New duality results for evenly convex optimization problems

ABSTRACT We present new results on optimization problems where the involved functions are evenly convex. By means of a generalized conjugation scheme and the perturbation theory introduced by Rockafellar, we propose an alternative dual problem for a general optimization one defined on a separated locally convex topological space. Sufficient conditions for converse and total duality involving the even convexity of the perturbation function and c-subdifferentials are given. Formulae for the c-subdifferential and biconjugate of the objective function of a general optimization problem are provided, too. We also characterize the total duality by means of the saddle-point theory for a notion of Lagrangian adapted to the considered framework.


Introduction
An important part of mathematical programming from both theoretical and computational points of view is the duality theory. Rockafellar (cf. [1]) developed the well-known perturbational approach (see also [2]), consisting in the use of a perturbation function as the keystone to obtain a dual problem for a general primal one by means of the Fenchel conjugation. Both problems always satisfy weak duality (the optimal value of the dual problem is less or equal to the optimal value of the primal one), as a direct consequence of the Fenchel-Young inequality, whereas conditions ensuring strong duality (no duality gap and dual problem solvable) can be found in many references in the literature. Another related interesting problem in conjugate duality theory is the notion of converse duality. It corresponds to the situation where there is no duality gap and the primal problem is solvable. This issue was investigated in the convex setting in [3] for Fenchel duality and later extended in [4].
In this paper we propose a new dual problem to a general primal one defined on locally convex spaces by means of a generalized conjugation scheme and study converse duality for this primal-dual pair. This pattern, which is inspired by a survey done by Martínez-Legaz where generalized convex duality theory is applied to quasiconvex programming, is called c-conjugation scheme and it was developed in [5]. In the same way like in the classical context convexity and lower semicontinuity of the perturbation function are required in most of the regularity conditions -see for instance [6] -the use of the c-conjugation scheme is associated with the even convexity of such a function. Evenly convex sets (functions) are a generalization of closed and convex sets (functions), see, for instance, [7]. Fenchel called in [8] a set evenly convex, e-convex in brief, if it is the intersection of an arbitrary family, possibly empty, of open halfspaces. Such sets have been employed to study the solvability of semi-infinite linear systems containing infinitely many strict inequalities in [9], whereas some important properties in terms of their sections and projections are given in [10]. Due to [7], an extended real-valued function defined on a locally convex space is said to be e-convex if its epigraph is e-convex. According to [5], the c-conjugation scheme is suitable for this class of functions in the sense that the double conjugate function equals the original one if it is proper and e-convex. This result can be seen as the e-convex counterpart of the celebrated Fenchel-Moreau theorem. Concerning the usage of e-convexity in finance mathematics and consumer theory, we refer to [11,12]. Some recent applications of the c-conjugate scheme in the direction of formulas for asymptotic functions and multi-marginal monotone sets can be found in [13,14], respectively.
The theory developed in [5] motivated in [15] the generalization of some wellknown properties of the sum of the epigraphs of two Fenchel conjugate functions and the infimal convolution, and, as an application, conditions for strong Fenchel duality were derived. Later, in [16], the perturbation approach was used to build a dual problem by means of the c-conjugate duality theory, and the counterparts of some regularity conditions, i.e. conditions ensuring strong duality, from the Fenchel conjugate setting were obtained. Moreover, in [17] regularity conditions for strong duality between an e-convex problem and its Lagrange dual were established. In [18] we analysed the problem of stable strong duality, and deduced Fenchel and Lagrange type duality statements for unconstrained and constrained optimization problems, respectively. In [19] the Fenchel-Lagrange dual problem of a (primal) minimization problem, whose involved functions do not need to be e-convex a priori, was derived. Furthermore, some relations between the optimal values of the Fenchel, Lagrange and Fenchel-Lagrange dual problems were presented. Finally, in [20] we used the formulation of the Fenchel-Lagrange dual problem from [19] to derive a characterization of strong Fenchel-Lagrange duality.
The purpose of the paper is twofold. First, we analyse the fulfillment of converse duality for a primal-dual pair expressed via the perturbation function, where its even convexity will play a fundamental role. In order to avoid repetitive arguments, we have decided to derive the aforementioned converse duality in the general form, leaving as an application the natural particularizations into the previously mentioned special cases. As a second target, we address the problem of total duality (no duality gap and both problems solvable) from this general perspective. Moreover and extending in this way results from the convex setting [21][22][23], we provide some formulae for the c-subdifferential and biconjugate of the objective function of a given general optimization problem. Motivated by [2], we highlight the analysis of saddle-point theory and its relation with total duality for e-convex problems through the study and application of Lagrangian functions.
The layout of this work is as follows. Section 2 contains preliminary results on e-convex sets and functions to make the paper self-contained. Section 3 is dedicated to sufficient conditions for strong converse duality and biconjugate formulae. Section 4 is devoted to new results on c-subdifferentials which allow to characterize total duality for a general primal-dual pair. Moreover, the ε-csubdifferential of the objective function of the considered problem is expressed via the ε-c-subdifferential of the considered perturbation function. Last but not least in Section 5 we extend the saddle-point theory from the classical framework to e-convex problems, showing its close relation with total duality and we close the paper with a short section dedicated to some final remarks, conclusions and ideas for future work.

