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On minimax programming problems involving right upperDiniderivative functions
Journal of Inequalities and Applications volume 2014, Article number: 326 (2014)
Abstract
In this paper, we derive necessary and sufficient optimality conditions for a general minimax programming problem involving some classes of generalized convexities with the toolright upperDiniderivative. Moreover, using the concept of optimality conditions, MondWeir type duality theory has been developed for such a minimax programming problem.
MSC:26A51, 49J35, 90C32.
1 Introduction
The minimax approach to optimization theory certainly is not new. It takes its origins in von Neumann’s game theory. The broad spectrum of existing results and applications of minimax theory in the field of optimization is captured in the book edited by Du et al. [1]. Starting with work of Schmittendorf [2], minimax programming problems have been studied by several authors; for example, see [3–9] and the references cited therein.
Convexity is a sufficient but not necessary condition for many important results of mathematical programming, since there are diverse extensions of the notion of convexity bearing the same properties. Moreover, it is well known that a function is convex iff its restriction to each line segment in its domain is convex. This property inspired Ortega and Rheinboldt [10] to introduce an important generalization of convex functions by replacing a line segment joining two points by a continuous arc and called them arcwise connected functions defined on arcwise connected sets.
Following the idea of arcwise convexity, Avriel and Zang [11] introduced Qconnected (QCN) functions and Pconnected (PCN) functions and also they have discussed necessary and sufficient localglobal minimum properties of these functions. Some elementary properties of these functions in terms of their directional derivatives have been studied by Bhatia and Mehra [12]. Bhatia and Mehra [12] also established optimality conditions for scalarvalued nonlinear programming problems involving these functions.
To relax the definition of arcwise convexity in terms of directional derivative recently Yuan and Liu [13] introduced the concept of (\alpha ,\rho )right upperDiniderivative locally arcwise connected with respect to the arc H and established optimality and duality results for a nonlinear multiobjective programming problem. In this paper, we use generalized convex functions, in terms of the right upperDiniderivative to derive necessary and sufficient optimality conditions for a general minimax programming problem and duality results for its MondWeir type dual model.
This paper is structured as follows: Some preliminary concepts and properties regarding generalized convex functions are given in Section 2. In Section 3, we establish necessary and sufficient optimality conditions for a general minimax programming problem involving generalized convex functions. In Section 4, we establish appropriate duality theorems for a MondWeir type dual problem. Finally, in Section 5 we summarize our main results and also point out some further research opportunities.
2 Preliminaries
Let {R}^{n} denote the ndimensional Euclidean space, {R}_{+}^{n} its nonnegative orthant and X\subset {R}^{n}. For a nonempty set Q in a topological vector pace E, \overline{Q} denote the closure of Q and
denotes the dual cone of Q, where {E}^{\ast} is the dual space of E.
