊂

*X*is defined by

*S*º = {

*u*

^{*}∈

*X*

^{*}: 〈

*u*

^{*},

*u*〉 ≤ 0, ∀

*u*∈

*S*} and the notation

*clS*represents the closure of

*S*.

Definition 2.1 (Mordukhovich^{18}). *Given a multifunction F: X**X ^{*} between a Banach space and its dual, the notation*

*signifies the sequential Painlevé-Kuratowski upper/outer limit with respect to the norm topology of X and the weak ^{*} topology of X^{*}, where the notation*

*indicates the convergence in the weak*

^{*}topology of X^{*}and N denotes the set of all natural numbers.Definition 2.2 (Mordukhovich^{18}). *Given S and ε ≥ 0, define the set of ε-normals to S at ū ∈ S by*

*where u**ū means that u → ū with u ∈ S. When ε = 0, the set*(ū, S) = 0(ū, S) in (1) is a cone called the Fréchet normal cone to S at ū. If ū ∉ S, we put ε(ū, S) = ∅ for all ε ≥ 0.

Definition 2.3 (Mordukhovich(^{18})). The limiting/Mordukhovich normal cone to S at ū ∈ S, denoted by N(ū, S), is obtained from (u, S) by taking the sequential Painlevé-Kuratowski upper limits as

If ū ∉ S, we put N(ū, S) = ∅. Note that one can put ε = 0 in (2) when S is (locally) closed around ū, i.e., there is a neighborhood U of ū such that S ∩ clU is closed (see Mordukhovich(^{18}), Theorem 1.6).

Definition 2.4 (Mordukhovich(^{18})). The limiting/Mordukhovich subdifferential of an extended real-valued function ψ: X → *= [-∞, ∞], at ū ∈ X with |ψ(ū)| < ∞ is defined by*

∂ψ(*ū*) = {*u*
^{*} ∈ *X*
^{*}: (*u*
^{*}, -1) ∈ *N*((*ū*, ψ(*ū*)), epiψ)}.

*where epiψ = {(u, μ) ∈ X × R: μ ≥ ψ(u)}.*

If |ψ(*ū*)| = ∞, one puts ∂ψ(*ū*) = ∅. It is known (cf. Mordukhovich^{18}) that when ψ is a convex function, the above-defined subdifferential coincides with the subdifferential in the sense of convex analysis (cf. Rockafellar^{25}).

Definition 2.5 (Mordukhovich^{18}). *A set S ⊂ X is sequentially normally compact (SNC) at ū ∈ S if for any sequence (ε _{n}, u_{n}, u_{n}
^{*}) ∈ [0, ∞) × S × X^{*} satisfying ε_{n} ↓ 0, u_{n}
*

*ū and u*

^{*}_{n}*0 with u*n ∈*(un, S), one has ||u*n|| → 0 as n → ∞.

In the above definition, εn can be omitted when S is closed around ū. Obviously, this property is automatically satisfied in finite dimensional spaces. The reader is referred to Mordukhovich(^{18}) for various sufficient conditions ensuring the fulfillment of the SNC property.

In the sequel of the paper, assume that S is a nonempty locally closed subset of X, and let I = {1, 2, ..., p}, J = {1, 2, ..., q} and K = {1, 2, ..., r} be index sets. In what follows, S is always assumed to be SNC at the point under consideration.

The problem to be considered in the present analysis is the minimax fractional programming problem of the form:

where the functions fi, gi, i ∈ I, hj, j ∈ J and ℓk, k ∈ K are locally Lipschitz on X. The region where the constraints are satisfied (feasibility region) is given by F = {x ∈ S: hj(x) ≤ 0, j ∈ J, ℓk = 0, k ∈ K} Hereafter, we use the notation, f = (f1, f2, ..., fp), g =(g1, g2, ..., gp), h = (h1, h2, ..., hq) and ℓ = (ℓ1, ℓ2, ..., ℓr).

Definition 2.6. Let Φ = (x) = , x ∈ X. A point ū ∈ F is termed to be a local optimal solution of problem (P) if there is a neighborhood U of ū such that

If the inequality in (3) holds for every u ∈ F, then ū is said to be a global optimal solution (or simply, optimal solution) of problem (P).