Preliminaries
Let X be a nontrivial separated locally convex space, lcs in brief, equipped with the σ (X, X * ) topology induced by X * , its continuous dual space endowed with the σ (X * , X) topology. The notation x, x * stands for the value at x ∈ X of the continuous linear functional x * ∈ X * . Let Y be another lcs. By R ++ we denote the set of positive real numbers. For a set D ⊆ X we denote its convex hull and its closure by convD and clD, respectively.
According to [24], a set C ⊆ X is evenly convex, e-convex in short, if for every point x 0 / ∈ C, there exists x * ∈ X * such that x − x 0 , x * < 0, for all x ∈ C. Furthermore, for a set C ⊆ X, the e-convex hull of C, e − convC, is the smallest e-convex set in X containing C. For a convex subset C ⊆ X, it always holds C ⊆ e − convC ⊆ clC. This operator is well defined because the class of e-convex sets is closed under arbitrary intersections. Since X is a separated lcs, X * = {0}. As a consequence of the Hahn-Banach theorem, it also holds that X is e-convex and every closed or open convex set is e-convex as well.
For a function f : X →R, we denote by domf = {x ∈ X : f (x) < +∞} the effective domain of f and by epif = {(x, r) ∈ X × R : f (x) ≤ r} and grhf = {(x, r) ∈ X × R : f (x) = r} its epigraph and its graph, respectively. We say that f is proper if epif does not contain vertical lines, i.e. f (x) > −∞ for all x ∈ X, and domf = ∅. By clf we denote the lower semicontinuous hull of f, which is the function whose epigraph equals cl(epif ). A function f is lower semicontinuous, lsc in brief, if for all x ∈ X, f (x) = clf (x), and e-convex if epif is e-convex in X × R. Clearly, any lsc convex function is e-convex, but the converse does not hold in general as one can see in [19,Ex. 2.1].
The e-convex hull of a function f : X → R, e − convf , is defined as the largest e-convex minorant of f. Based on the generalized convex conjugation theory introduced by Moreau [25], a suitable conjugation scheme for e-convex functions is provided in [5]. Let us consider the space W := X * × X * × R with the coupling functions c : X × W → R and c : W × X → R given by Given two functions f : X → R and g : W → R, the c-conjugate of f, f c : W → R, and the c -conjugate of g, g c : X → R, are defined Functions of the form x ∈ X → c(x, (x * , u * , α)) − β ∈R, with (x * , u * , α) ∈ W and β ∈ R are called c-elementary, and, in a similar way, functions of the form (x * , u * , α) ∈ W → c(x, (x * , u * , α)) − β ∈R with x ∈ X and β ∈ R are called celementary. In [5] it is shown that the family of proper e-convex functions from X toR along with the function identically equal to +∞ is actually the family of pointwise suprema of sets of c-elementary functions. Similarly, a function g : W →R is e -convex if it is the pointwise supremum of sets of c -elementary functions, and the e -convex hull of an extended real function g defined on W, denoted by e − convg, is the largest e -convex minorant of it. Moreover, a set D ⊂ W × R is e -convex if there exists an e -convex function g such that epig = D. The econvex hull of a set D ⊂ W × R, denoted by e − convD, is the smallest e -convex set containing D.
The following counterpart of the Fenchel-Moreau theorem for e-convex and e -convex functions was shown in [11,Prop. 6  (iv) f cc ≤ f ; g c c ≤ g.
The following lemma and proposition were shown in the finitely dimensional case in [7,26], respectively, and can be generalized for infinitely dimensional spaces easily. Recall that the recession cone of a nonempty convex subset A ⊆ X is defined by recA = {u ∈ X : a + u ∈ A, foralla ∈ A}. Lemma 2.2: Let C ⊆ X be a nonempty e-convex set and y ∈ X such that there exists x 0 ∈ X verifying x 0 + λy ∈ C, for all λ ≥ 0. Then y ∈ recC. Proposition 2.3: Let C ⊆ X × R be a nonempty e-convex set such that (0, 1) ∈ recC. Then h(x) = inf{a ∈ R : (x, a) ∈ C} is an e-convex function and epih = C ∪ grhh.
For convenience in Section 3, we need to ensure the fact that certain e-convex sets in X × R with (0, 1) ∈ recC must be epigraphs of e-convex functions. This requirement will be fulfilled via the following definition.