For some nonempty subset Y, let {R}^{Y}={\mathrm{\Pi}}_{Y}R denote the product space in a product topology. Then the topological dual space of {R}^{Y} is the generalized finite sequence space consisting of all the functions u:Y\to R with finite support [14]. The set {R}_{+}^{Y}={\mathrm{\Pi}}_{Y}{R}_{+} denote the convex cone of all nonnegative functions on Y. Then the topological dual of {R}_{+}^{Y} is given by
Now, we recall some wellknown results and concepts which will be used in the sequel.
Definition 2.1 [15]
A set X\subset {R}^{n} is said to be an arcwise connected set if, for every {x}_{1}\in X, {x}_{2}\in X, there exists a continuous vectorvalued function {H}_{{x}_{1},{x}_{2}}:[0,1]\to X, called an arc, such that
Definition 2.2 [13]
Let φ be a realvalued function defined on an arcwise connected set X\subset {R}^{n}. Let {x}_{1},{x}_{2}\in X, and {H}_{{x}_{1},{x}_{2}} be the arc connecting {x}_{1} and {x}_{2} in X. The right upperDiniderivative of φ with respect to {H}_{{x}_{2},{x}_{1}}(t) at t=0 is defined as follows:
Using this upperDiniderivative concept, Yuan and Liu [13] introduced a class of functions, which called (\alpha ,\rho )right upperDiniderivative function. For convenience, we use the following notations.
Definition 2.3 [13]
A set X\subset {R}^{n} is said to be locally arcwise connected at \overline{x} if for any x\in X and x\ne \overline{x} there exist a positive number a(x,\overline{x}), with 0<a(x,\overline{x})\le 1, and a continuous arc {H}_{\overline{x},x} such that {H}_{\overline{x},x}(t)\in X for any t\in (0,a(x,\overline{x})).
The set X is locally arcwise connected on X if X is locally arcwise connected at any x\in X.
Definition 2.4 [13]
Let X\subset {R}^{n} be a locally arcwise connected set and \phi :X\to R be a real function defined on X. The function φ is said to be (\alpha ,\rho )right upperDiniderivative locally arcwise connected with respect to H at \overline{x}, if there exist real functions \alpha :X\times X\to R, \rho :X\times X\to R such that
If φ is (\alpha ,\rho )right upperDiniderivative locally arcwise connected with respect to H at \overline{x} for any \overline{x}\in X, then φ is called (\alpha ,\rho )right upperDiniderivative locally arcwise connected with respect to H on X.
Remark 2.1 It revealed by an example given in [13] that there exists a function, which is (\alpha ,\rho )right upperDiniderivative locally arcwise connected but neither dρ(\eta ,\theta )invex [16] nor dinvex [17] nor directional differentially Barcwise connected [15].
Now we define the notions of ρgeneralizedpseudoright upperDiniderivative locally arcwise connected, strictly ρgeneralizedpseudoright upperDiniderivative locally arcwise connected and ρgeneralizedquasiright upperDiniderivative locally arcwise connected functions.
Definition 2.5 The function \phi :X\to R is said to be ρgeneralizedpseudoright upperDiniderivative locally arcwise connected (with respect to H) at \overline{x}, if there exists a real function \rho :X\times X\to R such that
equivalently
The function \phi :X\to R is said to be ρgeneralizedpseudoright upperDiniderivative locally arcwise connected (with respect to H) on X if it is ρgeneralizedpseudoright upperDiniderivative locally arcwise connected (with respect to H) at any \overline{x}\in X.
The following example shows that there exists a function which is ρgeneralizedpseudoright upperDiniderivative locally arcwise connected but not (\alpha ,\rho )right upperDiniderivative locally arcwise connected with respect to the arc H.
Example 2.1 Let X=(1,1) and the function \phi :X\to R be defined by
