Regularity conditions play a key role in optimization and nonsmooth analysis, as additional hypotheses that guarantee the fulfillment of strong or converse duality, necessary and/or optimality conditions and various formulae, respectively. In convex optimization there have been proposed a number of regularity conditions (also called constraint qualification when they involve only the constraints of the considered problem), depending on the initial assumptions. For instance, in the differentiable case one employs the regularity conditions due to Abadie, Guignard, or Mangasarian-Fromowitz, while for nondifferentiable constrained optimization problems one has the Slater constraint qualification. However, the latter is a priori violated for many classes of problems where the constraint cone has an empty interior, and several generalizations were proposed for it, the so-called interiority type regularity conditions, that involve notions of generalized interior of a set. But even these fail for large classes of problems and, inspired by Precupanu's pioneering work [1], Burachik, Jeyakumar and their coauthors, and on the other hand Boţ, Wanka and their coauthors have proposed in [2–9] a new class of regularity conditions, the closedness type ones. They have proven first to be sufficient conditions for guaranteeing duality statements in optimization and subdifferential formulae in convex analysis, delivering in the meantime results and formulae in some related research fields as well. They have proven thus to be viable alternatives to their more restrictive interiority type counterparts. In this review paper, that enhances and completes a similar study provided in the book [10], we provide a general look on the usage of the closedness type regularity conditions in the literature until now, showing that they arise naturally while dealing with optimization problems and pointing toward different assertions from the literature that can be rediscovered as special cases of the mentioned general results. To this end we deconstruct and then reconstruct closedness type regularity conditions formulated by means of epigraphs and subdifferentials, respectively, for general optimization problems, showing afterwards how to particularize them for constrained and unconstrained optimization problems.

As mentioned above, the first papers dealing with closedness type regularity conditions for convex optimization problems were [2–7], having as a starting point earlier statements from Precupanu [1]. Afterwards, it was noticed that other regularity conditions from the literature such as the Basic Constraint Qualification (see for instance, [11–13]) or the Farkas-Minkowski Constraint Qualification (cf. [14–16]) can be recovered as special cases of some general closedness type regularity conditions.

Let X and Y be locally convex Hausdorff vector spaces, whose topological dual spaces are denoted by X^* and Y^*, respectively. The dual spaces can be endowed with different topologies among which the weak^* ones, denoted by ω(·^*, ·) or shortly ω^*, will be considered for formulating closedness type regularity conditions. The natural topology on ℝ is denoted by R. By 〈x^*, x〉 = x^*(x) we denote the value at x ∈ X of the linear continuous functional x^* ∈ X^*. A cone K ⊆ X is a nonempty subset of X which fulfills αK ⊆ K for all α ≥ 0. A convex cone K ⊆ X induces on X the partial ordering “≦_K” defined by x ≦_K y whenever y − x ∈ K, where x, y ∈ X. If x ≦_K y and x ≠ y we write x ≤_K y. To X a greatest element with respect to “≦_K” denoted ∞_K ∉ X can be attached and we define X•=X∪{∞K}. Then for any x ∈ X one has x ≤_K ∞_K and we consider on X^• the operations x + ∞_K = ∞_K + x = ∞_K for all x ∈ X and α · ∞_K = ∞_K for all α ≥ 0. The dual cone of K is K^* = {x^* ∈ X^* : 〈x^*, x〉 ≥ 0 ∀x ∈ K}. By convention, 〈x*,∞K〉=+∞ for all x^* ∈ K^*.

