On the subdifferentiability of convex operators

Alexandru Nemeth
(original), ISI/JCR, paper

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A.B. Németh
Institute of Mathematics Cluj-Napoca, (ICTP)

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A.B. Nemeth, On the subdifferentiability of convex operators, J. London Math. Soc., s2-34 (1986) no. 3, pp. 552–558
https://doi.org/10.1112/jlms/s2-34.3.552

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Journal of the London Mathematical Society

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Oxford Academic

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10.1112/jlms/s2-34.3.552

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[1] J. M. BORWEIN, ‘Continuity and differentiability properties of convex operators’, Proc. London Math. Soc. (3) 44 (1982) 420-444.
[2] J. M. BORWEIN, ‘Subgradients of convex operators’, Math. Operationsforsch. Statist. Ser. Optim. 15 (1984) 179-191.
[3] W. W. BRECKNER and G. ORBAN, ‘On the continuity of convex mappings, Rev. Anal. Numer. Theor. Approx. 6 (1977) 117-123.
[4] M. M. FEL’DMAN, ‘About the sufficient conditions of the existence of supporting operators to sublinear mappings’, Sibirsk. Mat. t . 16 (1975) 132-138 (Russian).
[5] V. L. LEVIN, ‘The subdifferentials of convex mappings and composed functional’, Sibirsk. Mat. It. 13 (1972) 1295-1303 (Russian).
[6] C. W. MCARTHUR, ‘ In what spaces is every closed normal cone regular?’, Proc. Edinburgh Math. Soc. (2) 17 (1970) 121-125.
[7] C. W. MCARTHUR, ‘Convergence of monotone nets in ordered topological vector spaces’, Studia Math. 34(1970) 1-16.
[8] A. B. NGMETH, ‘Some differential properties of convex mappings’, Mathematica (Cluj) 22 (45) (1980) 107-114.
[9]  A. B. NEMETH, ‘Sequential regularity and directional differentiability of convex operators are equivalent’, preprint, Babes-Bolyai University 1984.
[10]  A. B. NEMETH, ‘On some universal subdifferentiability properties of ordered vector spaces’, preprint, Babes-Bolyai University 1985.
[11] N. S. PAPAGFORGIU, ‘Nonsmooth analysis on partially ordered vector spaces: part 1—convex case’ Pacific J. Math 107 (1983) 403-485.
[12] A. L. PERESSINI, Ordered topological vector spaces (Harper & Row, New York 1967).
[13] H. SCHAFFER, Banach lattices and positive operators (Springer, Berlin 1974).
[14] M. VALADIER, ‘Sous-differentiabilite de fonctions convexes a valuers dans un espace vectoriel ordonne’ Math. Scand. 30 (1972) 65-74.
[15] J. ZOWE, ‘Subdifferentiability of convex functions with values in ordered topological vector spaces’ Math. Scand. 34 (1974) 69-83.

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1987-Nemeth-, On the subdifferentiability of convex operators

ON THE SUBDIFFERENTIABILITY OF CONVEX OPERATORS

A. B. NÉMETH

Introduction

Since its inception in the papers of Valadier [14] and Levin [5], the theory of vector-valued convex analysis has been concerned with operators having values in order-complete vector lattices. The special interest of these spaces is motivated by the nice subdifferentiability properties of convex operators acting in such spaces. Besides considerable progress in this direction (see [11] and the references therein), there are results concerning more general ordered vector spaces. Zowe [15], and recently Borwein [1], have obtained results when conditions are imposed on the domain space of operators. Another approach is to build up the subgradients using directional minorants (or directional derivatives, when a topology is given). This method, initiated by Fel'dman [4] (see also [8,2]), has the advantage that conditions are imposed only on the range space. This approach suggests the natural question of how to characterize the ordered (topological) vector spaces in which every convex operator has directional minorants (directional derivatives) at each point and in every direction. Such a characterization can be done in terms of classical ordered (topological) vector spaces, and this is the principal result of the present note. It furnishes, using Fel'dman's theorem, a necessary and sufficient condition for every convex operator with values in an ordered vector space admitting isotone real functionals to have nice subdifferentiability properties.
The author expresses his gratitude to the referee and to J. M. Borwein for many valuable suggestions which improved the original version. Their remark, that the directional minorability of convex operators implies the Archimedian property of the space led to the present improved version of Theorem 1.

