Stable Zero Lagrange Duality for DC Conic Programming

Abstract

We consider the problems of minimizing a DC function under a cone-convex constraint and a set constraint. By using the infimal convolution of the conjugate functions, we present a new constraint qualification which completely characterizes the Farkas-type lemma and the stable zero Lagrange duality gap property for DC conical programming problems in locally convex spaces.

1. Introduction

Let 𝑋 and 𝑌 be real locally convex Hausdorff topological vector spaces and 𝐶⊆𝑋 be a nonempty convex set. Let 𝑆⊆𝑌 be a closed convex cone and 𝑆^⊕ the positive dual cone of 𝑆. Let 𝜑∶𝑋→ℝ∶=ℝ∪{+∞} be a proper function and ℎ∶𝑋→𝑌 be an 𝑆-convex mapping with respect to the cone 𝑆. Consider the conic programming problem (𝒫)𝜑(𝑥),(𝑥)Mins.t.𝑥∈𝐶,ℎ∈−𝑆. (1.1) Its Lagrange dual problem can be expressed as (𝒟)sup_{𝜆∈𝑆⊕}inf_𝑥∈𝐶𝜑(𝑥)+(𝜆ℎ)(𝑥). (1.2)

It is well know that the optimal values of these problems, 𝑣(𝒫), and 𝑣(𝒟) respectively, satisfy the so-called weak duality, that is, 𝑣(𝒫)≥𝑣(𝒟), but a duality gap may occur, that is, we may have 𝑣(𝒫)>𝑣(𝒟). A challenge in convex analysis has been to give sufficient conditions which guarantee the strong duality, that is, 𝑣(𝒫)=𝑣(𝒟) and the dual problem (𝒟) has at least an optimal solution. In the case when 𝜑 is a proper convex function, numerous conditions have been given in the literature ensuring the strong duality (see, e.g., [1–8] and the other references therein).

Recently, the zero duality, that is, only the situation when 𝑣(𝒫)=𝑣(𝒟), has received much attention (e.g., see [9–13] and references therein). Obviously, the strong duality implies the zero duality. However, the converse implication does not always hold. As mentioned in [10], the question of finding condition, which ensures the zero duality, is not only important for understanding the fundamental feature of convex programming but also for the efficient development of numerical schemes. Some sufficient conditions and characterizations in terms of the optimal value function of (𝒫) for the zero duality have been given in [12], and some convex programming problems which enjoy zero duality have been studied in [13]. Especially, in the case when 𝜑 is lower semicontinuous (lsc in brief) convex, ℎ is star lsc and 𝐶 is closed; Jeyakumar and Li in [10] presented some constraint qualifications which completely characterize the zero duality for convex programming problems in Banach spaces; they established necessary and sufficient dual conditions for the stable zero duality in [11] under the assumptions that 𝐶=𝑋 and ℎ is continuous.

Observe that most works dealing with problem (1.1) in the literature mentioned above were done under the assumptions that the involved functions are convex and lsc. In this paper, we consider the following DC conical programming: (𝑃)𝑓(𝑥)(𝑥),(𝑥)Mins.t.−𝑔𝑥∈𝐶,ℎ∈−𝑆, (1.3) and its dual problem (𝐷)inf_{𝑢∗∈dom𝑔∗}sup_{𝜆∈𝑆⊕}𝑔^∗𝑢^∗−𝑓+𝛿_𝐶+𝜆𝑔^∗𝑢^∗, (1.4) where 𝑓,𝑔∶𝑋→ℝ are proper convex functions. As pointed out in [14], problems of DC programming are highly important from both viewpoints of optimization theory and applications, and they have been extensively studied in the literature (cf. [14–21] and the references therein). Here and throughout the whole paper, following [22, page 39], we adopt the convention that (+∞)+(−∞)=(+∞)−(+∞)=+∞, 0⋅(+∞)=+∞ and 0⋅(−∞)=0. Then, for any two proper convex functions ℎ₁,ℎ₂∶𝑋→ℝ, we have that ℎ₁(𝑥)−ℎ₂(𝑥)∈ℝ,𝑥∈domℎ₁∩domℎ₂,=−∞,𝑥∈domℎ₁⧵domℎ₂,=+∞,𝑥∉domℎ₁; (1.5) hence, ℎ₁−ℎ₂isproper⟺domℎ₁⊆domℎ₂. (1.6)

The purpose of this paper is to study the stable zero duality. Our main contribution is to provide complete characterizations for the stable zero duality between (𝑃) and (𝐷) via the newly constraint qualifications. In general, we only assume that 𝑓,𝑔 are proper convex and ℎ is 𝑆-convex (not necessarily lsc).

The paper is organized as follows. The next section contains some necessary notations and preliminary results. The Farkas-type lemma and the stable zero duality between (𝑃) and (𝐷) are considered in Section 3.

2. Notations and Preliminary Results

The notation used in the present paper is standard (cf. [22]). In particular, we assume throughout the whole paper that 𝑋 and 𝑌 are real locally convex Hausdorff topological vector spaces, and let 𝑋^∗ denote the dual space, endowed with the weak^∗-topology 𝑤^∗(𝑋^∗,𝑋). By ⟨𝑥^∗,𝑥⟩, we will denote the value of the functional 𝑥^∗∈𝑋^∗ at 𝑥∈𝑋, that is, ⟨𝑥^∗,𝑥⟩=𝑥^∗(𝑥). Let 𝑍 be a set in 𝑋. The closure of 𝑍 is denoted by cl𝑍. If 𝑊⊆𝑋^∗, then cl𝑊 denotes the weak^∗-closure of 𝑊. For the whole paper, we endow 𝑋^∗×ℝ with the product topology of 𝑤^∗(𝑋^∗,𝑋) and the usual Euclidean topology.

