A Comparison of Some Recent Regularity Conditions for Fenchel Duality
7.3 Fenchel Duality
7.3.2 Interior-Point Regularity Conditions Expressed via Quasi Interior and Quasi-Relative Interior
Taking into account the relations that exist between the generalized interiority no-tions presented in Sect.7.2a natural question arises: is the condition 0∈qri(dom f− dom g)sufficient for strong duality? The following example (which can be found in [17]) shows that even if we impose a stronger condition, namely 0∈qi(dom f− dom g), the above question has a negative answer and this means that we need to look for additional assumptions in order to guarantee Fenchel duality.
Example 7.13. Consider the Hilbert space X=2(N)and the sets C={(xn)n∈N∈2: x2n−1+x2n=0∀n∈N}
and
S={(xn)n∈N∈2: x2n+x2n+1=0∀n∈N},
which are closed linear subspaces of2and satisfy C∩S={0}. Define the functions f,g :2→Rby f=δCand g(x) =x1+δS(x), respectively, for all x= (xn)n∈N∈2. One can see that f and g are proper, convex and lower semicontinuous func-tions with dom f =C and dom g=S. As v(PF) =0 and v(DF) =−∞(cf. [17, Example 3.3]), there is a duality gap between the optimal objective values of the primal problem and its Fenchel dual problem. Moreover, S−C is dense in 2 (cf. [17]), thus cl
cone(dom f−domg)
=cl(C−S) =2. The last relation implies 0∈qi(dom f−domg), hence 0∈qri(dom f−domg).
We notice that if v(PF) =−∞, by the weak duality result follows that for the primal-dual pair(PF)−(DF)strong duality holds. This is the reason why we suppose in what follows that v(PF)∈R.
Consider now the following regularity conditions expressed by means of the
Let us notice that these three regularity conditions were first introduced in [8].
We study in the following the relations between these conditions. We remark that epi f−epi(g −v(PF))
={(x−y,f(x) +g(y)−v(PF) +ε): x∈dom f,y∈dom g,ε≥0}, thus if the set epi f−epi(g−v(PF)is convex, then dom f−domg is convex, too.
Proposition 7.14. Let f,g : X →Rbe proper functions such that v(PF)∈Rand epi f−epi(g −v(PF))is a convex subset of X×R(the latter is the case if for instance
f and g are convex functions). The following statements are true:
(i) (RC7F)⇔(RC8F); if, moreover, f and g are convex, then (RC6F)⇒(RCF7)
can be equivalently written as(0,0)∈/qri (ii). Let us suppose that f and g are convex and(RC6F)is fulfilled. By applying Lemma7.7(i) with U :=dom g and V :=dom f we get 0∈qi(dom g−dom f) or, equivalently, 0∈qi(dom f−domg). This means that(RCF7)holds.
(ii) One can prove that the primal problem(PF)has an optimal solution if and only if(0,0)∈epi f−epi(g −v(PF)))and the conclusion follows.
(iii) See [8, Remark 3.4 (a)].
Remark 7.15. (a) The condition 0∈qi(dom f−domg)implies relation 0∈qi
(dom f−domg)−(dom f−domg)
in Proposition7.14(iii). This is a direct consequence of the inclusion dom f− dom g⊆(dom f−domg)−(dom f−domg).
(b) We have the following implication (0,0)∈qi trivially holds, we have cl
cone(dom f−domg)
=X , that is 0∈qi(dom f−domg).
Hence, the following implication is true 0∈qi(dom f−domg)⇒(0,0)∈/qi Nevertheless, in the regularity conditions given above one cannot substitute the condition(0,0)∈/qi but more handleable one 0∈qi(dom f−domg), since in all the regularity con-ditions(RCiF), i∈ {6,7,8}, the other hypotheses imply 0∈qi(dom f−domg) (cf. Proposition7.14(i)).
We give now the following strong duality result concerning the primal-dual pair (PF)−(DF). It was first stated in [8] under convexity assumptions for the functions involved.
Theorem 7.16. Let f,g : X→Rbe proper functions such that v(PF)∈Rand epi f−epi(g −v(PF))is a convex subset of X×R(the latter is the case if, for instance, f and g are convex functions). Suppose that either f and g are convex and(RC6F)is fulfilled, or one of the regularity conditions(RCiF), i∈ {7,8}, holds. Then v(PF) = v(DF)and(DF)has an optimal solution.
