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We consider a class of nonsmooth generalized semi-infinite programming problems. We apply results from parametric optimization to the lower level problems of generalized semi-infinite programming problems to get estimates for the value functions of the lower level problems and thus derive necessary optimality conditions for generalized semi-infinite programming problems. We also derive some new estimates for the value functions of the lower level problems in terms of generalized differentiation and further obtain the necessary optimality conditions.

Generalized semi-infinite programming problem (GSIP) is of the form

This first systematic study of GSIP was Hettich and Still [

GSIP in itself is of complex and exclusive structures such as the nonclosedness of the feasible set, nonconvexity, nonsmoothness, and bilevel structure and thus a difficult problem to solve see, for example, [

It is obvious that GSIP can be rewritten equivalently as the nonlinear programming problem

The bilevel problem (

In this paper we concentrate on the optimality conditions of nonsmooth GSIP whose defining functions are Lipschitz continuous. Similar works are [

In this section, we present some basic definitions and results from variational analysis [

The Clarke (convexified) normal cone can be defined by two different approaches. On the one hand, it can be defined by the polar cone of the Clarke’s tangent cone

The Clarke subgradients and Clarke horizon subgradients of

Let

The normal cone

If

The Clarke normal cone has the robustness property

It suffices to prove that

The following definitions are required for further development.

Let

Given

Here the concept of

Consider the maximum function of the form

Note that qualification (

The following two results are about continuity properties and estimates of subdifferentials of marginal functions which are crucial to our analysis for GSIP problems.

Consider the parametric optimization problem

One has the inclusions

Continue to consider the parametric problem (

Now we are prepared to develop the optimality conditions for GSIP problem (

Consider the GSIP problem (

Under regularity and Lipschitz continuity, since the following calculus rule for basic subgradients holds (see [

In addition to the assumptions in Theorem

Next we consider the case where the lower level value function

Consider the parametric optimization problem same as (

The proof is divided into two parts. First the set on the right hand of the required inclusion, denoted by

Let

Based on the assumption (iii), we have

Next, we justify that

Employing the sum rule from [

Consider

As mentioned, GSIP can be relaxed into the following bilevel programming problem:

Let

Qualification (

If

The cones

Let

If

Applying Proposition

So, noting that

This completes the proof.

The authors declare that there is no conflict of interests.

This work is supported by the National Science Foundation of China (11001289) and the Key Project of Chinese Ministry of Education (211151).