Converse duality and biconjugation
Let : X × Y → R be a perturbation function for (GP) and 0 ∈ Pr Y (dom ).
(2) Also in this point we recall the associated coupling function toc, which will be used in Section 3.2,c : Proceeding along the lines of [15], we consider the following primal-dual pair of problems, which verify weak duality, This dual problem can also be expressed via the infimum value function p : Y → R, defined by p(y) := inf x∈X (x, y), in the following way: From (GD c ) one can derive the minimization (primal) problem and, under appropriate regularity conditions, the optimal value of (GP c ) is equal to the optimal value of its dual, with the last problem being solvable as well. Let us calculate this dual problem, which we denote by (GD). If we consider the biconjugate function cc : Taking We consider the following dual problem for (GP c ) It follows that v(GD) ≤ v(GP c ) and, moreover, strong duality for (GP c ) − (GD) is equivalent to converse duality for (GP) − (GD c ). In [16], via the e-convexity of the perturbation function, some regularity conditions for a general primal problem (GP) and its general dual (GD c ) were obtained. In particular, assuming the properness and e-convexity of , the closedness-type one, was introduced to guarantee strong duality for the primal-dual pair (GP) − (GD c ). Due to [16,Lem. 5.3] this condition can be expressed (under the mentioned hypotheses) as Pr W×R (epi c ) = epi (·, 0) c . In order to be able to provide a counterpart of (C5) for the pair (GP c ) − (GD), we need first a suitable perturbation function for the new primal problem that plays the role of . Let us consider := c and the perturbation variable space to be X * × X * .
is always e -convex, an important feature which represents a difference from the standard context of strong duality on e-convex problems -see [16,18,20]. Recall that there the perturbation function is assumed to be e-convex in order to use the mentioned regularity condition. Naming G := c ((0, ·), (0, ·), ·) the new objective function, and defining the space this means that the function can be considered as a perturbation function for (GP c ) and, consequently, a dual problem for it would be In the case is e-convex, c = , and we obtain Since the biconjugate function is important to develop converse duality, let us analyse further properties of the function cc . Theorem 3.1: It always holds ( (·, 0)) cc ≥ cc (·, 0). If is proper and e-convex and (C5) is satisfied, one obtains ( (·, 0)) cc = cc (·, 0).
Proof: Let us take (x * , u * , α) ∈ W. Then, Taking now the c -conjugate of the first and last function in (5), we obtain, When is proper and e-convex and (C5) holds, one has by [16,Prop. 5.4] strong duality for the primal-dual pair (6) is then a chain of equalities, and one obtains ( (·, 0)) cc = cc (·, 0).
Next we present the counterparts of Lemma 5.1, 5.2 and 5.3 from [16] that are necessary for proving the converse duality statement.