For any, x,y\in R, defining the arc H:[0,1]\to R by
Note that, by the definition of right upperDiniderivative by (1), for x\in (1,0)\cup (0,1) we have
Let \rho :X\times X\to R be defined by
Now, for \overline{x}=0, it follows that
This means that φ is ρgeneralizedpseudoright upperDiniderivative locally arcwise connected (with respect to H) at \overline{x}=0. But φ is not a (\alpha ,\rho )right upperDiniderivative locally arcwise connected with respect to same arc H and ρ at \overline{x}=0 because for x\in (1,0)\cup (0,1) and \alpha (x,\overline{x})=1, we can see that
Definition 2.6 The function \phi :X\to R is said to be strictly ρgeneralizedpseudoright upperDiniderivative locally arcwise connected (with respect to H) at \overline{x}, if there exists a real function \rho :X\times X\to R such that
equivalently
The function \phi :X\to R is said to be strictly ρgeneralizedpseudoright upperDiniderivative locally arcwise connected (with respect to H) on X if it is strictly ρgeneralizedpseudoright upperDiniderivative locally arcwise connected (with respect to H) at any \overline{x}\in X.
Definition 2.7 The function \phi :X\to R is said to be ρgeneralizedquasiright upperDiniderivative locally arcwise connected with respect to H at \overline{x}, if there exists a real function \rho :X\times X\to R such that
equivalently
The function \phi :X\to R is said to be ρgeneralizedquasiright upperDiniderivative locally arcwise connected (with respect to H) on X if it is ρgeneralizedquasiright upperDiniderivative locally arcwise connected (with respect to H) at any \overline{x}\in X.
The next example shows that there exists a function which is ρgeneralizedquasiright upperDiniderivative locally arcwise connected but neither (\alpha ,\rho )right upperDiniderivative locally arcwise connected nor ρgeneralizedpseudoright upperDiniderivative locally arcwise connected with respect to the arc H.
Example 2.2 Let X=(\frac{\pi}{2},\frac{\pi}{2}) and the function \phi :X\to R be defined by
For any, x,y\in R, defining the arc H:[0,1]\to R by
Clearly, for x\in (\frac{\pi}{2},0)\cup (0,\frac{\pi}{2}) we have
Let \rho :X\times X\to R be defined by
Now, we can easily verify that φ is ρgeneralizedquasiright upperDiniderivative locally arcwise connected (with respect to H) at \overline{x}=0. However, for x\in (\frac{\pi}{2},0)\cup (0,\frac{\pi}{2}), \alpha (x,\overline{x})=1 and \overline{x}=0, we can deduce that
and
Hence, φ is neither (\alpha ,\rho )right upperDiniderivative locally arcwise connected nor ρgeneralizedpseudoright upperDiniderivative locally arcwise connected with respect to same arc H and ρ at \overline{x}=0.
Definition 2.8 [13]
A function f:X\to R is called preinvex (with respect to \eta :X\times X\to {R}^{n}) on X if there exists a vectorvalued function η such that
holds for all x,u\in X and any t\in [0,1].
Definition 2.9 [13]
A function f:X\to R is said to be convexlike if for any x,y\in X and 0\le \theta \le 1, there is z\in X such that
Remark 2.2 The convex and the preinvex functions are convexlike functions.
In the next section we will use the following version of Theorem 2.3 from [9].
Lemma 2.1 Let G:X\times Y\to R and \psi :X\to R, where X and Y are arbitrary nonempty sets. Let the pair (G,\psi ) be convexlike on X. Assume that for some neighborhood U of 0 in {R}^{Y} and a constant \nu >0, the set {\mathrm{\Omega}}_{0}\cap \overline{U}\times (\mathrm{\infty},\nu ] is a nonempty closed subset of {R}^{Y}\times R, where
Then exactly one of the following systems is solvable:

(I)
G(x,y)\le 0, \mathrm{\forall}y\in Y, \psi (x)<0,

(II)
∃ an integer s>0, scalars {\lambda}_{i}\ge 0, 1\le i\le s, \mu \ge 0 and vectors {y}^{i}\in Y, 1\le i\le s, such that ({\lambda}_{1},\dots ,{\lambda}_{s},\mu )\ne 0 and {\sum}_{i=1}^{s}{\lambda}_{i}G(x,{y}^{i})+\mu \psi (x)\ge 0.
3 Optimality conditions
Consider the following general minimax programming problem:
where f:X\times Y\to R, g=({g}_{1},{g}_{2},\dots ,{g}_{m}):X\to {R}^{m}, X is an open arcwise connected subset of {R}^{n}, Y is a compact subset of {R}^{m} and f(x,\cdot ) is continuous on Y for every x\in X. {X}_{0}=\{x\in X{g}_{j}(x)\le 0,\mathrm{\forall}1\le j\le m\} denote the set of feasible solutions of (P).
For x\in X, we define
In view of the continuity of f(x,\cdot ) on Y and compactness of Y, it is clear that Y(x) is nonempty compact subset of Y, \mathrm{\forall}x\in X. Throughout this paper we assume that the right upperDiniderivatives of the functions f(\cdot ,y), {g}_{j}(\cdot ), j\in \{1,2,\dots ,m\} with respect to an arc {H}_{\overline{x},x} at t=0 exist \mathrm{\forall}\overline{x}, x\in X, \mathrm{\forall}y\in Y, and {(df)}^{+}({H}_{\overline{x},x}({0}^{+}),\cdot ) is continuous on Y, \mathrm{\forall}\overline{x}, x\in X. Also assume that {g}_{j}(\cdot ), 1\le j\le m is continuous on X.
The following lemma can be proved without difficulty on the same lines as in Lemma 3.1 (Mehra and Bhatia [9]).
Lemma 3.1 Let \overline{x} be an optimal solution of (P). Then the system
has no solution x\in X.
We now prove the following theorem by using Lemmas 2.1 and 3.1, which gives the necessary optimality conditions for an optimal solution of problem (P).
Theorem 3.1 (Necessary optimality conditions)
Let \overline{x} be an optimal solution of (P). Further, let {(df)}^{+}({H}_{\overline{x},x}({0}^{+})), {(d{g}_{j})}^{+}({H}_{\overline{x},x}({0}^{+})), j\in I(\overline{x}) be convexlike functions of x on X and let there exist a neighborhood U of 0 in {R}^{Y(\overline{x})} and a constant \nu ={({\nu}_{j})}_{j\in I(\overline{x})} such that \mathrm{\Omega}(\overline{x})\cap \overline{U}\times {\mathrm{\Pi}}_{j\in I(\overline{x})}(\mathrm{\infty},{\nu}_{j}] is a nonempty closed set, where
Then there exist an integer s>0, scalars {\lambda}_{i}\ge 0, 1\le i\le s, {\mu}_{j}\ge 0, 1\le j\le m, and vectors {y}^{i}\in Y(\overline{x}), 1\le i\le s, such that
Proof If \overline{x} is an optimal solution of (P) then, by Lemma 3.1, the system (2) has no solution x\in X. But the assumption of Lemma 2.1 also holds and since the system (2) has no solution x\in X, we obtain the result that there exist an integer s>0, scalars {\lambda}_{i}\ge 0, 1\le i\le s, {\mu}_{j}\ge 0, j\in I(\overline{x}), and vectors {y}^{i}\in Y(\overline{x}), 1\le i\le s, such that
and
If we put {\mu}_{j}=0, for j\in J(\overline{x}), by (3) and (4) we obtain the required result. □
Now, we prove the following sufficient optimality conditions for the considered minimax problem (P) under generalized convexity with upperDiniderivative concept.
Theorem 3.2 (Sufficient optimality conditions)
Let \overline{x}\in {X}_{0} and there exist an integer s>0, scalars {\lambda}_{i}\ge 0, 1\le i\le s, {\sum}_{i=1}^{s}{\lambda}_{i}\ne 0, {\mu}_{j}\ge 0, 1\le j\le m, and vectors {y}^{i}\in Y(\overline{x}), 1\le i\le s, such that \alpha (x,\overline{x})>0,
Also, assume that

(i)
for 1\le i\le s, f(\cdot ,{y}^{i}) is (\alpha ,{\overline{\rho}}_{i})right upperDiniderivative locally arcwise connected (with respect to H) at \overline{x},

(ii)
for 1\le j\le m, {g}_{j}(\cdot ) is (\alpha ,{\stackrel{\u02d8}{\rho}}_{j})right upperDiniderivative locally arcwise connected (with respect to H) at \overline{x},