The indicator function 𝛿_𝑍 of the nonempty set 𝑍 is defined by 𝛿_𝑍(𝑥)∶=0𝑥∈𝑍,+∞otherwise. (2.1) Let 𝑓∶𝑋→ℝ be a proper convex function. The effective domain, the conjugate function, and the epigraph of 𝑓 are denoted by dom𝑓, 𝑓^∗ and epi𝑓, respectively; they are defined by (𝑥),𝑓dom𝑓∶=𝑥∈𝑋∶𝑓<+∞_∗𝑥^∗∶=sup𝑥^∗,𝑥−𝑓(𝑥)∶𝑥∈𝑋foreach𝑥^∗∈𝑋^∗,epi𝑓∶=(𝑥,𝑟)∈𝑋×ℝ∶𝑓(𝑥)≤𝑟. (2.2) It is well known and easy to verify that epi𝑓^∗ is weak^∗-closed. The lsc hull of 𝑓, denoted by cl𝑓, is defined by epicl𝑓=clepi𝑓. (2.3) Then (cf. [22, Theorems 2.3.1]), 𝑓^∗=cl𝑓^∗. (2.4) By definition, the Young-Fenchel inequality below holds: 𝑓(𝑥)+𝑓^∗𝑥^∗≥𝑥,𝑥^∗foreachpair𝑥,𝑥^∗∈𝑋×𝑋^∗. (2.5) If 𝑔,ℎ are proper, then epi𝑔^∗+epiℎ^∗⊆epi𝑔+ℎ^∗,𝑔≤ℎ⇒𝑔^∗≥ℎ^∗⟺epi𝑔^∗⊆epiℎ^∗. (2.6)(2.7) Moreover, if 𝑔 is convex and lsc on domℎ, then the same argument for the proof of [21, Lemma 2.3] shows that epiℎ−𝑔^∗=_{𝑥∗∈dom𝑔∗}epiℎ^∗−𝑥^∗,𝑔^∗𝑥^∗. (2.8) Furthermore, we define the infimal convolution of 𝑔 and ℎ as the function 𝑔□ℎ∶𝑋→ℝ∪{±∞} given by 𝑔□ℎ(𝑥)∶=inf_𝑧∈𝑋𝑔(𝑧)+ℎ(𝑥−𝑧). (2.9) If 𝑔 and ℎ are lsc and dom𝑔∩domℎ≠∅, then by [22], we have that 𝑔□ℎ^∗=𝑔^∗+ℎ^∗,𝑔+ℎ^∗=cl𝑔^∗□ℎ^∗. (2.10) Moreover, we also have epi𝑔^∗+epiℎ^∗⊆epi𝑔^∗□ℎ^∗⊆clepi𝑔^∗+epiℎ^∗. (2.11) Note that an element 𝑝∈𝑋^∗ can be naturally regarded as a function on 𝑋 in such a way that 𝑝(𝑥)∶=𝑝,𝑥foreach𝑥∈𝑋. (2.12) Thus, the following facts are clear for any 𝑎∈ℝ and any function ℎ∶𝑋→ℝ: ℎ+𝑝+𝑎^∗𝑥^∗=ℎ^∗𝑥^∗−𝑝−𝑎foreach𝑥^∗∈𝑋^∗;epiℎ+𝑝+𝑎^∗=epiℎ^∗+𝑝,−𝑎. (2.13)(2.14)

We end this section with a lemma, which is known in [3, 22].

Lemma 2.1. Let 𝑔,ℎ∶𝑋→ℝ be proper convex functions satisfying dom𝑔∩domℎ≠∅.(i)If 𝑔,ℎ are lsc, then epi𝑔+ℎ^∗=clepi𝑔^∗+epiℎ^∗. (2.15)(ii)If either 𝑔 or ℎ is continuous at some point of dom𝑔∩domℎ, then epi𝑔+ℎ^∗=epi𝑔^∗+epiℎ^∗. (2.16)