Given a subset U of X, by cl U, int U and cone U we denote its closure, interior and conical hull, respectively. Moreover, if U is convex, by sqri sqri U={x∈U:  cone(U−x) is a closed linear subspace} we denote its strong quasi-relative interior. When U ⊆ ℝⁿ we denote by ri U its relative interior, which coincides in this case with sqri U. The indicator function of U is δU:X→ℝ¯=ℝ∪{±∞}, defined as δ_U(x) = 0 if x ∈ U and δ_U(x) = +∞ otherwise, while its support function σU:X*→ℝ¯ is given by σU(x*)=supx∈U〈x*,x〉. The normal cone associated to the set U at x ∈ U is NU(x)={x∗∈X∗:〈x∗,y−x〉≤0 ∀y∈U} and, for ε ≥ 0, the ε-normal set of U at x ∈ U is NUε(x)={x∗∈X∗:〈x∗,y−x〉≤−ε ∀y∈U}. The projection function of X is Pr_X : X × Y → X, defined by Pr_X(x, y) = x for (x, y) ∈ X × Y, the identity function of X is id : X → X, id(x) = x for x ∈ X and, for n ∈ ℕ, we use the notation ΔXn={(x,…,x):x∈X}⊆Xn.

For being able to deliver by means of duality very general regularity conditions for the problem of guaranteeing the maximal monotonicity of the sum of two maximal monotone operators defined in reflexive Banach spaces, we introduced in Boţ et al. [17] the notion of a set closed regarding a subspace (see also [18–20]). For the investigations in this study even more general closedness notions are necessary, that were first proposed in Boncea and Grad [21] (and when Z = X × ℝ, in [22]; see also [10]).

Definition 1.1. Given ε ≥ 0, a set U ⊆ X × ℝ is said to be (0, ε)-vertically closed regarding the set Z ⊆ X × ℝ if (cl U) ∩ Z ⊆ (U ∩ Z) − (0, ε), while when Z = X × ℝ, U is called simply (0, ε)-vertically closed. Moreover, a set U ⊆ X that fulfills (cl U) ∩ W = U ∩ W, where W ⊆ X, is said to be closed regarding the set W.

Remark 1. A set U ⊆ X is closed if and only if it is closed regarding the whole space X. Each closed set U ⊆ X is closed regarding any other set W ⊆ X, but a set U ⊆ X that is closed regarding some W ⊆ X is not necessarily closed, as shown below in Example 1.

Example 1. The interval [0, 1) ⊆ ℝ is closed regarding the set {0}, but not closed.

Example 2. The set {0} × (0, +∞) ⊆ ℝ² is (0, ε)-vertically closed regarding the set [−1, 1] × (−1, +∞) for all ε > 0, but there is no ε ≥ 0 for it to be (0, ε)-vertically closed regarding the set [−1, 1] × (−∞, 0]. On the other hand, the set [0, 1] × (0, +∞) ⊆ ℝ² is (0, ε)-vertically closed for all ε > 0, while the set [0, 1] × (0, 1) ⊆ ℝ² is not (0, ε)-vertically closed for any ε ≥ 0.

Remark 2. In the literature one can find other definitions of ε-closed (see for instance [23, 24]) and vertically closed (see [25]) sets, respectively, introduced for purposes that have basically nothing in common with our present investigations.

Let us present now some preliminary notions and results involving functions. Given a function f:X→ℝ¯, its domain is dom f = {x ∈ X : f(x) < +∞}, its epigraph epi f = {(x, r) ∈ X × ℝ : f(x) ≤ r} and conjugate function f*:X*→ℝ¯, f^*(x^*) = sup{〈x^*, x〉 − f(x) : x ∈ X}. If U ⊆ X, the conjugate function of f regarding U ⊆ X is fU*:X→ℝ¯, fU*=(f+δU)*. We call f proper when f(x) > − ∞ for all x ∈ X and dom f ≠ ∅. When f is proper and ε ≥ 0, if f(x) ∈ ℝ the (convex) ε-subdifferential of f at x is ∂εf(x)={x∗∈X∗:f(y)−f(x)≥〈x∗,y−x〉−ε ∀y∈X} ∀y ∈ X}, while if f(x) = +∞ we take by convention ∂_εf(x) = ∅. The ε-subdifferential of f becomes in case ε = 0 its classical (convex) subdifferential denoted by ∂f. When U ⊆ X and ε ≥ 0 one has ∂εδU=NUε. The Young-Fenchel inequality says that fU*(x*)+f(x)≥〈x*,x〉 for all x ∈ U and x^* ∈ X^*. If U = X, this inequality is fulfilled as equality if and only if x^* ∈ ∂f(x) and in general one has f^*(x^*) + f(x) ≤ 〈x^*, x〉 + ε if and only if x*∈∂εf(x). When f,g:X→ℝ¯ are proper, their infimal convolution is f□g:X→ℝ¯, f□g(a)=infx∈X[f(x)+g(a-x)], that is called exact at a ∈ X when there exists an x ∈ X such that f□g(a) = f(x) + g(a − x). For α ∈ ℝ, defining the function αf:X→ℝ¯, (αf)(x) = αf(x) for x ∈ X, we take 0f = δ_{dom f} when α = 0. Given a linear continuous mapping A : X → Y, its adjoint is A^* : Y^* → X^*, 〈A^*y^*, x〉 = 〈y^*, Ax〉 for any (x, y^*) ∈ X × Y^*, while ImA = {Ax : x ∈ X} denotes its image.