For ū ∈ S we put

J(ū) = {j ∈ J: hj(ū) = 0}, K(ū) = {k ∈ K: ℓk(ū) = 0}.

Definition 2.7. The problem (P) is said to satisfy the Constraint qualification (CQ) at ū ∈ S if there do not exist βj ≥ 0, j ∈ J(ū) and γk ≥ 0, k ∈ K(ū) such that βj + γk ≠ 0 and

0 ∈ βj∂hj(ū) + γk(∂ℓk(ū) ∪ ∂(-ℓk)(ū)) + N(ū, S).

Remark 2.1. If we consider ū ∈ F, S = X and all the functions are continuously differentiable, then the above-defined (CQ) reduces to Mangasarian-Fromovitz constraint qualification; see e.g., Mordukhovich(^{18}) for more details.

Now, we define the concept of generalized convexity-affineness type for locally Lipschitz functions as follows on the lines of Chuong(^{7}).

Definition 2.8. We say that (f, -g, h; ℓ) is L-(strictly) invex-infine on S at ū ∈ S if for any x ∈ S, xi* ∈∂fi(ū), yi* ∈ ∂(-gi)(ū), i ∈ I, zj* ∈∂hj(ū), j ∈ J and ξk* ∈ ∂ℓk(ū) ∪ ∂(-ℓk)(ū), k ∈ K there exists v ∈ N(ū, s)º such that

fi(x) - fi(ū)(>) ≥ 〈xi*, v〉, i ∈ I, (x ≠ ū),

-gi(x) + gi(ū) ≥ 〈yi*, v〉, i ∈ I,

hj(x) - hj(ū) ≥ 〈zj*, v〉, j ∈ J,

ℓk(x) - ℓk(ū) = wk〈ξk*, v〉, k ∈ K,

where wk = 1 (respectively, wk = -1) whenever ξk* = ∂ℓk(ū) (respectively, ξk* = ∂(-ℓk)(ū)).

In the subsequent part of this paper, we assume that wk = 1 (respectively, wk = -1) whenever ξk* = ∂ℓk(ū) (respectively, ξk* = ∂(-ℓk)(ū)) and ū ∈ S.

It is well known that the problem (P) is equivalent (see(^{27})) to the following nonfractional parametric problem:

where v ∈ R+ is a parameter.

Lemma 2.1 (Zalmai(^{26})). Problem (P) has an optimal solution at ū with the optimal value *if and only if V(**) = 0 and ū is an optimal solution of (**).*

Lemma 2.2 (Zalmai^{26}). *For each x ∈ F, one has*

3 OPTIMALITY CONDITIONS

In this section, we first derive Karush-Kuhn-Tucker type necessary conditions for (local) optimal solutions of problem (P) and then using the notion of generalized convexity-affineness-type for locally Lipschitz functions, we also establish sufficient optimality conditions.

Theorem 3.1 (Karush-Kuhn-Tucker Type Necessary Conditions). If *is a local optimal solution of problem (P), and the constraints qualification (CQ) is satisfied at*, then there exist *= Φ(**) ∈ R _{+},*

*∈ R*

^{p}_{+}\{0},*∈ R*

^{q}_{+}, and*∈ R*

^{r}_{+}such that

Proof. If is a local optimal solution of problem (P), by Lemma , it is a local optimal solution of () with optimal value . By Theorem 3.3(^{8}), there exist ∈ *R ^{p}
*

_{+}\{0}, ∈

*R*

^{q}_{+}, and ∈

*R*

^{r}_{+}such that the conditions (4)-(6) are satisfied.