3. Characterizations for the Stable Zero Duality

Throughout this section, let 𝑋,𝑌 be locally convex spaces and 𝐶⊆𝑋 be a nonempty convex set. Let 𝑆⊆𝑌 be a closed convex cone. Its dual cone 𝑆^⊕ is defined by 𝑆^⊕∶=𝑦^∗∈𝑌^∗∶𝑦^∗,𝑦≥0foreach𝑦∈𝑆. (3.1) Define an order on 𝑌 by saying that 𝑦≤_𝑆𝑥 if 𝑦−𝑥∈−𝑆. We attach a greatest element ∞ with respect to ≤_𝑆 and denote 𝑌^•∶=𝑌∪{+∞}. The following operations are defined on 𝑌^•: for any 𝑦∈𝑌, 𝑦+∞=∞+𝑦=∞ and 𝑡∞=∞ for any 𝑡≥0. Let 𝑓,𝑔∶𝑋→ℝ be proper convex functions such that cl𝑔 and 𝑓−𝑔 are proper, and ℎ∶𝑋→𝑌^• be 𝑆-convex in the sense that for every 𝑢,𝑣∈dom𝐺 and every 𝑡∈[0,1], ℎ(𝑡𝑢+(1−𝑡)𝑣)≤_𝑆𝑡ℎ(𝑢)+(1−𝑡)ℎ(𝑣), (3.2) (see [6]). Let 𝜆∈𝑆^⊕ and let domℎ∶={𝑥∈𝑋∶ℎ(𝑥)∈𝑌}≠∅. As in [3], we define for each 𝜆∈𝑆^⊕, (𝑥)⟨(𝑥𝜆ℎ)(∶=𝜆,ℎ)⟩if𝑥∈domℎ,+∞otherwise. (3.3) It is easy to see that ℎ is 𝑆-convex if and only if (𝜆ℎ)(⋅)∶𝑋→ℝ is a convex function for each 𝜆∈𝑆^⊕. Following [10], we define the function ℎ^⋄∶𝑋^∗→ℝ by ℎ^⋄𝑥^∗=inf_{𝜆∈𝑆⊕}(𝜆ℎ)^∗𝑥^∗foreach𝑥^∗∈𝑋^∗. (3.4) Let ℎ⁻¹(−𝑆)∶={𝑥∈dom𝑔∶ℎ(𝑥)∈−𝑆}. Recall from [19, 23] that 𝐺 is said to be star lsc if 𝜆𝐺 is lsc on 𝑋 for each 𝜆∈𝑆^⊕ and to be 𝑆-epi-closed if epi_𝑆(𝐺) is closed, where epi_𝑆(𝐺)∶=𝑥,𝑦∈𝑋×𝑌∶𝑦∈𝐺(𝑥)+𝑆. (3.5) It is known (cf. [23]) that if 𝐺 is star lsc, then it is 𝑆-epi-closed. Let 𝐴 denote the solution set of the system {𝑥∈𝐶;ℎ(𝑥)∈−𝑆}, that is, 𝐴∶={𝑥∈𝐶∶ℎ(𝑥)∈−𝑆}. (3.6) To avoid trivially, we always assume that 𝐴≠∅.

The following lemma, which is taken from [10, Theorem 3.1], will be useful in our study.

Lemma 3.1. Suppose that ℎ is a proper star lsc and 𝑆-convex mapping with ℎ⁻¹(𝑆)≠∅. Then(i)ℎ^⋄ is a proper convex function on 𝑋^∗.(ii)epiℎ^⋄ is a convex cone.(iii)epi𝛿^∗_{ℎ−1(−𝑆)}=cl(epiℎ^⋄) and epi𝛿^∗_𝐴=cl(epi𝛿^∗_𝐶+epiℎ^⋄).

Let 𝑝∈𝑋^∗. Consider the primal problem 𝑃_𝑝𝑓(𝑥)(𝑥)−,(𝑥)Mins.t.−𝑔𝑝,𝑥𝑥∈𝐶,ℎ∈−𝑆 (3.7) and its dual problem of (𝑃_𝑝)𝐷_𝑝inf_{𝑢∗∈dom𝑔∗}sup_{𝜆∈𝑆⊕}𝑔^∗𝑢^∗−𝑓+𝛿_𝐶+𝜆ℎ^∗𝑝+𝑢^∗. (3.8) In the case when 𝑝=0, problem (𝑃_𝑝) and its dual problem (𝐷_𝑝) are reduced to problem (𝑃) and its dual problem (𝐷) defined in (1.1) and (1.2), respectively. Let 𝑣(𝑃_𝑝) and 𝑣(𝐷_𝑝) denote the optimal values of (𝑃_𝑝) and (𝐷_𝑝), respectively. Let 𝑟∈ℝ, then by the definition of conjugate function, one has that 𝑝,𝑟∈epi𝑓−𝑔+𝛿_𝐴^∗⟺𝑣𝑃_𝑝≥−𝑟. (3.9) Moreover, in the case when 𝑔 is lsc, then for each 𝑥∈𝑋, 𝑔(𝑥)=𝑔^∗∗(𝑥)=sup_{𝑥∗∈dom𝑔∗}𝑥^∗,𝑥−𝑔^∗𝑥^∗; (3.10) thus, it is easy to see that the following inequality holds: 𝑣𝐷_𝑝≤𝑣𝑃_𝑝foreach𝑝∈𝑋^∗, (3.11) that is, the stable weak Lagrange duality holds. However, (3.11) does not necessarily hold in general as showed in the following example.

Example 3.2. Let 𝑋=𝑌=𝐶∶=ℝ and 𝑆=[0,+∞). Define 𝑓,𝑔,ℎ,𝑝∶ℝ→ℝ by 𝑓=ℎ∶=𝛿_(−∞,0], 𝑝=0 and for each 𝑥∈ℝ, 𝑔(𝑥)∶=0𝑥<0,2𝑥=0,+∞𝑥>0, (3.12) (note that 𝑔 is not lsc at 𝑥=0). Then 𝐴=(−∞,0] and 𝑣(𝑃)=−2. Note that for each 𝑆^⊕=[0,+∞), 𝑓+𝛿_𝐶+𝜆ℎ^∗=𝑔^∗=𝛿_[0,+∞). (3.13) Then 𝑣(𝐷)=0>𝑣(𝑃). This means that (3.11) does not hold.

Below we give a sufficient condition to ensure that (3.11) holds.

Lemma 3.3. Suppose that the following condition holds: epi𝑓−𝑔+𝛿_𝐴^∗=epi𝑓−cl𝑔+𝛿_𝐴^∗. (3.14) Then (3.11) holds.