Lemma 3.2: It holds
We continue with a Moreau-Rockafellar-type result involving the c -conjugate of the function ((0, ·), (0, ·), ·) that is also of interest per se.
Finally, we show that for all y ∈ domp, h(y) ≤ p(y). Notice that from the e-convexity of h and the definition of the e-convex hull of a function, we deduce h(y) ≤ e − convp(y).
Let y ∈ domp, with p(y) > −∞, denote a = p(y). Then inf x∈X (x, y) = a, and there exists a sequence {x r } ⊂ X such that lim r→+∞ (x r , y) = a and (x r , y), for all r ∈ N, and h(y) ≤ a.
In the case p(y) = −∞, there exists a sequence {x r } ⊂ X such that (x r , y) < −1/r, for all r ∈ N, and, since (y, (x r , y)) ∈ C, for all r ∈ N, we have h(y) ≤ (x r , y), for all r ∈ N, concluding h(y) = −∞.
We give the following sufficient condition to converse duality, assuming that is a proper and e-convex function, Pr Y×R epi c is e − convex and functionally representable. (C5)

Remark 3.1:
Although under e-convexity of the perturbation function it holds c = , we kept in the formulation of (C5) the function c , in order to have a similar formulation to (C5).
On the other hand, it is clear that Due to Lemma 3.2, it is enough to show that, for all y ∈ Y, one has inf x∈X c (x, y) ≤ ((0, ·), (0, ·), ·) c (y).

Proposition 3.6: If is proper and e-convex, and e − convPr Y×R (epi c ) is functionally representable, then Pr Y×R (epi c ) is e-convex if and only if
The following corollary arises from Lemma 3.2 and Proposition 3.6.

Remark 3.2:
Despite the similarity between converse duality and strong duality, see for instance [18], the property of being functionally representable makes a difference between them. It is totally necessary since the e-convex envelope of a given epigraph is not, in general, an epigraph anymore.

Example 3.8:
Let us consider the function f, defined on R: It is clear that which is not an epigraph.

C-subdifferentiability 4.1. New results and application to total duality
The subdifferentiability of a function at a point associated with the c-conjugation scheme was considered in [5] as a particular case of the notion of csubdifferentiability introduced in [11, p. 246]. We also denote by ∂ the classical (convex) subdifferential.

Definition 4.1: Let
The set of all the c-subgradients of f at x 0 is denoted by ∂ c f (x 0 ) and is called the c-subdifferential set of f at x 0 . In the case The notion of c -subdifferentiability that we introduce comes also from [11, p. 246].

Definition 4.2:
Let g : W →R be a function.

Lemma 4.4: Let g : W →R be a function and (x
Next is the counterpart to [1, Cor. 23.5.1] via the c-conjugation scheme.

) and the converse statement holds if f is e-convex.
Proof: As which means that x ∈ ∂ c f c (x * 0 , v * 0 , α 0 ) according to Lemma 4.4. Now, in the case f is e-convex, we have (f c ) c = f , hence (14) and (15) are equivalent. We apply these results to total duality for (GP) − (GD c ), that is i.e. the situation when both the primal and the dual have optimal solutions and their optimal values coincide. In the classical setting (see [2]), total duality for (GP) − (GD) and finiteness of both optimal values amounts to the existence of a point (x,ȳ * ) ∈ X × Y * satisfying (0,ȳ * ) ∈ ∂ (x, 0), or, equivalently, see [1,Th. 23.5], (x, 0) + * (0,ȳ * ) = 0, being, in that case,x an optimal solution of (GP) andȳ * an optimal solution of (GD).

Remark 4.2:
Notice that for e-convex, a necessary and sufficient condition for total duality for (GP) − (GD c ) is (by Propositions 4.5 and 4.6) the existence

ε-c-subdifferentiability
Next we extend the characterizations of ε-subdifferential formulae for convex functions from [22] to the current e-convex setting. First, recall the definition of the ε-c-subdifferential of a function f : X →R from [15,Def. 4].

Definition 4.7:
The set of all the ε-c-subgradients of f atx is denoted by ∂ c,ε f (x) and is called the ε-c-subdifferential set of f atx.