Let K ⊆ X be a convex cone f:X→ℝ¯ is said to be K-increasing if f(x) ≤ f(y) for all x, y ∈ X such that x ≦_K y. A vector function F : Y → X^• is said to be proper if its domain dom F = {y ∈ Y : F(y) ∈ X} is nonempty and K-convex if F(tx + (1 − t)y) ≦_K tF(x) + (1 − t)F(y) for all x, y ∈ Y and all t ∈ [0, 1]. When K is closed, it is called K-epi-closed if its K-epigraph epi_K F = {(y, x) ∈ Y × X : x ∈ F(y) + K} is closed. For x^* ∈ K^* the function (x*F):Y→ℝ¯ is defined by (x^*F)(y) = 〈x^*, F(y)〉, y ∈ Y.

For an attained infimum (supremum) instead of inf (sup) we write min (max), while the optimal objective value of the optimization problem (P) is denoted by v(P).

The two main classical duality approaches in convex optimization, the Lagrange type for the constrained optimization problems and the Fenchel type for the unconstrained ones, have been unified in the general theory of conjugate duality by means of perturbations that can be found, for instance, in Rockafellar [26], Zălinescu [27], Ekeland and Temam [28], and Boţ et al. [29]. Note that also other duality concepts from the literature can be reconsidered in the framework of this general theory, as we have done for the Wolfe and Mond-Weir duality concepts in Boţ et al. [29], Boţ and Grad [30, 31], and Grad and Pop [32] and the geometric duality in Boţ et al. [33]. In the following we present briefly this general duality scheme before proceeding with the ε-duality investigations that lead to constructing closedness type regularity conditions for guaranteeing strong and stable duality, respectively.

Consider two locally convex Hausdorff vector spaces X and Y and the proper function F:X→ℝ¯. Let the general optimization problem

Using a proper perturbation function Φ:X×Y→ℝ¯, fulfilling Φ(x, 0) = F(x) for all x ∈ X, a hypothesis that guarantees that 0 ∈ Pr_YdomΦ, the problem (PG) can be rewritten as

We call Y the perturbation space and its elements perturbation variables. Note that the way Φ is defined guarantees that 0 ∈ Pr_Y(domΦ). To (PG) one attaches the following conjugate dual problem (cf., for instance, [19, 26, 27, 29])

and for this primal-dual pair of optimization problems weak duality always holds, i.e., v(DG) ≤ v(PG). In order to investigate further duality properties of these optimization problems, for each x^* ∈ X^* we consider the following problem

obtained by linearly perturbing the objective function of (PG). Thus (PG) is embedded in the family of optimization problems {(PGx*):x*∈X*}, where it coincides with (PG₀). To each problem in the mentioned family one can attach the corresponding conjugate dual problem, namely, for x^* ∈ X^*,

By construction, whenever x^* ∈ X^* one has v(DGx*)≤v(PGx*), but more important is to find out when the optimal objective values of the primal and its corresponding dual problem coincide or they are different with less than a given small ε ≥ 0.

