Closedness-Type Regularity Conditions

Besides the generalized interior-point regularity conditions, there exist in the liter-ature so-called closedness-type regularity conditions for conjugate duality. In the following, we will recall two sufficient conditions of this type for Fenchel duality and we will relate them to the ones investigate in the previous subsection. Let these two conditions be:

(RC₉^F) f and g are lower semicontinuous and epi f^∗+epi g^∗is closed in(X^∗,w(X^∗,X))×R and

(RC₁₀^F) f and g are lower semicontinuous, f^∗2g^∗is w(X^∗,X)-lower semicontinuous on X^∗and exact at 0.

The condition(RC^F₉) has been first considered by Burachik and Jeyakumar in Banach spaces (cf. [10]) and by Bot¸ and Wanka in separated locally convex spaces (cf. [7]), while the second one, (RC₁₀^F), has been introduced in [7]. We have the following duality results (cf. [7]).

Theorem 7.23. Let f,g : X →Rbe proper and convex functions such that dom f∩ dom g=/0. If(RC₉^F)is fulfilled, then

(f+g)^∗(x^∗) =min{f^∗(x^∗−y^∗) +g^∗(y^∗): y^∗∈X^∗} ∀x^∗∈X^∗. (7.3) Theorem 7.24. Let f,g : X →Rbe proper and convex functions such that dom f∩ dom g= /0. If (RC^F₁₀) is fulfilled, then v(PF) =v(DF) and (DF) has an optimal solution.

Remark 7.25. (a) Let us notice that condition (7.3) is referred in the literature as stable strong duality (see [6,11,22] for more details) and obviously guarantees strong duality for(PF)−(DF). When f,g : X→Rare proper, convex and lower semicontinuous functions with dom f∩dom g=/0 one has in fact that(RC₉^F)is fulfilled if and only if (7.3) holds (cf. [7, Theorem 3.2]).

(b) If f,g are proper, convex and lower semicontinuous such that dom f∩dom g= /0, then(RC^F₉)⇒(RC₁₀^F)(cf. [7, Sect. 4]). Moreover, there are examples showing that in general (RC^F₁₀) is weaker than(RC₉^F)(see[7]). Finally, let us mention that (under the same hypotheses) f^∗2g^∗ is a w(X^∗,X)-lower semicontinuous function on X^∗if and only if(f+g)^∗=f^∗2g^∗. This is a direct consequence of the equality(f+g)^∗=cl(f^∗2g^∗), where the closure is considered with respect to the weak^∗topology on X^∗(cf. [7, Thorem 2.1]).

(c) In case X is a Fr´echet space and f,g are proper, convex and lower semicontinu-ous functions, we have the following relations between the regularity conditions considered for the primal-dual pair(PF)−(DF)(cf. [7], see also [17] and [26, Theorem 2.8.7])

(RC^F1)⇒(RC2^F)⇔(RC3^F)⇒(RC4^F)⇔(RC^F5)⇒(RC9^F)⇒(RC10^F).

We refer to [6,7,10,22] for several examples showing that in general the impli-cations above are strict. The implication(RC₁^F)⇒(RC₉^F)⇒(RC^F₁₀)holds in the general setting of separated locally convex spaces (in the hypotheses that f,g are proper, convex and lower semicontinuous).

We observe that if X is a finite-dimensional space and f,g are proper, convex and lower semicontinuous, then(RC^F₆)⇒(RC₇^F)⇔(RC₈^F)⇒(RC₉^F)⇒(RC₁₀^F). How-ever, in the infinite-dimensional setting this is no longer true. In the following two examples, the conditions(RC^F₉)and(RC₁₀^F)are fulfilled, unlike(RC_i^F), i∈ {6,7,8} (we refer to [6,7,10,19,22] for examples in the finite-dimensional setting).

Example 7.26. Consider the same setting as in Example7.22. We know that(RC^F₅) is fulfilled, hence also(RC₉^F)and(RC₁₀^F)(cf. Remark7.25(c)). This is not surprising, since epi f^∗+epi g^∗= (B_∗(0,1)+Rx^∗₀)×[0,∞), which is closed in(X^∗,w(X^∗,X))×

R(note that by the Banach–Alaoglu Theorem, the unit ball B_∗(0,1) is compact in(X^∗,w(X^∗,X))). As shown in Example 7.22, none of the regularity conditions (RC_i^F), i∈ {6,7,8}, is fulfilled.