Proof. Let 𝑝∈𝑋^∗. Then for each 𝑢^∗∈dom𝑔^∗ and 𝜆∈𝑆^⊕, one has by (2.5) that for each 𝑥∈𝑋, 𝑔^∗𝑢^∗−𝑓+𝛿_𝐶+𝜆𝑔^∗𝑝+𝑢^∗≤𝑔^∗𝑢^∗−𝑝+𝑢^∗,𝑥+𝑓+𝛿_𝐶+𝜆𝑔(𝑥)≤𝑔^∗𝑢^∗−𝑢^∗,𝑥+𝑓+𝛿_𝐴−𝑝(𝑥), (3.15) where the last inequality holds because 𝛿_𝐶+𝜆𝑔≤𝛿_𝐴 for each 𝜆∈𝑆^⊕. Note that the above inequalities hold for each 𝑢^∗∈dom𝑔^∗. Then for each 𝑥∈𝑋, 𝑣𝐷_𝑝≤inf_{𝑢∗∈dom𝑔∗}𝑔^∗𝑢^∗−𝑢^∗,𝑥+𝑓+𝛿_𝐴−𝑝(𝑥)=𝑓−cl𝑔+𝛿_𝐴−𝑝(𝑥), (3.16) where the last equality holds by (3.10). Hence, 𝑣𝐷_𝑝≤inf_𝑥∈𝑋𝑓−cl𝑔+𝛿_𝐴−𝑝(𝑥), (3.17) which implies that (𝑝,−𝑣(𝐷_𝑝))∈epi(𝑓−cl𝑔+𝛿_𝐴)^∗ and (𝑝,−𝑣(𝐷_𝑝))∈epi(𝑓−𝑔+𝛿_𝐴)^∗ by (3.14). Hence, by (3.9), one has 𝑣(𝑃_𝑝)≥𝑣(𝐷_𝑝). Therefore, (3.11) holds by the arbitrary of 𝑝∈𝑋^∗. The proof is complete.

Remark 3.4. Condition (3.14) was introduced in [21] and was called (LSC) there. Obviously, if 𝑔 is lsc on 𝐴, then (3.14) holds. But the converse is not true in general as showed by [21, Example 4.1].

This section is devoted to the study of the zero dualities between (𝑃) and (𝐷), which is defined as follows.

Definition 3.5. We say that(a)the zero duality holds between (𝑃) and (𝐷) if 𝑣(𝑃)=𝑣(𝐷);(b)the stable zero duality holds between (𝑃) and (𝐷) if for each 𝑝∈𝑋^∗, the zero duality holds between (𝑃_𝑝) and (𝐷_𝑝).

Definition 3.6. We say the family {𝑓,𝑔,𝛿_𝐶,ℎ} satisfies the constraint qualification (CQ) if epi𝑓−𝑔+𝛿_𝐴^∗=_{𝑢∗∈dom𝑔∗}epi𝑓^∗□𝛿^∗_𝐶□ℎ^⋄−𝑢^∗,𝑔^∗𝑢^∗. (3.18)

The following proposition provides an equivalent condition for (CQ) to hold.

Proposition 3.7 3.7. Suppose that (3.14) holds (e.g., 𝑔 is lsc) and that 𝑓islsc,ℎisstarlsc,𝐶isclosed. (3.19) Then the family {𝑓,𝑔,𝛿_𝐶,ℎ} satisfies (𝐶𝑄) if and only if epi𝑓−𝑔+𝛿_𝐴^∗⊆_{𝑢∗∈dom𝑔∗}epi𝑓^∗□𝛿^∗_𝐶□ℎ^⋄−𝑢^∗,𝑔^∗𝑢^∗. (3.20)

Proof. To show the equivalence of (CQ) and (3.20), we only need to show that epi𝑓_{𝑢∗∈dom𝑔∗}^∗□𝛿^∗_𝐶□ℎ^⋄−𝑢^∗,𝑔^∗𝑢^∗⊆epi𝑓−𝑔+𝛿_𝐴^∗. (3.21) To do this, by (3.19) and the fact (2.11), it is easy to see that the following inclusion holds: epi𝑓^∗□𝛿^∗_𝐶□ℎ^⋄⊆clepi𝑓^∗+epi𝛿^∗_𝐶+epiℎ^⋄. (3.22) Note that 𝐴 is closed and so 𝛿_𝐴 is lsc. Then by Lemma 3.1(c), one has that epi𝑓^∗□𝛿^∗_𝐶□ℎ^⋄⊆clepi𝑓^∗+epi𝛿^∗_𝐴=epi𝑓+𝛿_𝐴^∗, (3.23) where the last inclusion holds by Lemma 2.1(i). Therefore, epi𝑓_{𝑢∗∈dom𝑔∗}^∗□𝛿^∗_𝐶□ℎ^⋄−𝑢^∗,𝑔^∗𝑢^∗⊆_{𝑢∗∈dom𝑔∗}epi𝑓+𝛿_𝐴^∗−𝑢^∗,𝑔^∗𝑢^∗=epi𝑓−cl𝑔+𝛿_𝐴^∗=epi𝑓−𝑔+𝛿_𝐴^∗, (3.24) where the first equality holds by (2.8) and the last equality holds by (3.14). Hence, (3.21) holds and the proof is complete.

Below we give another sufficient conditions ensuring (CQ). For the study of the Lagrange duality and the Fenchel-Lagrange duality, the authors in [3] introduced the following condition: 𝐶₁𝑓,𝐴epi𝑓+𝛿_𝐴^∗=epi𝑓^∗+epi𝛿^∗_𝐶+_{𝜆∈𝑆⊕}epi(𝜆ℎ)^∗. (3.25) This condition was also introduced independently but with different terminologies “𝐶₁(𝑓,𝐴)" and “(𝐶𝐶)" in [24], under the assumptions 𝑓islsc,𝐶isclosed,ℎis𝑆-epi-closed, (3.26) and [19, 20] (under the assumptions (3.26) together with the star lsc of ℎ), respectively.