Definition 2.1. Let ε ≥ 0. We say that there is ε-duality gap for the problems (PG) and (DG) if v(PG) − v(DG) ≤ ε. If v(PGx*)-v(DGx*)≤ε for all x^* ∈ X^*, we say that for (PG) and (DG) one has stable ε-duality gap.

Definition 2.2. We say that there is strong duality for the problems (PG) and (DG) if v(PG) = v(DG) and (DG) has an optimal solution. When there is strong duality for (PG) and (DG) and (PG) has an optimal solution, too, one speaks about total duality. If v(PGx*)=v(DGx*) and (DGx*) has an optimal solution for all x^* ∈ X^*, we say that for (PG) and (DG) one has stable strong duality.

Definition 2.3. Let ε ≥ 0. An element x ∈ X is said to be an ε-optimal solution to (PG) if 0 ∈ ∂_ε Φ (·, 0)(x).

In order to ensure strong duality for (PG) and (DG) one usually assumes that Φ is convex and a certain regularity condition is fulfilled. Various such additional hypotheses were considered in the literature (some of which are mentioned later in Remark 8), the most relevant being the interiority type ones (mentioned in [19, 27]) and the closedness type ones (cf. [3, 4, 6, 7, 10, 19, 29]). Moreover, the situation of total duality is closely related to some subdifferential formulae and the regularity conditions can be used to guarantee these, too. However, in some situations one can only show that the difference between the optimal objective values of the primal and dual problem is less than an ε ≥ 0, situation coined in Boncea and Grad [21, 22] as ε-duality gap. Using as a basis our investigations in these articles, where the ε-duality gap for composed and constrained optimization problems, respectively, were characterized via epigraph and subdifferential inclusions, we provide in the following section characterizations via epigraph inclusions of stable ε-duality gap (for an ε ≥ 0) for very general optimization problems, with the involved functions not necessary convex, only proper. Endowing then the functions with the classical convexity and topological properties, we derive new important equivalences as well as sufficient conditions, from which when ε = 0 closedness type regularity conditions are derived. These can then be employed, for instance, for subdifferential formulae, as done in Boţ et al. [18], Hiriart-Urruty [34], and Boţ and Grad [35] or, like in Boţ and Grad [36] and Grad and Wanka [37], for providing formulae for biconjugates of combinations of functions.

After presenting these investigations for general optimization problems, we deal with both constrained and unconstrained optimization problems, showing how the mentioned results can be specialized for them, too, by means of the perturbation theory (cf. [26, 27]). In this way some of our results from Grad [10], Boţ et al. [18, 20, 29, 38, 39], Boncea and Grad [21, 22], and Boţ and Grad [35] as well as different others from the literature (e.g., from [2–4, 6, 7, 19]) can be obtained as special cases of the general statements presented below.

Before proceeding, we recall a statement that will play an important role later in our investigations when additional convexity and topological hypotheses will be considered on the function Φ in order to derive closedness type regularity conditions for various duality situations.

Lemma 2.1. (cf. [40, Theorem 2.2 and Theorem 2.3], see also [19, Theorem 5.1 and Theorem 5.2] and [1]) Let the function Φ be also convex and lower semicontinuous. Then one has (Φ(·,0))*=clω*infy*∈Y*Φ*(·,y*) and

To make this review paper as self-contained as possible, we recall the definition of the Lagrangian function for the pair of primal-dual problems (PGx*)-(DGx*), where x^* ∈ X^*, and a nice connection between it and the optimal objective values of the mentioned problems (cf. [26, 27, 29]).

Definition 2.4. Let x^* ∈ X^*. The function L(PGx*):X×Y*→ℝ¯ defined by

is called the Lagrangian function of the pair of primal-dual problems (PGx*)-(DGx*) relative to the perturbation function Φ.

Remark 3. Given x^* ∈ X^*, one can rewrite the primal-dual pair of problems (PGx*)-(DGx*) by means of the Lagrangian L(PGx*) as follows. The dual problem (DGx*) is equivalent to supy*∈Y*infx∈XL(PGx*), while if for any x ∈ X the function Φ(x, ·) is convex, lower semicontinuous and nowhere equal to −∞, (PGx*) actually means infx∈Xsupy*∈Y*L(PGx*). Note that even without the additional hypotheses, v(PGx*)≥infx∈Xsupy*∈Y*L(PGx*).