(iii)
{\sum}_{i=1}^{s}{\lambda}_{i}{\overline{\rho}}_{i}(x,\overline{x})+{\sum}_{j=1}^{m}{\mu}_{j}{\stackrel{\u02d8}{\rho}}_{j}(x,\overline{x})\ge 0.
Then \overline{x} is an optimal solution of (P).
Proof Suppose to the contrary that \overline{x} is not an optimal solution of (P). Then there exists an \tilde{x}\in {X}_{0} such that
Further, as {y}^{i}\in Y(\overline{x}), we have
Also, {y}^{i}\in Y, 1\le i\le s, we have
Thus, from the above three inequalities, we get
Using {\lambda}_{i}\ge 0, 1\le i\le s and {\sum}_{i=1}^{s}{\lambda}_{i}\ne 0, we obtain
For \tilde{x}\in {X}_{0}, {\mu}_{j}\ge 0, 1\le j\le m, we have {\mu}_{j}{g}_{j}(\tilde{x})\le 0, which in view of (6) implies that
Now, by (7) and (8) we obtain
On the other hand, from the assumptions that f(\cdot ,{y}^{i}), 1\le i\le s and {g}_{j}(\cdot ), 1\le j\le m are (\alpha ,{\overline{\rho}}_{i}) and (\alpha ,{\stackrel{\u02d8}{\rho}}_{j})right upperDiniderivative locally arcwise connected (with respect to H) at \overline{x}, we have
From (10) and (11) together with {\lambda}_{i}\ge 0, 1\le i\le s, and {\mu}_{j}\ge 0, 1\le j\le m, we get
By (5) and using \alpha (\tilde{x},\overline{x})>0, {\sum}_{i=1}^{s}{\lambda}_{i}{\overline{\rho}}_{i}(\tilde{x},\overline{x})+{\sum}_{j=1}^{m}{\mu}_{j}{\stackrel{\u02d8}{\rho}}_{j}(\tilde{x},\overline{x})\ge 0, it follows that
which is a contradiction to (9). Hence \overline{x} is an optimum solution for (P) and the theorem is proved. □
Theorem 3.3 (Sufficient optimality conditions)
Let \overline{x}\in {X}_{0} and there exist an integer s>0, scalars {\lambda}_{i}\ge 0, 1\le i\le s, {\sum}_{i=1}^{s}{\lambda}_{i}\ne 0, {\mu}_{j}\ge 0, 1\le j\le m and vectors {y}^{i}\in Y(\overline{x}), 1\le i\le s, such that the conditions (5) and (6) of Theorem 3.2 hold. Also, assume that

(i)
{\sum}_{i=1}^{s}{\lambda}_{i}f(\cdot ,{y}^{i}) is \overline{\rho}generalizedpseudoright upperDiniderivative locally arcwise connected (with respect to H) at \overline{x},

(ii)
{\sum}_{j=1}^{m}{\mu}_{j}{g}_{j}(\cdot ) is \stackrel{\u02d8}{\rho}generalizedquasiright upperDiniderivative locally arcwise connected (with respect to H) at \overline{x},

(iii)
\overline{\rho}(x,\overline{x})+\stackrel{\u02d8}{\rho}(x,\overline{x})\ge 0.
Then \overline{x} is an optimal solution of (P).
Proof Suppose to the contrary that \overline{x} is not an optimal solution of (P) and following the proof of Theorem 3.2, we have
which by \overline{\rho}generalizedpseudoright upperDiniderivative locally arcwise connected (with respect to H) of {\sum}_{i=1}^{s}{\lambda}_{i}f(\cdot ,{y}^{i}) at \overline{x}, we have
For \tilde{x}\in {X}_{0}, {\mu}_{j}\ge 0, 1\le j\le m, we have {\mu}_{j}{g}_{j}(\tilde{x})\le 0, which in view of (6) implies that
which by \stackrel{\u02d8}{\rho}generalizedquasiright upperDiniderivative locally arcwise connected (with respect to H) of {\sum}_{j=1}^{m}{\mu}_{j}{g}_{j}(\cdot ) at \overline{x}, we have
By (12) and (13), we get
where the last inequality is according to \overline{\rho}(\tilde{x},\overline{x})+\stackrel{\u02d8}{\rho}(\tilde{x},\overline{x})\ge 0. Therefore,
which is a contradiction to (5). Hence \overline{x} is an optimum solution for (P) and the theorem is proved. □
Theorem 3.4 (Sufficient optimality conditions)
Let \overline{x}\in {X}_{0} and there exist an integer s>0, scalars {\lambda}_{i}\ge 0, 1\le i\le s, {\sum}_{i=1}^{s}{\lambda}_{i}\ne 0, {\mu}_{j}\ge 0, 1\le j\le m, and vectors {y}^{i}\in Y(\overline{x}), 1\le i\le s, such that the conditions (5) and (6) of Theorem 3.2 hold. Also, assume that