Proposition 3.8. Suppose that (3.14) and (3.19) hold. Then 𝐶₁𝑓,𝐴⇒(𝐶𝑄). (3.27)

Proof. Suppose that 𝐶₁(𝑓,𝐴) holds. Note by the definition ℎ^⋄ that (𝜆ℎ)^∗≥ℎ^⋄ for each 𝜆∈𝑆^⊕. Then ∪_𝜆∈𝑆^⊕epi(𝜆ℎ)^∗⊆epiℎ^⋄ and epi𝑓^∗+epi𝛿^∗_𝐶+_{𝜆∈𝑆⊕}epi(𝜆ℎ)^∗⊆epi𝑓^∗+epi𝛿^∗_𝐶+epiℎ^⋄. (3.28) Hence, by (2.11), one has that epi𝑓^∗+epi𝛿^∗_𝐶+_{𝜆∈𝑆⊕}epi(𝜆ℎ)^∗⊆epi𝑓^∗+epi𝛿^∗_𝐶□ℎ^⋄⊆epi𝑓^∗□𝛿^∗_𝐶□ℎ^⋄. (3.29) This together with the 𝐶₁(𝑓,𝐴) implies that epi𝑓+𝛿_𝐴^∗⊆epi𝑓^∗□𝛿^∗_𝐶□ℎ^⋄. (3.30) Thus, by (2.8) and (3.14), we can obtain that epi𝑓−𝑔+𝛿_𝐴^∗=epi𝑓−cl𝑔+𝛿_𝐴^∗=_{𝑢∗∈dom𝑔∗}epi𝑓+𝛿_𝐴^∗−𝑢^∗,𝑔^∗𝑢^∗⊆_{𝑢∗∈dom𝑔∗}epi𝑓^∗□𝛿^∗_𝐶□ℎ^⋄−𝑢^∗,𝑔^∗𝑢^∗. (3.31) Hence, by Proposition 3.7, (CQ) holds and the proof is complete.

The converse of Proposition 3.8 does not necessarily hold, even in the case when 𝑔=0, as showed in the following example.

Example 3.9. Let 𝑋=𝑌=𝐶∶=ℝ and 𝑆=[0,+∞). Let 𝑓,𝑔,ℎ∶ℝ→ℝ be defined by 𝑓∶=𝛿_[0,+∞), 𝑔∶=0 and ℎ(𝑥)=𝑥² for each 𝑥∈ℝ. Then epi𝑓^∗=(−∞,0]×[0,+∞) and 𝐴∶={𝑥∈𝐶∶ℎ(𝑥)∈−𝑆}={0}. Hence, epi(𝑓+𝛿_𝐴)^∗=ℝ×[0,+∞). Moreover, it is easy to see that for each 𝑥^∗∈ℝ, (𝜆ℎ)^∗𝑥^∗𝑥∶=^∗2𝛿4𝜆,𝜆>0,_{0}𝑥^∗,𝜆=0. (3.32) Then, ℎ^⋄=0. This implies that epiℎ^⋄=ℝ×[0,+∞). Hence, epi𝑓^∗□𝛿^∗_𝐶□ℎ^⋄=ℝ×[0,+∞)=epi𝑓+𝛿_𝐴^∗, (3.33) This means that (CQ) holds (noting that dom𝑔^∗={0}). However, note that epi(_𝜆≥0𝜆ℎ)^∗=ℝ×(0,+∞)∪{(0,0)}. (3.34) Then epi𝑓^∗+epi𝛿^∗_𝐶+_𝜆≥0epi(𝜆ℎ)^∗=ℝ×(0,+∞)∪{(0,0)}≠epi𝑓+𝛿_𝐴^∗ (3.35) and so the 𝐶₁(𝑓,𝐴) does not hold.

Proposition 3.10. Let 𝑔=0. Suppose that (3.19) holds. Then the family {𝑓,𝑔,𝛿_𝐶,ℎ} satisfies (𝐶𝑄) if and only if epi(𝑓^∗□𝛿^∗_𝐶□ℎ^⋄) is 𝑤𝑒𝑎𝑘^∗-closed.

Proof. Since 𝑓 is lsc and 𝐴 is closed, it follows from Lemma 2.1(i) that epi𝑓+𝛿_𝐴^∗=clepi𝑓^∗+epi𝛿^∗_𝐴=clepi𝑓^∗+epi𝛿^∗_𝐶+epiℎ^⋄, (3.36) while the last equality holds by Lemma 3.1(c). Note by (2.11) that epi𝑓^∗+epi𝛿^∗_𝐶+epiℎ^⋄⊆epi𝑓^∗□𝛿^∗_𝐶□ℎ^⋄. (3.37) Hence, by (3.36), one has that epi𝑓+𝛿_𝐴^∗⊆clepi𝑓^∗□𝛿^∗_𝐶□ℎ^⋄. (3.38) This together with (3.23) implies that epi𝑓+𝛿_𝐴^∗=clepi𝑓^∗□𝛿^∗_𝐶□ℎ^⋄. (3.39) Thus, the result is seen to hold.

The following theorem provides a Farkas-type lemma for the DC optimization problem (3.7) in terms of the condition (CQ).

