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We consider the strong and total Lagrange dualities for infinite quasiconvex optimization problems. By using the epigraphs of the

Consider the following infinite optimization problem:

Observe that most works in the literature mentioned above were done under the assumptions that the involved functions are convex. Indeed, in mathematical programming, many of the problems naturally involve nonconvex functions. Recently, the quasiconvex programming, for which the involved functions are quasiconvex, has received much attention (cf. [

Constraint qualifications involving epigraphs of the conjugate functions have been studied extensively. Our main aim in the present paper is to use these constraint qualifications (or their variations) to provide complete characterizations for the strong Lagrange duality and for the total Lagrange duality. It is well known that the Fenchel conjugate provides dual problems of convex minimization problems. In a similar way, different notions of conjugate for quasiconvex functions can be introduced in order to obtain dual problems of quasiconvex minimization problems. Note that the

The paper is organized as follows. The next section contains the necessary notations and preliminary results. In Section

The notations used in this paper are standard (cf. [

Let

Let

Note that (

The Greenberg-Pierskalla subdifferential of

We also define

Let

If

If

The following lemma is known in [

Let

The following example shows that (

Let

On the other hand, take

Unless explicitly stated otherwise, let

The family

the quasi-

the quasi-

the quasi-

Note that

The following implication holds:

Suppose that the quasi-

Conversely, suppose that

Consider the following quasiconvex programming:

Its dual problem is defined by

The following statements are equivalent.

For each

The family

The strong Lagrange duality holds between

It is evident that (i)

Suppose that (i) holds. Let

Conversely, suppose that the strong Lagrange duality holds. To show (i), by Remark

The following two examples illustrate Theorem

Let

Let

(a) In [

(b) Recall from [

Let

In the remainder of this section, we study the total Lagrange duality problem; that is, when does the strong duality hold between

The following assertions are equivalent.

The total Lagrange duality holds between

For each

For each

The family

It is evident that (ii)

Suppose that there exists

Let

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

Donghui Fang was supported in part by the National Natural Science Foundation of China (Grant 11101186) and supported in part by the Scientific Research Fund of Hunan Provincial Education Department (Grant 13B095). Xianfa Luo was supported in part by the Natural Science Foundation of Zhejiang Province (Grant LY12A01029).