Most of the closedness type regularity conditions that can be found in the literature are constructed by means of epigraphs and this section is dedicated to them.

Let ε ≥ 0. We begin our presentation with a characterization via epigraph inclusions of a situation of partial stable ε-duality gap for the problems (PG) and (DG) that holds for very general proper functions, no convexity or topological assumptions being necessary in order to derive it. Despite its simple formulation and proof, this statement is a key result for the following presentation and one shall see that using it virtually all the statements involving closedness type regularity conditions from the literature on duality for convex optimization problems, some of which have quite involved proofs in the original sources, can be rediscovered.

Theorem 3.1. Given a subset W of X^*, one has

if and only if for each x^* ∈ W there exists a ȳ^* ∈ Y^* such that

PROOF. If W is empty or for x^* ∈ W it holds (Φ(·, 0))^*(x^*) = +∞, there is nothing to prove, since subtracting anything from an empty set gives again the empty set and (Φ(·,0))*≤infy*∈Y*Φ*(·,y*), respectively.

Let x^* ∈ W such that (Φ(·, 0))^*(x^*) ∈ ℝ. Then (x^*, (Φ(·, 0))^*(x^*)) ∈ epi(Φ(·, 0))^*. For ȳ^* ∈ Y^* one has (x^*, ȳ^*, (Φ(·, 0))^*(x^*)) ∈ epi Φ^* − (0, 0, ε) if and only if Φ^*(x^*, ȳ^*) ≤ (Φ(·, 0))^*(x^*) + ε, hence the desired equivalence.          □

Remark 4. One can rewrite the inequality (Equation 2) as −(Φ(·, 0))^*(x^*) ≤ −Φ^*(x^*, ȳ^*) + ε, where in the left-hand side there is actually v(PGx*). However, in the right-hand side one does not necessarily have v(DGx*)+ε, since there is no guarantee that the supremum in (DGx*) is attained at ȳ^*. Consequently, Equation (2) implies v(PGx*)≤v(DGx*)+ε, i.e., condition (1) implies that there is a partial (because x^* ∈ W ⊆ X^*) stable ε-duality gap for the problems (PG) and (DG). Note also that employing (Equation 2), one can characterize the stable ε-duality gap for the primal-dual pair (PG) − −(DG) via the epigraph inclusion

When ε = 0 the epigraph inclusion and the inequality from Theorem 3.1 collapse both into equalities.

Corollary 3.2. Given a subset W of X^*, one has

if and only if for each x^* ∈ W there exists a ȳ^* ∈ Y^* such that

Taking W = X^*, one obtains an epigraph inclusion that ensures the stable ε-duality gap for (PG) and (DG), that (taking into consideration Corollary 3.2) becomes stable strong duality when ε = 0.

Corollary 3.3. It holds

if and only if for each x^* ∈ X^* there exists a ȳ^* ∈ Y^* such that

Corollary 3.4. It holds

if and only if for each x^* ∈ X^* there exists a ȳ^* ∈ Y^* such that

Using Theorem 3.1, one can derive necessary and sufficient ε-optimality conditions for primal-dual pairs (PGx*)-(DGx*), where x^* ∈ X^*.

Theorem 3.5. Let W be a subset of X^* and x^* ∈ W.

                 □

Remark 5. Note that in Theorem 3.5 one does not obtain the kind of equivalence usually delivered in optimality conditions statements as in a) one assumes x¯ to be an optimal solution to (PGx*), while in b) x¯ turns out to be only an ε-optimal solution to (PGx*).

When ε = 0, relations (Equations 6 and 7) become optimality conditions for (PGx*) and (DGx*).

Corollary 3.6. Let W be a subset of X^* and x^* ∈ W.

Remark 6. When W = X^*, Corollary 3.6 delivers stable optimality conditions for (PG) and (DG).