Theorem 3.11. Let 𝑝∈𝑋^∗ and 𝑟∈ℝ. Suppose that the family {𝑓,𝑔,𝛿_𝐶,ℎ} satisfies (CQ). Consider the following statements.(i)For each 𝑥∈𝐴, 𝑓(𝑥)−𝑔(𝑥)−⟨𝑝,𝑥⟩≥−𝑟.(ii)(𝑝,𝑟)+epi𝑔^∗⊆epi(𝑓^∗□𝛿^∗_𝐶□ℎ^⋄). (iii)For each 𝜖>0 and for each 𝑢^∗∈dom𝑔^∗, there exists 𝜆∈𝑆^⊕ such that 𝑔^∗𝑢^∗−𝑓+𝛿_𝐶+𝜆𝑔^∗𝑝+𝑢^∗≥−𝑟−𝜖. (3.40)Then (i)⇒ (ii)⇒ (iii). Furthermore, if (3.14) holds, then (𝑖)⇔(𝑖𝑖)⇔(𝑖𝑖𝑖).

Proof. Consider (i)⇒(ii). Suppose that (i) holds. Then for each 𝑥∈𝐴, 𝑓(𝑥)+𝛿_𝐴(𝑥)≥𝑔(𝑥)+⟨𝑝,𝑥⟩−𝑟. Thus, by (2.7) and the assumed (CQ), epi𝑔+𝑝−𝑟^∗⊆epi𝑓+𝛿_𝐴^∗⊆epi𝑓^∗□𝛿^∗_𝐶□ℎ^⋄; (3.41) while by (2.14), one has that epi𝑔+𝑝−𝑟^∗=epi𝑔^∗+𝑝,𝑟. (3.42) Hence, (ii) holds.
Consider (ii)⇒(iii). Suppose that (ii) holds. Let 𝑢^∗∈dom𝑔^∗ be arbitrary. Then, 𝑝+𝑢^∗,𝑟+𝑔^∗𝑢^∗∈epi𝑓^∗□𝛿^∗_𝐶□ℎ^⋄, (3.43) that is, inf_{𝑥∗1,𝑥∗2∈𝑋∗}𝑓^∗𝑥^∗₁+𝛿^∗_𝐶𝑥^∗₂+ℎ^⋄𝑝+𝑢^∗−𝑥^∗₁−𝑥^∗₂≤𝑟+𝑔^∗𝑢^∗. (3.44) This means that for each 𝜖>0, there exist 𝑥^∗₁,𝑥^∗₂∈𝑋^∗ such that 𝑓^∗𝑥^∗₁+𝛿^∗_𝐶𝑥^∗₂+ℎ^⋄𝑝+𝑢^∗−𝑥^∗₁−𝑥^∗₂≤𝑟+𝑔^∗𝑢^∗+𝜖2. (3.45) Moreover, by the definition of the function ℎ^⋄, there exists 𝜆∈𝑆^⊕ such that (𝜆ℎ)^∗𝑝+𝑢^∗−𝑥^∗₁−𝑥^∗₂≤ℎ^⋄𝑝+𝑢^∗−𝑥^∗₁−𝑥^∗₂+𝜖2. (3.46) Hence, 𝑓^∗𝑥^∗₁+𝛿^∗_𝐶𝑥^∗₂+(𝜆ℎ)^∗𝑝+𝑢^∗−𝑥^∗₁−𝑥^∗₂≤𝑟+𝑔^∗𝑢^∗+𝜖. (3.47) Therefore, by the Young-Fenchel inequality (2.5), one sees that for each 𝑥∈𝑋, −𝑟−𝜖≤𝑔^∗𝑢^∗−𝑓^∗𝑥^∗₁−𝛿^∗_𝐶𝑥^∗₂−(𝜆ℎ)^∗𝑝+𝑢^∗−𝑥^∗₁−𝑥^∗₂≤𝑔^∗𝑢^∗−𝑥^∗₁,𝑥+𝑓(𝑥)−𝑥^∗₂,𝑥+𝛿_𝐶(𝑥)−𝑝+𝑢^∗−𝑥^∗₁−𝑥^∗₂,𝑥+(𝜆ℎ)(𝑥)=𝑔^∗𝑢^∗−𝑝+𝑢^∗,𝑥+𝑓+𝛿_𝐶+𝜆ℎ(𝑥). (3.48) Note that the above inequalities and the equality hold for each 𝑥∈𝑋, it follows that −𝑟−𝜖≤𝑔^∗𝑢^∗−sup_𝑥∈𝑋𝑝+𝑢^∗,𝑥−𝑓+𝛿_𝐶+𝜆ℎ(𝑥)=𝑔^∗𝑢^∗−𝑓+𝛿_𝐶+𝜆ℎ𝑝+𝑢^∗. (3.49) Hence, (ii) holds.
Furthermore, suppose that (3.14) holds. Then the weak duality holds between (𝑃_𝑝) and (𝐷_𝑝), that is, 𝑣(𝐷_𝑝)≤𝑣(𝑃_𝑝). Below we show that (iii)⇒(i). To do this, assume that (iii) holds. Then by the definition of 𝑣(𝐷_𝑝), one has that 𝑣(𝐷_𝑝)≥−𝑟−𝜖 and 𝑣(𝐷_𝑝)≥−𝑟 by the arbitrary of 𝜖. Thus, by the weak duality holds between (𝑃_𝑝) and (𝐷_𝑝), one has that 𝑣(𝑃_𝑝)≥−𝑟. Hence, (i) holds and the proof is complete.

Let contℎ denote the set of all points at which ℎ is continuous, that is, contℎ={𝑥∈𝑋∶ℎiscontinuousat𝑥}. (3.50) The following theorem shows that the condition (CQ) is equivalent to the stable zero duality.