Proof. One has to use the techniques employed in the proof of [8, Theorem 3.5].
When the condition (0,0)∈/ qri
co
(epi f−epi(g −v(PF)))∪ {(0,0)}
is removed, the duality result given above may fail. In the setting of Example7.13, strong duality does not hold. Moreover, it has been proved in [8, Example 3.12(b)]
that(0,0)∈qri
co
(epi f−epi(g −v(PF)))∪ {(0,0)}
.
The following example (given in [8, Example 3.13]) justifies the study of the regularity conditions expressed by means of the quasi interior and quasi-relative interior.
Example 7.17. Consider the real Hilbert space2=2(N). We define the functions f,g :2→Rby the contrary, one would have that the set cl
holds r≥0. This leads to the desired contradiction.
Hence, the regularity condition(RC6F)is fulfilled, thus strong duality holds (cf.
Theorem7.16). On the other hand,2is a Fr´echet space (being a Hilbert space), the functions f and g are proper, convex and lower semicontinuous and, as sqri(dom f− dom g) =sqri(x0−2+) = /0, none of the conditions(RCFi), i∈ {1,2,3,4,5}, listed at the beginning of this section, can be applied for this optimization problem.
As for all x∗∈2it holds G∗(x∗) =δc−2
+(x∗)and (cf. [26, Theorem 2.8.7]) f∗(−x∗) = inf
x∗1+x∗2=−x∗{ · ∗(x∗1) +δx∗0−l+2(x∗2)}= inf
x∗1+x∗2=−x∗, x∗1≤1,x∗2∈2+
x∗2,x0,
the optimal objective value of the Fenchel dual problem is v(DF) = sup
x∗2∈2+−c−x∗1, x∗1≤1,x∗2∈2+
−x∗2,x0= sup
x∗2∈2+−x∗2,x0=0 and x∗2=0 is the optimal solution of the dual.
The following example (see also [14, Example 2.5]) underlines the fact that in general the regularity condition(RCF7) (and automatically also (RC8F)) is weaker than(RC6F)(see also Example7.28below).
Example 7.18. Consider the real Hilbert space 2(R) and the functions f,g : 2(R)→Rdefined for all s∈2(R)by
f(s) =
s(1),if s∈2+(R), +∞, otherwise and
g(s) =
s(2),if s∈2+(R), +∞, otherwise,
respectively. The optimal objective value of the primal problem is v(PF) = inf
s∈2+(R){s(1) +s(2)}=0
and s=0 is an optimal solution (let us notice that(PF)has infinitely many optimal solutions). We have qri(dom g) =qri(2+(R)) =/0 (cf. Example7.5), hence the con-dition(RC6F)fails. In the following, we show that(RCF7)is fulfilled. One can prove that dom f−domg=2+(R)−2+(R) =2(R), thus 0∈qi(dom f−domg). Like in the previous example, we have
epi f−epi(g −v(PF)) ={(s−s,s(1) +s(2) +ε): s,s∈2+(R),ε≥0}
and with the same technique one can show that(0,0)∈/qri
epi f−epi(g −v(PF)) , hence the condition(RC7F)holds.
Let us take a look at the formulation of the dual problem. To this end we have to calculate the conjugates of f and g. Let us recall that the scalar product on2(R),
·,·:2(R)×2(R)→Ris defined bys,s=supF⊆R,Ffinite
r∈Fs(r)s(r), for s,s∈ 2(R)and that the dual space
2(R)∗
is identified with2(R). For an arbitrary u∈2(R), we have
f∗(u) = sup
s∈2+(R){u,s −s(1)}= sup
s∈2+(R)
F⊆R,Fsupfiniter∈F
u(r)s(r)−s(1)
= sup
F⊆R,Ffinite
⎧⎨
⎩ sup
s∈2+(R) r∈F
u(r)s(r)−s(1)⎫
⎬
⎭.
If there exists r∈R\ {1}with u(r)>0 or if u(1)>1, then one has f∗(u) = +∞.
Assuming the contrary, for every finite subset F of R, independently from the fact that 1 belongs to F or not, it holds sups∈2
+(R){
r∈Fu(r)s(r)−s(1)}=0.
Consequently, f∗(u) =
0, if u(r)≤0∀r∈R\ {1}and u(1)≤1, +∞,otherwise.
Similarly, one can provide a formula for g∗and in this way we obtain that v(DF) =0 and that u=0 is an optimal solution of the dual ((DF)has actually infinitely many optimal solutions).