Example 7.27. Consider the real Hilbert space ²(R) and the functions f,g : ²(R) → R defined by f = δ²

+(R) and g = δ₋²

+(R), respectively. We have qri(dom f−dom g) =qri

²₊(R)

= /0 (cf. Example7.5), hence all the general-ized interior-point regularity conditions (RC_i^F), i∈ {1,2,3,4,5,6,7,8}, fail (see also Proposition7.14(i)). The conjugate functions of f and g are f^∗=δ₋²

+(R)

and g^∗=δ²

+(R), respectively, hence epi f^∗+epi g^∗=²(R)×[0,∞), that is the condition(RC₉^F)holds (hence also (RC^F₁₀), cf. Remark7.25(b)). One can see that v(PF) =v(DF) =0 and y^∗=0 is an optimal solution of the dual problem.

The next issue we investigate concerns the relation between the generalized interior-point conditions(RC_i^F), i∈ {6,7,8} and the closedness-type ones(RC^F₉) and(RC₁₀^F). In the last two examples the conditions(RC^F₉)and(RC₁₀^F)are fulfilled, while(RC_i^F), i∈ {6,7,8}, fail. In the following we provide an example for which (RC₇^F)is fulfilled, unlike(RC_i^F), i∈ {9,10}. In this way we give a negative answer to

an open problem stated in [19, Remark 4.3], concomitantly proving that in general (RC₇^F)(and automatically also(RC^F₈)) and(RC^F₉)are not comparable.

Example 7.28. (See also [14, Example 2.7]) Like in Example7.13, consider the real Hilbert space X=²(N)and the sets

C={(xn)n∈N∈²: x_2n−1+x2n=0∀n∈N}

and

S={(xn)n∈N∈²: x_2n+x_2n+1=0∀n∈N},

which are closed linear subspaces of²and satisfy C∩S={0}. Define the functions f,g :²→Rby f=δCand g=δS, respectively, which are proper, convex and lower semicontinuous. The optimal objective value of the primal problem is v(PF) =0 and x=0 is the unique optimal solution of v(PF). Moreover, S–C is dense in ² (cf.

[17, Example 3.3]), thus cl

cone(dom f−dom g)

=cl(C−S) =². This implies 0∈qi(dom f−domg). Further, one has

epi f−epi(g−v(PF)) ={(x−y,ε): x∈C,y∈S,ε≥0}= (C−S)×[0,+∞) and cl

cone

epi f−epi(g −v(PF))

=²×[0,+∞), which is not a linear sub-space of²×R, hence(0,0)∈/ qri

epi f −epi(g −v(PF))

. All together, we get that the condition(RC₇^F)is fulfilled, hence strong duality holds (cf. Theorem7.16).

One can prove that f^∗=δ_C⊥and g^∗=δ_S⊥, where

C^⊥={(xn)n∈N∈²: x_2n−1=x2n∀n∈N}

and

S^⊥={(xn)n∈N∈²: x1=0,x2n=x_2n+1∀n∈N}.

Further, v(D_F) =0 and the set of optimal solutions of the dual problem is exactly C^⊥∩S^⊥={0}.

We show that (RC^F₁₀) is not fulfilled (hence (RC₉^F) fails too, cf. Remark 7.25(b)). Let us consider the element e¹∈², defined by e¹₁=1 and e¹_k =0 for all k∈N\ {1}. We compute(f+g)^∗(e¹) =sup_x∈2{e¹,x − f(x)−g(x)}=0 and(f^∗2g^∗)(e¹) =δ_C⊥+S^⊥(e¹). If we suppose that e¹∈C^⊥+S^⊥, then we would have(e¹+S^⊥)∩C^⊥=/0. However, it has been proved in [17, Example 3.3] that (e¹+S^⊥)∩C^⊥=/0. This shows that(f^∗2g^∗)(e¹) = +∞>0= (f+g)^∗(e¹). Via Remark 7.25(b) it follows that the condition (RC^F₁₀) is not fulfilled and, conse-quently,(RC_i^F), i∈ {1,2,3,4,5,9}, fail, too (cf. Remark7.25(c)), unlike condition (RC₇^F). Concerning(RC₆^F), one can see that this condition is not fulfilled, since 0∈qi(dom g−domg)does not hold.

In the next example, the conditions(RC_i^F), i∈ {6,7,8}, are fulfilled and(RC^F₉) fails.