Theorem 3.12. Suppose that (3.14) holds. Consider the following statements.(i)The family {𝑓,𝑔,𝛿_𝐶,ℎ} satisfies (CQ).(ii)The stable zero duality holds between (𝑃) and (𝐷).Then (i)⇒ (ii). Furthermore, (i)⇔ (ii) if (3.19) holds and one of the following conditions holds:(a)cont𝑓∩𝐴≠∅ and contℎ∩𝐴≠∅;(b)contℎ∩𝐴∩int𝐶≠∅.

Proof. Consider (i)⇒ (ii). Suppose that (i) holds. Let 𝑝∈𝑋^∗. If 𝑣(𝑃_𝑝)=−∞, then the stable zero duality holds between (𝑃) and (𝐷) trivially. Below we assume that −𝑟∶=𝑣(𝑃_𝑝)∈ℝ. Then by the implication (i)⇒(ii) of Theorem 3.11, one has that 𝑣(𝐷_𝑝)≥−𝑟. Hence, 𝑣(𝐷_𝑝)≥𝑣(𝑃_𝑝) and, by Lemma 3.3, 𝑣(𝑃_𝑝)=𝑣(𝐷_𝑝). Thus, (ii) holds.
Furthermore, suppose that (3.19) holds and one of the conditions (a) and (b) holds. Then, by Lemma 2.1(b), one has that _{𝜆∈𝑆⊕}epi𝑓+𝛿_𝐶+𝜆ℎ^∗=epi𝑓^∗+epi𝛿^∗_𝐶+_{𝜆∈𝑆⊕}epi(𝜆ℎ)^∗. (3.51) To show (i), by Proposition 3.7, it suffices to show that (3.20) holds. To do this, let (𝑝,𝑟)∈epi(𝑓−𝑔+𝛿_𝐴)^∗. Then, by (3.9), 𝑣(𝑃_𝑝)≥−𝑟 and hence 𝑣(𝐷_𝑝)≥−𝑟 by the stable zero duality between (𝑃) and (𝐷). Let 𝜖>0 and 𝑢^∗∈dom𝑔^∗, then there exists 𝜆∈𝑆^⊕ such that 𝑔^∗𝑢^∗−𝑓+𝛿_𝐶+𝜆𝑔^∗𝑝+𝑢^∗≥−𝑟−𝜖. (3.52) This implies that (𝑓+𝛿_𝐶+𝜆ℎ)^∗(𝑝+𝑢^∗)≤𝑟+𝜖+𝑔^∗(𝑢^∗). Hence, 𝑝+𝑢^∗,𝑟+𝜖+𝑔^∗𝑢^∗∈epi𝑓+𝛿_𝐶+𝜆ℎ^∗⊆_{𝜆∈𝑆⊕}𝑓+𝛿_𝐶+𝜆ℎ^∗, (3.53) and by the arbitrary of 𝜖, one has that 𝑝+𝑢^∗,𝑟+𝑔^∗𝑢^∗∈_{𝜆∈𝑆⊕}epi𝑓+𝛿_𝐶+𝜆ℎ^∗=epi𝑓^∗+epi𝛿^∗_𝐶+_{𝜆∈𝑆⊕}epi(𝜆ℎ)^∗, (3.54) where the equality holds by (3.51). This together with (3.29) and (2.11) implies that 𝑝+𝑢^∗,𝑟+𝑔^∗𝑢^∗∈epi𝑓^∗□𝛿^∗_𝐶□ℎ^⋄, (3.55) that is, 𝑝,𝑟∈epi𝑓^∗□𝛿^∗_𝐶□ℎ^⋄−𝑢^∗,𝑔^∗𝑢^∗. (3.56) Hence, by the arbitrary of 𝑢^∗, we have that 𝑝,𝑟∈_{𝑢∗∈dom𝑔∗}epi𝑓^∗□𝛿^∗_𝐶□ℎ^⋄−𝑢^∗,𝑔^∗𝑢^∗. (3.57) Therefore, (3.20) holds and the proof is complete.

Recall that in the case when 𝐶=𝑋 and 𝑔=0, under the assumptions that 𝑓 is lsc and ℎ is continuous, the authors establish in [11, Theorem 3.1] the equivalence between the stable zero duality and the following regularity condition: epi𝑓^∗□ℎ^⋄isweak^∗closed. (CQ2) In this case, by Proposition 3.10, the following equivalence holds: (CQ)⟺(CQ2). (3.58) Hence, the following corollary, which follows from Theorem 3.12, improves the result in [11, Theorem 3.1].

Corollary 3.13. Suppose that epi𝑓−𝑔+𝛿_{𝑔−1(−𝑆)}^∗=epi𝑓−cl𝑔+𝛿_{𝑔−1(−𝑆)}^∗. (3.59) Consider the following statements.(i)The family {𝑓,𝑔,ℎ} satisfies (𝐶𝑄), that is, epi𝑓−𝑔+𝛿_{𝑔−1(−𝑆)}^∗=_{𝑢∗∈dom𝑔∗}epi𝑓^∗□ℎ^⋄−𝑢^∗,𝑔^∗𝑢^∗. (3.60)(ii)For each 𝑝∈𝑋^∗, inf_{𝑥∈𝑔−1()−𝑆}𝑓(𝑥)−𝑔(𝑥)−𝑝,𝑥=sup_{𝜆∈𝑆⊕}inf_𝑥∈𝑋𝑓(𝑥)−𝑔(𝑥)+(𝜆ℎ)(𝑥)−𝑝,𝑥. (3.61)Then (𝑖)⇒(𝑖𝑖). Furthermore, if (3.19) holds and cont𝑔∩𝐴≠∅, then (𝑖)⇔(𝑖𝑖).