Remark 7. Taking x^* = 0 in Theorem 3.5 one obtains ε-optimality conditions for the primal-dual pair of optimization problems (PG) − (DG) and, moreover, (or, alternatively, directly from Corollary 3.3) that the satisfaction of the condition (5) guarantees that there is ε-duality gap for these problems.

So far we have deconstructed the closedness type regularity conditions formulated by means of epigraph inclusions, showing that such inclusions are intimately connected to ε-duality gap statements. In the following we will reconstruct them and for this we add convexity and topological properties to the function Φ.

We begin with an characterization of Equation (2) by means of the notion of the (0, ε)-vertical closedness of the conjugate of Φ regarding a product set that can be obtained via Lemma 2.1 and Theorem 3.1.

Theorem 3.7. Let W be a subset of X^* and take the function Φ also convex and lower semicontinuous. Then the set PrX*×ℝepiΦ* is (0, ε)-vertically closed regarding W × ℝ in the topology ω(X^*, X) × R if and only if for every x^* ∈ W there exists a ȳ^* ∈ Y^* such that Equation (2) holds.

When ε = 0 Theorem 3.7 collapses to Boţ [19, Theorem 9.1].

Corollary 3.8. Let W be a subset of X^* and the function Φ be also convex and lower semicontinuous. Then the set PrX*×ℝepiΦ* is closed regarding the set W×ℝ in the topology ω(X^*, X) × R if and only if for each x^* ∈ W there exists a ȳ^* ∈ Y^* such that

Taking in Theorem 3.7 W = X^* one gets the following statement.

Corollary 3.9. Let the function Φ be also convex and lower semicontinuous. Then the set PrX*×ℝepiΦ* is (0, ε)-vertically closed in the topology ω(X^*, X) × R if and only if for each x^* ∈ X^* there exists a ȳ^* ∈ Y^* such that

Taking in Corollary 3.8 moreover W = X^* (or ε = 0 in Corollary 3.9, noticing also the comment preceding Corollary 3.1), one obtains a characterization of the stable strong duality for (PG) and (DG), rediscovering thus [29, Theorem 3.2.2] and [4, Theorem 3.1].

Corollary 3.10. Let the function Φ be also convex and lower semicontinuous. Then the set PrX*×ℝepiΦ* is closed in the topology ω(X^*, X) × R if and only if for each x^* ∈ X^* there exists a ȳ^* ∈ Y^* such that

A crucial consequence of Theorem 3.7, via Corollary 3.10, is the strong duality statement for (PG) and (DG) that follows (see also [1, 3, 4, 19, 29]) and contains the weakest hypotheses that guarantee this outcome.

Corollary 3.11. Assume that Φ is convex and lower semicontinuous. If PrX*×ℝ epi Φ^* is a closed set in the topology ω(X^*, X) × R, then v(PG) = v(DG) and the dual problem (DG) has an optimal solution ȳ^* ∈ Y^*.

Remark 8. Several regularity conditions were proposed in the literature in order to achieve strong duality for (PG) and (DG). We list in the following the most important of those considered when the function Φ is convex (cf. [19, 27, 29]), namely the one involving continuity

a weak generalized interiority type one,

another one applicable when the dimension of the linear hull of Pr_Y(dom Φ) is finite,

and finally the closedness type regularity condition already mentioned in Corollary 3.11,

Worth noticing is that that all these four regularity conditions ensure actually stable strong duality for the primal-dual pair of optimization problems (PG) − −(DG). One can thus notice that (RCiG), i = 1, 2, 3, imply (RC4G), which is equivalent to the stable strong duality for (PG) and (DG). An example to show that the closedness type regularity condition (RC4G) is indeed weaker than its counterparts of continuity or interiority type follows, others being available for instance in Boţ et al. [18, 20].

Example 3. (cf. [6]) Let X = Y = ℝ and Φ:ℝ×ℝ→ℝ¯, Φ(x, y) = δ_ℝ+(x)+δ_ℝ−(x + y). Then PrX*×ℝepiΦ*=ℝ×ℝ+ is closed, thus (RC4G) is satisfied, while neither is Φ(0, ·) continuous at 0 nor is 0 ∈ ri Pr_Y(dom Φ) = (−∞, 0) fulfilled, hence (RCiG), i = 1, 2, 3, fail in this case.