(i)
{\sum}_{i=1}^{s}{\lambda}_{i}f(\cdot ,{y}^{i}) is strictly \overline{\rho}generalizedpseudoright upperDiniderivative locally arcwise connected (with respect to H) at \overline{x},

(ii)
{\sum}_{j=1}^{m}{\mu}_{j}{g}_{j}(\cdot ) is \stackrel{\u02d8}{\rho}generalizedquasiright upperDiniderivative locally arcwise connected (with respect to H) at \overline{x},

(iii)
\overline{\rho}(x,\overline{x})+\stackrel{\u02d8}{\rho}(x,\overline{x})\ge 0.
Then \overline{x} is an optimal solution of (P).
Proof The proof follows along similar lines as the proof of Theorem 3.3 and hence is omitted. □
Theorem 3.5 (Sufficient optimality conditions)
Let \overline{x}\in {X}_{0} and there exist an integer s>0, scalars {\lambda}_{i}\ge 0, 1\le i\le s, {\sum}_{i=1}^{s}{\lambda}_{i}\ne 0, {\mu}_{j}\ge 0, 1\le j\le m, and vectors {y}^{i}\in Y(\overline{x}), 1\le i\le s, such that the conditions (5) and (6) of Theorem 3.2 hold. Also, assume that

(i)
{\sum}_{i=1}^{s}{\lambda}_{i}f(\cdot ,{y}^{i}) is \overline{\rho}generalizedquasiright upperDiniderivative locally arcwise connected with respect to H at \overline{x},

(ii)
for j\in I(\overline{x}), j\ne l, {g}_{j}(\cdot ) is {\stackrel{\u02d8}{\rho}}_{j}generalizedquasiright upperDiniderivative locally arcwise connected and {g}_{l}(\cdot ) is strictly {\stackrel{\u02d8}{\rho}}_{l}generalizedpseudoright upperDiniderivative locally arcwise connected with respect to H at \overline{x}, with {\mu}_{l}>0,

(iii)
\overline{\rho}(x,\overline{x})+{\sum}_{j=1}^{m}{\mu}_{j}{\stackrel{\u02d8}{\rho}}_{j}(x,\overline{x})\ge 0.
Then \overline{x} is an optimal solution of (P).
Proof Suppose to the contrary that \overline{x} is not an optimal solution of (P) and following the proof of Theorem 3.2, we have
which by \overline{\rho}generalizedquasiright upperDiniderivative locally arcwise connected of {\sum}_{i=1}^{s}{\lambda}_{i}f(\cdot ,{y}^{i}) with respect to H at \overline{x}, we have
Since for \tilde{x}\in {X}_{0} and for j\in I(\overline{x}), we have
which by {\stackrel{\u02d8}{\rho}}_{j}generalizedquasiright upperDiniderivative locally arcwise connected of {g}_{j}(\cdot ), j\in I(\overline{x}), j\ne l and strictly {\stackrel{\u02d8}{\rho}}_{l}generalizedpseudoright upperDiniderivative locally arcwise connected of {g}_{l}(\cdot ) with respect to H at \overline{x}, we have
Since {\mu}_{j}\ge 0, \mathrm{\forall}j\in I(\overline{x}), {\mu}_{l}>0, and {\mu}_{j}=0 for j\in J(\overline{x}), from (15) and (16), we get
By (14) and (17), we get
where the last inequality is according to \overline{\rho}(\tilde{x},\overline{x})+{\sum}_{j=1}^{m}{\mu}_{j}{\stackrel{\u02d8}{\rho}}_{j}(x,\overline{x})\ge 0. Therefore,
which is a contradiction to (5). Hence \overline{x} is an optimum solution for (P) and the theorem is proved. □
4 Duality
This section deals with the duality theorems for the following MondWeir type dual (D) of minimax problem (P):
where \mathbb{K} = {(s,\lambda ,y)s is an integer, \lambda \in {R}_{+}^{s}, {\sum}_{i=1}^{s}{\lambda}_{i}=1, y=({y}^{1},{y}^{2},\dots ,{y}^{s}), {y}^{i}\in Y(x) for some x\in X, 1\le i\le s}, and \mathbb{H}(s,\lambda ,y) denotes the set of all (z,\mu )\in X\times {R}_{+}^{m} satisfying
If for a triplet (s,\lambda ,y) in \mathbb{K} the set \mathbb{H}(s,\lambda ,y) is empty then we define the supremum over it to be −∞.
Theorem 4.1 (Weak duality)
Let x and (z,\mu ,s,\lambda ,y) be feasible solutions of (P) and (D), respectively. Assume that