In the case when 𝑔=0, the authors introduce in [10] the following condition: clepi𝛿^∗_𝐶+epiℎ^⋄=epi𝛿^∗_𝐶□ℎ^⋄ (CQ1) to study the zero duality between (𝑃) and (𝐷). Under the assumptions that (3.19) holds and int(dom𝑓)∩𝐴≠∅, the authors in [10] establish the zero duality using the regularity condition (CQ1). In this case, by Lemma 2.1(b) and Lemma 3.1, we have that epi𝑓+𝛿_𝐴^∗=epi𝑓^∗+epi𝛿^∗_𝐴=epi𝑓^∗+clepi𝛿^∗_𝐶+epiℎ^⋄. (3.62) This together with Proposition 3.7 implies that (CQ1)⇒(CQ). (3.63) By Theorem 3.12, we get the following corollary straightforwardly, which improves the corresponding result in [10, Theorem 4.1], since we do not need to assume that (3.19) holds and int(dom𝑓)∩𝐴≠∅.

Corollary 3.14. Suppose that the family {𝑓,𝑔,𝛿_𝐶,ℎ} satisfies (𝐶𝑄). Then the zero duality holds between (𝑃) and (𝐷).

By Theorem 3.12, we have the following result, where the equivalences of (i), (iii), and (iv) are given in [10, Theorem 4.1].

Corollary 3.15. Suppose that 𝐶 is closed, ℎ is star lsc and that contℎ∩𝐴≠∅. Then the following statements are equivalent.(i)The condition (CQ1) holds.(ii)If the proper lsc convex function 𝜑 is such that epi𝜑+𝛿_𝐴^∗=epi𝜑^∗+epi𝛿^∗_𝐴, (3.64)then inf_𝑥∈𝐴𝜑(𝑥)=sup_{𝜆∈𝑆⊕}inf_𝑥∈𝐶𝜑(𝑥)+(𝜆ℎ)(𝑥). (3.65)(iii) If the proper lsc convex function 𝜑 is continuous at some point in 𝐴, then (3.65) holds.(iv) If 𝑝∈𝑋^∗, then inf_𝑥∈𝐴𝑝(𝑥)=sup_{𝜆∈𝑆⊕}inf_𝑥∈𝐶𝑝(𝑥)+(𝜆ℎ)(𝑥). (3.66)

Proof. Consider (i)⇒(ii). Suppose that (i) holds and let 𝜑 be such that (3.64) is satisfied. Then, it follows from Lemma 3.1(c) that epi𝜑+𝛿_𝐴^∗=epi𝜑^∗+clepi𝛿^∗_𝐶+epiℎ^⋄=epi𝜑^∗+epi𝛿^∗_𝐶□ℎ^⋄⊆epi𝜑^∗□𝛿^∗_𝐶□ℎ^⋄, (3.67) where the second equality holds by the condition (CQ1) and the last inclusion holds by (2.11). Hence, by Proposition 3.7(a) (note that 𝑔=0), the (CQ) holds. Applying Corollary 3.14 to 𝜑 in place of 𝑓, we complete the proof of the implication (i)⇒(ii).
Consider (ii)⇒(iii). Note that (3.64) is satisfied if 𝜑 is continuous at some point in 𝐴 (see Lemma 2.1(ii)). Thus, it is immediate that (ii)⇒(iii).
Consider (iii)⇒(iv). It is trivial.
Consider (iv)⇒(i). Suppose that (iv) holds. Then applying Theorem 3.12 to 𝑓=0, one has that epi𝛿^∗_𝐴=epi𝛿^∗_𝐶□ℎ^⋄. (3.68) Hence, by Lemma 3.1(c), we obtain that epi𝛿^∗_𝐶□ℎ^⋄=clepi𝛿^∗_𝐶+epiℎ^⋄, (3.69) that is, the (CQ1) holds.

Using the same argument, one can obtain a sufficient and necessary condition to ensure the zero duality between the primal problem and the Fenchel-Lagrange duality.

Theorem 3.16. Suppose that (3.14) holds. Consider the following statements.(i)The family {𝑓,𝑔,𝛿_𝐶,ℎ} satisfies the following condition: epi𝑓−𝑔+𝛿_𝐴^∗=_{𝑢∗∈dom𝑔∗}epi𝑓^∗+epi𝛿^∗_𝐶□ℎ^⋄−𝑢^∗,𝑔^∗𝑢^∗. (3.70)(ii)For 𝑝∈𝑋^∗, the following equality holds: 𝑣𝑃_𝑝=inf_{𝑢∗∈dom𝑔∗}sup_{𝜆∈𝑆⊕,𝑥∗∈dom𝑔∗}𝑔^∗𝑢^∗−𝑓^∗𝑥^∗−𝛿_𝐶+𝜆ℎ^∗𝑝+𝑢^∗−𝑥^∗. (3.71)Then (i)⇒ (ii). Furthermore, if (3.19) holds and either cont𝑔∩𝐶≠∅ or dom𝑔∩int𝐶≠∅, then (𝑖)⇔(𝑖𝑖).

Acknowledgments

The author is grateful to the anonymous reviewer for valuable suggestions and remarks helping to improve the quality of the paper. This author was supported in part by the National Natural Science Foundation of China (grant 11101186).