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We are interested in a nonsmooth minimax programming Problem (SIP). Firstly, we establish the necessary optimality conditions theorems for Problem (SIP) when using the well-known Caratheodory's theorem. Under the Lipschitz

Convexity plays a central role in many aspects of mathematical programming including analysis of stability, sufficient optimality conditions, and duality. Based on convexity assumptions, nonlinear programming problems can be solved efficiently. There have been many attempts to weaken the convexity assumptions in order to treat many practical problems. Therefore, many concepts of generalized convex functions have been introduced and applied to mathematical programming problems in the literature [

Recently, Antczak and Stasiak [

Due to a growing number of theoretical and practical applications, semi-infinite programming has recently become one of the most substantial research areas in applied mathematics and operations research. For more details on semi-infinite programming we refer to the survey papers [

The rest of the paper is organized as follows. In Section

In this section, we provide some definitions and results that we shall use in the sequel. Let

A real-valued function

Let

Note that if a function is locally Lipschitz, then its Clarke subgradient must exist.

The definition of the locally Lipschitz

Let

In order to define an analogous class of (strictly) locally Lipschitz

In this paper, we deal with the nonsmooth semi-infinite minimax programming Problem

If

Consider the nonlinear programming problem

If

To deal with the nonsmooth Problem

We assume that (a) the sets

For any finite subset

Clarke [

Similarly, the maximum function

In this section, we establish not only the necessary optimality conditions theorems but also the sufficient optimality conditions theorems for Problem

Let

Here, one allow the case, where if

Let

By Condition

Let

By Theorem

Now one obtains from the assumptions of Condition

Next, we derive a sufficient optimality conditions theorem for Problem

Let

Suppose, contrary to the result, that

Thus,

Now, we can write the following statement:

By the generalized invexity assumptions of

Employing (

By (

This, together with (

This is a contradiction to condition (

Let

Then,

Note that

Consider

Making use of the optimality conditions of the preceding section, we present dual Problem

Our dual problem

Note that if

Let

Suppose to the contrary that

Thus, we obtain

Note that

We obtain that

Therefore,

Similar to the proof of Theorem

This follows that

Thus, we have a contradiction to (

Let Problem

By Theorem

Let

Suppose to the contrary that

Therefore, we obtain from (

Thus

From the above inequality, we can conclude that there exists

It follows that

On the other hand, we know from Theorem

This contradicts to (

In this paper, we have discussed a nonsmooth semi-infinite minimax programming Problem

The authors declare that there is no conflict of interests regarding the publication of this paper.

This research is supported by the Natural Science Foundation of Guangdong Province (Grant no. S2013010013101) and the Foundation of Hanshan Normal University (Grant nos. QD20131101 and LQ200905).