(i)
{\sum}_{i=1}^{s}{\lambda}_{i}f(\cdot ,{y}^{i}) is \overline{\rho}generalizedpseudoright upperDiniderivative locally arcwise connected (with respect to H) at z,

(ii)
{\sum}_{j=1}^{m}{\mu}_{j}{g}_{j}(\cdot ) is \stackrel{\u02d8}{\rho}generalizedquasiright upperDiniderivative locally arcwise connected (with respect to H) at z,

(iii)
\overline{\rho}(x,z)+\stackrel{\u02d8}{\rho}(x,z)\ge 0.
Then
Proof Suppose to the contrary that
Thus, we have
It follows from {\lambda}_{i}\ge 0, 1\le i\le s, and {\sum}_{i=1}^{s}{\lambda}_{i}=1, that
which by \overline{\rho}generalizedpseudoright upperDiniderivative locally arcwise connected (with respect to H) of {\sum}_{i=1}^{s}{\lambda}_{i}f(\cdot ,{y}^{i}) at z, we have
For x\in {X}_{0}, {\mu}_{j}\ge 0, 1\le j\le m, we have {\mu}_{j}{g}_{j}(x)\le 0, which in view of (19) implies that
which by \stackrel{\u02d8}{\rho}generalizedquasiright upperDiniderivative locally arcwise connected (with respect to H) of {\sum}_{j=1}^{m}{\mu}_{j}{g}_{j}(\cdot ) at z, we have
By (20) and (21), we get
where the last inequality is according to \overline{\rho}(x,z)+\stackrel{\u02d8}{\rho}(x,z)\ge 0. Therefore,
which is a contradiction to (18). Hence the theorem is proved. □
Theorem 4.2 (Strong duality)
Let {x}^{\ast} be an optimal solution of (P). Assume that the conditions of Theorem 3.1 are satisfied. Then there exist ({s}^{\ast},{\lambda}^{\ast},{y}^{\ast})\in \mathbb{K}, and ({x}^{\ast},{\mu}^{\ast})\in \mathbb{H}({s}^{\ast},{\lambda}^{\ast},{y}^{\ast}) such that ({x}^{\ast},{\mu}^{\ast},{s}^{\ast},{\lambda}^{\ast},{y}^{\ast}) is a feasible solution of (D) and the two objectives have same values. If, in addition, the assumption of weak duality Theorem 4.1 hold for all feasible solutions of (D), then ({x}^{\ast},{\mu}^{\ast},{s}^{\ast},{\lambda}^{\ast},{y}^{\ast}) is an optimal solution of (D).
Proof Since {x}^{\ast} is an optimal solution for (P) and all the conditions of Theorem 3.1 are satisfied, there exist ({s}^{\ast},{\lambda}^{\ast},{y}^{\ast})\in \mathbb{K}, and ({x}^{\ast},{\mu}^{\ast})\in \mathbb{H}({s}^{\ast},{\lambda}^{\ast},{y}^{\ast}) such that ({x}^{\ast},{\mu}^{\ast},{s}^{\ast},{\lambda}^{\ast},{y}^{\ast}) is a feasible solution of (D) and the two objective values are equal. The optimality of ({x}^{\ast},{\mu}^{\ast},{s}^{\ast},{\lambda}^{\ast},{y}^{\ast}) for (D) thus follows from Theorem 4.1. □
Theorem 4.3 (Strict converse duality)
Let {x}^{\ast} and ({z}^{\ast},{\mu}^{\ast},{s}^{\ast},{\lambda}^{\ast},{y}^{\ast}) be optimal solutions of (P) and (D), respectively. Assume that the hypothesis of Theorem 4.2 is fulfilled. Also, assume that