Let us compare in the following the regularity conditions expressed by means of the quasi interior and quasi relative interior with the classical ones form the litera-ture, mentioned at the beginning of the section. To this end, we need an auxiliary result.
Proposition 7.19. Suppose that for the primal-dual pair(PF)−(DF)strong duality holds. Then(0,0)∈/qi
co
(epi f−epi(g −v(PF)))∪ {(0,0)}
.
Proof. By the assumptions we made, there exists x∗ ∈X∗ such that v(PF) =
−f∗(−x∗)−g∗(x∗) =infx∈X{x∗,x+f(x)}+infx∈X{−x∗,x+g(x)}, hence v(PF)≤ x∗,x+f(x) +−x∗,y+g(y)∀(x,y)∈X×Y,
that is
−x∗,x−y −(f(x) +g(y)−v(PF))≤0∀(x,y)∈dom f×domg.
We obtain
(−x∗,−1),(z,r) ≤0∀(z,r)∈epi f−epi(g −v(PF)), hence
(−x∗,−1),(z,r) ≤0∀(z,r)∈co
(epi f−epi(g −v(PF)))∪ {(0,0)}
.
The last relation ensures (−x∗,−1) ∈ N[co((epi f−epi(g−v(P F)))∪{(0,0)})](0,0) and Proposition7.2implies that(0,0)∈qi/
co
(epi f−epi(g −v(PF)))∪ {(0,0)}
. A comparison of the above regularity conditions is provided in the following.
Proposition 7.20. Suppose that X is a Fr´echet space and f,g : X→Rare proper, convex and lower semicontinuous functions. The following relations hold
(RC1F)⇒(RC2F)⇔(RC3F)⇒(RC7F)⇔(RC8F).
Proof. In view of Remark 7.12 and Proposition 7.14(i) we have to prove only the implication(RC3F)⇒(RCF7). Let us suppose that (RC3F) is fulfilled. We ap-ply (7.2) and obtain 0∈qi
dom f−dom g
. Moreover, the regularity condition (RC3F) ensures strong duality for the pair(PF)−(DF)(cf. Theorem 7.11), hence (0,0)∈/qi
co
(epi f−epi(g −v(PF)))∪ {(0,0)}
(cf. Proposition7.19). Apply-ing Proposition7.14(iii) (see also Remark7.15(a)) we get that the condition(RCF7)
holds and the proof is complete.
Remark 7.21. One can notice that the implications (RCF1)⇒(RCF7)⇔(RC8F)
hold in the framework of separated locally convex spaces and for f,g : X→Rproper and convex functions (nor completeness for the space neither lower semicontinuity for the functions is needed here).
Next we show that, in general, the conditions(RCiF), i∈ {4,5}, cannot be com-pared with(RCiF), i∈ {6,7,8}. Example7.17provides a situation for which(RCiF), i∈ {6,7,8}, are fulfilled, unlike(RCiF), i∈ {4,5}. In the following example, the conditions(RCiF), i∈ {4,5}, are fulfilled, while(RCFi), i∈ {6,7,8}, fail.
Example 7.22. Consider(X, · )a nonzero real Banach space, x∗0∈X∗\ {0}and the functions f,g : X→Rdefined by f=δker x∗0and g= · +δkerx∗0, respectively.
The optimal objective value of the primal problem is v(PF) = inf
x∈ker x∗0x=0
and ¯x=0 is the unique optimal solution of (PF). The functions f and g are proper, convex and lower semicontinuous. Further, dom f−dom g=ker x∗0, which is a closed linear subspace of X , hence(RCiF), i∈ {4,5}, are fulfilled. Moreover, dom g−domg=dom f−domg=ker x∗0and it holds cl(ker x∗0) =ker x∗0=X . Thus, 0∈/qi(dom g−dom g)and 0∈/qi(dom f−dom g)and this means that all the three regularity conditions(RCiF), i∈ {6,7,8}, fail.
The conjugate functions of f and g are f∗=δ(ker x∗
0)⊥=δRx∗0 and g∗=δB∗(0,1)2δRx∗0=δB∗(0,1)+Rx∗0,
respectively (cf. [26, Theorem 2.8.7]), where B∗(0,1)is the closed unit ball of the dual space X∗. Hence, v(DF) =0 and the set of optimal solutions of(DF)coincides withRx∗0. Finally, let us notice that instead of ker x∗0 one can consider any closed linear subspace S of X such that S=X .