Saddle-point theory on e-convex problems
In the classical setting there exists a connection between saddle-point theory and total duality. This relation comes due to the fact that saddle-points can be characterized in terms of optimal solutions for the primal and the dual problem -see [2,Sect. 3.3]. In the following we extend the definition of Lagrangian function and saddle-point theory into the application of the c-conjugation scheme. The following definitions are the counterpart of Definitions 3.1 and 3.2 in [2], respectively. For more on Lagrangian functions in the classical (convex) case we refer the reader to [27,Sect. 3.3].
where Y x = dom (x, ·), for each x ∈ X, is called the c-Lagrangian of the problem (GP) relative to . In the classical setting, the Lagrangian function of (GP) relative to , L : X × Y * →R, satisfies that L x : Y * →R, defined for all x ∈ X by L x (y * ) = L(x, y * ), is a concave and upper semicontinuous function. Nevertheless, the function L y * : X →R given by L y * (x) = L(x, y * ) for all y * ∈ Y * is convex when is convex -see [2,Sect. 3.3]. In our context, the function L x : Y * × Y * × R ++ →R, which is defined for all x ∈ X by verifies that −L x = (x, ·) c and it is e -convex. However, we cannot guarantee that for all (y * , v * , α) ∈ Y * × Y * × R ++ , the function L (y * ,v * ,α) : X →R given by is convex when is so.
Taking into account that, if x > −1 and y ≤ −x then x − y ≥ x − 1 > −2, we conclude which is not convex.
The c-Lagrangian of the problem (GP) relative to is related to both optimal values in the following way (see [11,Sec.3]). If (x, ·) is e-convex at 0, for all x ∈ X, it also holds L(x, (y * , v * , α)).

Remark 5.2:
In a more general framework, Penot and Rubinov in [28] related the Lagrangian and the pertubational approach (or parametrization approach, as it is named in that paper) to duality for optimization problems. In order to allow a more comprehensive comparison between their work and ours, we have adapted their notation to the one used in this work. Given a set Z, a Lagrangian for (GP) is a function L : X × Z →R which must verify that F(·) = sup z∈Z L(·, z), in which case, the optimal value of (GP) satisfies v(GP) = inf x∈X sup z∈Z L(x, z).
Note that no convexity or topological assumptions were imposed on the involved functions in this case. Defining a dual functional for (GP), as d L (z) = inf x∈X L(x, z), for every z ∈ Z, a dual problem for (GP) is and for this primal-dual pair of optimization problems there holds weak duality. According to [28,Prop. 1 (Sect. 3.3)], if : X × Y →R is a perturbation function for (GP), and, for all x ∈ X, (x, ·) is H c -convex at 0 (see [28,Sect. 2] for a definition), then is a Lagrangian for (GP). Here c : Y × Z →R is any coupling function. As it can be observed, the function L in Definition 5.1 is not the same as (20): if we take the infimum on Y, the function in Definition 5.1 is always −∞, except perhaps for points (x, (0, 0,ᾱ)) ∈ X × (Y * × Y * × R ++ ), because of the special structure of the coupling functions we considered.

Final remarks, conclusions and future work
In this paper we present new results regarding evenly convex (e-convex) functions, in particular converse and total duality statements for e-convex problems that extend their counterparts from the (classical) convex case. Other results can be generalized to the current setting as well, for instance the ε-duality statements from [29], however the proofs work straightforwardly and present no difficulty so we opted not to include them here. On the other hand, some results known at the moment for proper, convex and lower semicontinuous functions, such as the maximal monotonicity of their subdifferentials or the fact that their proximal point operators are single valued, do not hold in general for e-convex functions -check for instance the function considered in [19,Ex. 2.1]. We extend in this article the notion of converse duality from the convex setting to e-convex optimization problems, providing a sufficient closednesstype regularity condition for it and an alternative formulation using the infimum value function. In order to prove the mentioned results we introduced the notion of functionally representable functions and we also gave a new Moreau-Rockafellar type result for e -convex functions. We introduced the concept of the c -subdifferential of a function, providing novel characterizations of the elements of c-subdifferentials and c -subdifferentials, respectively, and studying how total duality is connected with them. Formulae for the c-subdifferential and biconjugate of the objective function of a given general optimization problem are provided, too. On the other hand, we extend the definition of the classical Lagrangian towards the e-convex setting by means of the c-conjugation scheme and relate the corresponding saddle-points to total duality. The results for general optimization problems can be specialized for constrained and unconstrained optimization problems as well.
We have restricted ourselves to closedness type regularity conditions, but taking into consideration [18,Sect. 3] one can alternatively provide interiority type ones to the same end as well. Using the connection between the c-and csubdifferentials and the notion of c-conjugation, other paths can be followed along this direction. On the other hand, the investigations from [30] on subdifferentials of e-convex functions can be continued in the vein of this paper, too. Last but not least other properties of proper, convex and lower semicontinuous functions and formulae involving them could be extendable to the e-convex setting.