Example 7.29. The example we consider in the following is inspired by [22, Example 11.3]. Consider X an arbitrary Banach space, C a convex and closed subset of X and x0an extreme point of C which is not a support point of C. Taking for instance X =², 1<p<2 and C :=

x∈²:_∞

n=1|xn|^p≤1

one can find extreme points in C that are not support points (see [22]). Consider the functions f,g : X →R defined as f =δx₀−C and g=δC−x₀, respectively. They are both proper, convex and lower semicontinuous and fulfill, as x0 is an extreme point of C, f+g=δ_{0}. Thus v(PF) =0 and x=0 is the unique optimal solution of (PF). We show that, different to the previous example,(RC^F₆)is fulfilled and this will guarantee that both(RC^F₇)and(RC₈^F)are valid, too (cf. Proposition7.14(i)).

To this end, we notice first that x0∈qi(C). Assuming the contrary, one would have that there exists x^∗∈X^∗\ {0} such thatx^∗,x0=sup_x∈Cx^∗,x(cf. Propo-sition7.2), contradicting the hypothesis that x0 is not a support point of C. This means that x0∈qri(C), too, and so 0∈dom f∩qri(dom g). Further, since it holds consequence the fact that(RC₆^F)is fulfilled. Hence strong duality holds (cf. Theorem 7.16), v(DF) =0 and 0 is an optimal solution of the dual problem.

We show that(RC^F₉)is not fulfilled. Assuming the contrary, one would have that the equality in (7.3) holds for all x^∗∈X^∗. On the other hand, in [22, Example 11.3]

it is proven that this is the case only when x^∗=0 and this provides the desired contradiction.

Remark 7.30. Consider the following optimization problem (P_F^A) inf

x∈X{f(x) + (g◦A)(x)},

where X and Y are separated locally convex spaces with topological dual spaces X^∗and Y^∗, respectively, A : X→Y is a linear continuous mapping, f : X→Rand of the primal and the dual problem, respectively, and suppose that v(P_F^A)∈R. We consider the set

A×id_R(epi f) ={(Ax,r)∈Y×R: f(x)≤r}.

By using the approach presented in the previous section one can provide similar discussions regarding strong duality for the primal-dual pair(P_F^A)−(D^A_F). To this end, we introduce the following functions: F,G : X×Y →R, F(x,y) = f(x) + δ{u∈X:Au=y}(x)and G(x,y) =g(y) for all (x,y)∈X×Y . The functions F and G are proper and their domains fulfill the relation

dom F−domG=X×

A(dom f)−domg .

Since epi F={(x,Ax,r): f(x)≤r}andepi(G− v(P_F^A)) ={(x,y,r): r≤ −G(x,y)+

v(P_F^A)}=X×epi(g −v(P_F^A)), we obtain epi F−epi(G −v(PF^A)) =X×

A×id_R(epi f)−epi(g −v(PF^A)) .

Moreover,

(x,y)∈X×Yinf {F(x,y) +G(x,y)}=inf

x∈X{f(x) + (g◦A)(x)}=v(PF^A).

On the other hand, for all(x^∗,y^∗)∈X^∗×Y^∗we have F^∗(x^∗,y^∗) =f^∗(x^∗+A^∗y^∗)and G^∗(x^∗,y^∗) =

g^∗(y^∗),if x^∗=0, +∞, otherwise.

Therefore, sup

x^∗∈X^∗ y^∗∈Y^∗

{−F^∗(−x^∗,−y^∗)−G^∗(x^∗,y^∗)}= sup

y^∗∈Y^∗{−f^∗(−A^∗y^∗)−g^∗(y^∗)}=v(D^AF).

For more details concerning this approach, we refer to [8,14].

We remark that Borwein and Lewis gave in [3] some regularity conditions by means of the quasi-relative interior, in order to guarantee strong duality for(P_F^A)and (D^A_F). However, they considered a more restrictive case, namely when the codomain of the operator A is finite-dimensional. Here we have considered the more general case, when both spaces X and Y are infinite-dimensional.

Finally, let us notice that several regularity conditions by means of the quasi inte-rior and quasi-relative inteinte-rior were introduced in the literature in order to guarantee strong duality between a primal optimization problem with geometric and cone con-straints and its Lagrange dual problem. However, they have either contradictory assumptions, like in [12], or superfluous conditions, like in [16]. For a detailed ar-gumentation of these considerations and also for correct alternative strong duality results in the case of Lagrange duality we refer to [8,9].

Acknowledgements The research of the first author was partially supported by DFG (German Research Foundation), project WA 922/1-3.

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