Necessary and sufficient optimality conditions for (PGx*) and (DGx*), where x^* ∈ W ⊆ X^* follow by means of Theorem 3.5 via Corollary 3.10 (see also [19, 27, 29]).

Corollary 3.12. Let W be a subset of X^* and x^* ∈ W.

Remark 9. When W = X^*, Corollary 3.12 delivers what may be called stable optimality conditions for (PG) and (DG). Taking there x^* = 0 one obtains necessary and sufficient optimality conditions for (PG) and (DG) (see also [19, 27, 29]).

As byproducts of the duality investigations presented in this subsection one can also derive ε-Farkas statements and results involving (η, ε)-saddle points, inspired for instance by Boţ and Wanka [41], as follows. We begin with the ε-Farkas type results for (PGx*) and (DGx*), where x^* ∈ W ⊆ X^*. They extend some recent Farkas type statements from the literature that generalize the classical Farkas Lemma. For more on the latest developments in the literature on Farkas type statements we refer the reader to the survey [42] together with the references therein and the additional discussion from the same issue of the journal.

Theorem 3.13. Let W be a subset of X^*.

                                        □□

Using Equation (3) as a regularity condition other ε-Farkas type results for (PGx*) and (DGx*), where x^* ∈ W ⊆ X^*, can be formulated and proven analogously to the ones in Theorem 3.13.

Theorem 3.14. Let W be a subset of X^*.

If ε = 0, the ε-Farkas type results become equivalences, as follows.

Corollary 3.15. Let W be a subset of X^* and suppose that Equation (4) holds. Given x^* ∈ W, one has Φ(x, 0) − 〈x^*, x〉 ≥ 0 for all x ∈ X if and only if there exists a ȳ^* ∈ Y^* such that Φ^*(x^*, ȳ^*) ≤ 0.

In the following statement we assume that Equation (3) holds as an equality for ε = 0.

Corollary 3.16. Let W be a subset of X^* and suppose that epi(Φ(·,0))*=epiinfy*∈Y*Φ*(·,y*). Given x^* ∈ W, one has Φ(x, 0) − 〈x^*, x〉 ≥ 0 for all x ∈ X if and only if infy*∈Y*Φ*(x*,y*)≤0.

In order to deal with statements involving (η, ε)-saddle points, we generalize the classical notion of a saddle point (cf. [10, 21]).

Definition 3.1. Let η ≥ 0 and x^* ∈ X^*. We say that (x¯,ȳ*)∈X×Y* is an (η, ε)-saddle point of the Lagrangian L(PGx*) if

Remark 10. The notion of an ε-saddle point of a function with two variables was already considered in the literature, see for instance [43, 44].

Slightly weakening the properness hypothesis imposed on Φ and adding to it convexity and topological assumptions, one obtains the following statement connecting the (η, ε)-saddle points of L(PGx*) with the (ε+η)-duality gap for the problems (PGx*) and (DGx*), and the existence of some (ε+η)-optimal solutions to them.

Theorem 3.17. Let η ≥ 0 and x^* ∈ X^*.

                                         □

If one takes in Theorem 3.17 η = ε = ν = 0, the two assertions become equivalent, rediscovering [29, Theorem 3.3.2].

Corollary 3.18. Let x^* ∈ X^* and assume that Φ is a convex and lower semicontinuous function taking nowhere the value −∞. Then (x¯,ȳ*)∈X×Y* is a saddle point of L(PGx*) if and only if x¯ is an optimal solution to (PGx*), ȳ^* is an optimal solution to (DGx*) and v(PGx*)=v(DGx*).

The general scalar optimization problem (PG) encompasses as special cases different classes of scalar optimization problems. We dedicate the next subsections to writing constrained and unconstrained optimization problems as special cases of (PG) and dual problems will be assigned to them by employing carefully chosen perturbation functions (see [19, 29] for more details).