(i)
{\sum}_{i=1}^{{s}^{\ast}}{\lambda}_{i}^{\ast}f(\cdot ,{y}^{\ast i}) is strictly \overline{\rho}generalizedpseudoright upperDiniderivative locally arcwise connected (with respect to H) at {z}^{\ast},

(ii)
{\sum}_{j=1}^{m}{\mu}_{j}^{\ast}{g}_{j}(\cdot ) is \stackrel{\u02d8}{\rho}generalizedquasiright upperDiniderivative locally arcwise connected (with respect to H) at {z}^{\ast},

(iii)
\overline{\rho}({x}^{\ast},{z}^{\ast})+\stackrel{\u02d8}{\rho}({x}^{\ast},{z}^{\ast})\ge 0.
Then {z}^{\ast}={x}^{\ast}.
Proof Suppose to the contrary that {z}^{\ast}\ne {x}^{\ast}. According to Theorem 4.2, we know that there exist ({s}^{\ast},{\lambda}^{\ast},{y}^{\ast})\in \mathbb{K}, and ({x}^{\ast},{\mu}^{\ast})\in \mathbb{H}({s}^{\ast},{\lambda}^{\ast},{y}^{\ast}) such that ({x}^{\ast},{\mu}^{\ast},{s}^{\ast},{\lambda}^{\ast},{y}^{\ast}) is a feasible solution of (D) and
Thus, we have
It follows from {\lambda}_{i}^{\ast}\ge 0, 1\le i\le {s}^{\ast}, and {\sum}_{i=1}^{{s}^{\ast}}{\lambda}_{i}^{\ast}=1, that
Now proceeding on the same lines as in Theorem 4.1, we get
which is a contradiction to (18). Hence the theorem is proved. □
5 Conclusion
In this study we have established necessary and sufficient optimality conditions under generalized convexity using the toolright upperDiniderivative for a general minimax programming problem. MondWeir type duality theory is also obtained. These results can be extended to the following semiinfinite minimax programming problem (SIP) with the toolright upperDiniderivative:
where X\subset {R}^{n} is a nonempty open arcwise connected set, Y is a compact metrizable topological space, f(\cdot ,y) is a realvalued function defined on X. {T}_{j} and {S}_{k} are compact subsets of complete metric spaces, for each j\in \underline{q}, {G}_{j}(\cdot ,t) is a realvalued function defined on X for all t\in {T}_{j}, for each k\in \underline{r}, {H}_{k}(\cdot ,s) is a realvalued function defined on X for all s\in {S}_{k}, for each j\in \underline{q} and k\in \underline{r}, {G}_{j}(x,\cdot ), and {H}_{k}(x,\cdot ) are continuous realvalued functions defined, respectively, on {T}_{j} and {S}_{k} for all x\in X. We shall investigate this semiinfinite programming problem in subsequent papers.
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Acknowledgements
The research of the second and fourth author is financially supported by King Fahd University of Petroleum and Minerals, Saudi Arabia under the Internal Research Project No. IN131026.
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Jayswal, A., Ahmad, I., Kummari, K. et al. On minimax programming problems involving right upperDiniderivative functions. J Inequal Appl 2014, 326 (2014). https://doi.org/10.1186/1029242X2014326
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DOI: https://doi.org/10.1186/1029242X2014326
Keywords
 minimax programming
 upperDiniderivative
 optimality
 duality