Theorem 3.2 (Sufficient Optimality Conditions). *Let (**,*, , ) ∈ F × Rp+\{0} × Rq+ × Rr+ satisfy the relations (4)-(6) and Φ() = *∈ R _{+}. Assume also that (f, -g, h; ℓ) is L-invex-infine on S at*. Then

*is a global optimal solution of problem (P).*

Proof. Since (, , , ) ∈ F × Rp+\{0} × Rq+ × Rr+ satisfy the relations (4)-(6), there exist, x*i ∈ ∂fi(), y*i ∈ ∂(-gi)(), i ∈ I, z*j ∈ ∂hj(), j ∈ J and ξ*k ∈ ∂ℓk() ∪∂(-ℓk)(), k ∈ K such that

Suppose to the contrary that is not a global optimal solution of (P). Then, there exists a feasible solution *x*
_{0} ∈ *F* such that

Using this inequality along with Lemma 2.2 and Φ() = , we get

Consequently, relations (8) and (10) yield

By assumption, (f, -g, h; ℓ) is L-invex-infine on S at . Then, by Definition 2.8, there exists v ∈ N(, s)º such that the following inequalities

hold for any x0 ∈ F, x*i ∈ ∂fi(), y*i ∈ ∂(-gi)(), i ∈ I, z*j ∈ ∂hj() and ξ*k ∈ ∂ℓk() ∪ ∂(-ℓk)(), k ∈ K.

Since ∈ *R*
_{+}, then inequalities (12) and (13) together yield

Multiplying each inequality (16) by , i ∈ I, each inequality (14) by , j ∈ J and each inequality (15) by , k ∈ K, then summing resultant inequalities, we get

Now using the definition of polar cone, it follows from (7) and v ∈ N(, s)º that

By (9), (17), (18) and the fact x0 ∈ F, ∈ *F* we see that

which contradicts (11). This completes the proof.

Now we give an example of minimax fractional programming problem, where to prove optimality the concept of *L*-invexity-infiness may be applied.

Example 3.1. *Consider the problem*

*and let S = R. Note that the set of feasible solutions of (P) is F = R and for**= 0 ∈ F, we have N(**, S) = {0} and N(**, S)º = R. It is easy to see that there exist**∈ R _{+},*

*∈ R*

^{2}_{+}\{0},*∈ R*

_{+}and*∈ R such that the relations*(4)

*-*(6)

*hold. Also, for any x ∈ S, x*

^{*}_{i}∈ ∂f_{i}(*) = {1, -1}, y*

^{*}_{i}= ∈ ∂(-g_{i})(*) = {1, -1}, i = 1, 2; z*

^{*}∈ ∂h(*) = {1, -1} and ξ*

^{*}∈ ∂ℓ(*) ∪ ∂(-ℓ)(*

*) = {0}, by taking v =*

*∈ N(*

*, S)º, it is not difficult to prove that (f, -g, h; ℓ) is L-invex-infine on S at*

*= 0. However, (f, -g, h; ℓ) is not invex-infine*

^{21}

*on S at*

*= 0 (see Chuong*

^{7}

*, Example 3.3). Since all hypotheses of Theorem 3.2 are satisfied, then*

*= 0 is optimal in the considered minimax fractional programming problem.*

4 DUALITY

In this section, we study the following parametric duality model for (P):

(D) max *v*

subject to

where Ω(0, ||γ||) = {σ ∈ *R ^{r}
*: ||σ|| = ||γ||}. We denote by

*W*the set of all feasible solutions (

*y*, α, β, γ, υ) ∈

*S*× Λ×

*R*

^{q}_{+}×

*R*

^{r}_{+}×

*R*

_{+}of problem (D).

The following theorems show that (D) is a dual problem for (P).

Theorem 4.1 (Weak Duality). *Let x ∈ F and (y, α, β, γ, υ) ∈ W. Assume also that (f, -g, h; ℓ) is L-invex-infine on S at y, then Φ(x) ≥ v.*

Proof. Since (*y*, α, β, γ, υ) ∈ *W* satisfy the relations (18)-(22), there exist *x*
^{*}
*
_{i}
* ∈ ∂

*f*(),

_{i}*y*

^{*}

*∈ ∂(-*

_{i}*g*)(),

_{i}*i*∈

*I*,

*z*

^{*}

*∈ ∂*

_{j}*h*(),

_{j}*j*∈

*J*and ξ

^{*}

*∈ ∂ℓ*

_{k}*() ∪ ∂(-ℓ*

_{k}*)(),*

_{k}*k*∈

*K*such that

Suppose to the contrary that

Φ(*x*) < *v*.

Using this inequality along with Lemma 2.2, as in the proof of Theorem 3.2, we get

By assumption, (*f*, -*g*, *h*; ℓ) is *L*-invex-infine on *S* at *y*. Then, by Definition 2.8, there exists *v* ∈ *N*(*y*, *s*)º such that the following inequalities

hold for any *x* ∈ *F*, *x*
^{*}
*
_{i}
* ∈ ∂

*f*(

_{i}*y*),

*y*

^{*}

*∈ ∂(-*

_{i}*g*)(

_{i}*y*),

*i*∈

*I*,

*z*

^{*}

*∈ ∂*

_{j}*h*(

_{j}*y*),

*j*∈

*J*and ξ

^{*}

*∈ ∂ℓ*

_{k}*(*

_{k}*y*) ∪ ∂(-ℓ

*)(*

_{k}*y*),

*k*∈

*K*.

As seen in the proof of Theorem 3.2, the above relations (29)-(32) leads to the following inequality

Thus, by setting σ = , k ∈ K, we have

From (25), (26) and the fact that x ∈ F, above inequality yields

equivalently,

where σ = (σ1, σ2, ..., σr) ∈ Rr Notice that ||σ|| = ||γ|| and thus, by (27), (28) and (34), we get

which contradicts (29). This completes the proof.

Theorem 4.2 (Strong Duality). If *is a local optimal solution of (P), and the constraint qualification (CQ) is satisfied at*, then there exist (, , , ) ∈ Λ × Rq+ × Rr+ × R+ such that (, , , , ) is a feasible solution of (D) and the two objectives have the same values. Assume also that the conditions of Theorem 4.1 hold for all feasible solutions of (D), then (, , *,**,*) is a global optimal solution of (D).

Proof. By assumption, is a local optimal solution of problem (P), and the constraint qualification (CQ) is satisfied at . Then, there exist ∈ *R ^{p}
*

_{+}\{0}, ∈

*R*

^{q}_{+}, ∈

*R*

^{r}_{+}and ∈

*R*

_{+}such that the Karush-Kuhn-Tucker type necessary conditions (Theorem 3.1) are fulfilled at . Thus, we have

Take

It is easy to see that = (_{1}, 2. ..., r) ∈ Λ, = (_{1}, 2, ..., q) ∈ Rq+ and = (_{1}, 2, ..., r) ∈ Rr+.

Observe that the conditions (35)-(37) are also valid when 's, 's, and 's are replaced by 's, 's, and 's, respectively.

Since, ℓk() = 0, k ∈ K for ∈ *F*. Consequently this gives that 〈 - σ, ℓ()〉 for all σ ∈ *R ^{r}
* with ||σ|| = ||||. That is ℓ() ∈ ( - Ω(0, ||||))º. Therefore, (, , , , ) is a feasible solution of (D), moreover, the corresponding objective values of (P) and (D) are equal. The global optimality of (, , , , ) for (D) follows from weak duality Theorem 4.1.

Theorem 5 (Strict Converse Duality). Let *and (**,*, , , ) be optimal solutions of (P) and (D), respectively, and assume that the assumptions of Theorem are fulfilled. Also, assume that (f, -g, h; ℓ) is L-strictly invex-infine on S at , then *=*; that is, *is an optimal solution of (P).*

Proof. Suppose to the contrary that ≠ . By Theorem 4.2, it follows that

Now, proceeding as in Theorem 4.1, we see that the *L*-strictly invex-infine of (*f*, -*g*, *h*; ℓ) on *S* at , yields the following inequality

Using this inequality along with Lemma , we see that

which contradicts (38). This completes the proof.

5 CONCLUSION

In this paper, we have established optimality conditions and duality results for a class of nonsmooth minimax fractional programming problems possessing *L*-invex-infiness property. This paper extends entirely earlier works, in which optimality conditions and duality results have been obtained in terms of the Clarke generalized gradient for a generalized optimization problems (for example, the results of Ahmad^{2}, Antczak^{3}, Zalmai^{26} and Zheng and Cheng^{27}). We are going to extend the results established in the paper to a larger class of nonsmooth variational and nonsmooth control problems. This will orient the future research of the author.