1. Prerequisites

An ordered vector space is, by definition, a pair ( Y , K Y , K Y,KY, KY,K ), where Y Y YYY is a real vector space and K K KKK is a cone in it, that is, a subset having the properties (i) K + K ⊂ K K + K ⊂ K K+K sub KK+K \subset KK+K⊂K, (ii) t K ⊂ K t K ⊂ K tK sub Kt K \subset KtK⊂K for each non-negative real number t t ttt and (iii) K ∩ ( − K ) = { 0 } K ∩ ( − K ) = { 0 } K nn(-K)={0}K \cap(-K)=\{0\}K∩(−K)={0}. The order relation induced by K K KKK in Y Y YYY is defined by putting u ⩽ v u ⩽ v u <= vu \leqslant vu⩽v whenever v − u ∈ K v − u ∈ K v-u in Kv-u \in Kv−u∈K. Then ⩽ ⩽ <=\leqslant⩽ is a reflexive, transitive and antisymmetric relation which is invariant with respect to translations and to multiplication by non-negative real numbers. The cone K K KKK is called the positive cone of the ordered vector space.
The ordered vector space ( Y , K Y , K Y,KY, KY,K ) is Archimedian if y ⩽ 0 y ⩽ 0 y <= 0y \leqslant 0y⩽0 whenever y ⩽ t z y ⩽ t z y <= tzy \leqslant t zy⩽tz for some z ∈ K z ∈ K z in Kz \in Kz∈K and all t > 0 t > 0 t > 0t>0t>0.
The space ( Y , K Y , K Y,KY, KY,K ) is said to admit an isotone functional if there exists f : Y → R f : Y → R f:Y rarrRf: Y \rightarrow \mathbb{R}f:Y→R having the property that for any u u uuu and v v vvv in Y Y YYY, the relations u ⩽ v , u ≠ v u ⩽ v , u ≠ v u <= v,u!=vu \leqslant v, u \neq vu⩽v,u≠v imply that f ( u ) < f ( v ) f ( u ) < f ( v ) f(u) < f(v)f(u)<f(v)f(u)<f(v).
The space ( Y , K Y , K Y,KY, KY,K ) is said to have the monotone sequence (monotone net) property if every decreasing sequence (net) in Y Y YYY with a lower bound has an infimum.
If we suppose that Y Y YYY is a topological vector space, then ( Y , K Y , K Y,KY, KY,K ) will be called an ordered topological vector space. All the topological vector spaces we shall deal with are supposed Hausdorff.
The ordered topological vector space ( Y , K Y , K Y,KY, KY,K ) is called normal if there is a base of neighbourhoods V V VVV of 0 with
V = ( V − K ) ∩ ( K − V ) . V = ( V − K ) ∩ ( K − V ) . V=(V-K)nn(K-V).V=(V-K) \cap(K-V) .V=(V−K)∩(K−V).
The ordered topological vector space ( Y , K Y , K Y,KY, KY,K ) is called regular (sequentially regular) if every decreasing net (sequence) having a lower bound has a limit in Y Y YYY. If ( Y , K Y , K Y,KY, KY,K ) is regular (sequentially regular) and if K K KKK is closed, then ( Y , K Y , K Y,KY, KY,K ) has the monotone net (sequence) property by [12, Corollary II.3.2]. Observe that in this case ( Y , K Y , K Y,KY, KY,K ) is Daniell space. An ordered topological vector space is called (countably) Daniell if it has the monotone net (sequence) property and every decreasing net (sequence) with a lower bound converges to its infimum.
Let F F FFF be an operator from a real vector space X X XXX to ( Y , K ) ( Y , K ) (Y,K)(Y, K)(Y,K). Then F F FFF is called convex if
F ( t x 1 + ( 1 − t ) x 2 ) ⩽ t F ( x 1 ) + ( 1 − t ) F ( x 2 ) F t x 1 + ( 1 − t ) x 2 ⩽ t F x 1 + ( 1 − t ) F x 2 F(tx_(1)+(1-t)x_(2)) <= tF(x_(1))+(1-t)F(x_(2))F\left(t x_{1}+(1-t) x_{2}\right) \leqslant t F\left(x_{1}\right)+(1-t) F\left(x_{2}\right)F(tx1+(1−t)x2)⩽tF(x1)+(1−t)F(x2)
for all x 1 , x 2 x 1 , x 2 x_(1),x_(2)x_{1}, x_{2}x1,x2 in x x xxx and all t t ttt in [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1][0,1].
The directional minorant of F F FFF at x 0 x 0 x_(0)x_{0}x0 in the direction h h hhh is defined by
∇ F ( x 0 ; h ) = inf t > 0 t − 1 ( F ( x 0 + t h ) − F ( x 0 ) ) ∇ F x 0 ; h = inf t > 0   t − 1 F x 0 + t h − F x 0 grad F(x_(0);h)=i n f_(t > 0)t^(-1)(F(x_(0)+th)-F(x_(0)))\nabla F\left(x_{0} ; h\right)=\inf _{t>0} t^{-1}\left(F\left(x_{0}+t h\right)-F\left(x_{0}\right)\right)∇F(x0;h)=inft>0t−1(F(x0+th)−F(x0))
when this infimum exists.
Suppose that ( Y , K Y , K Y,KY, KY,K ) is an ordered topological vector space. Then the directional derivative of F F FFF at x 0 x 0 x_(0)x_{0}x0 in the direction h h hhh is the limit
F ( x 0 ; h ) = lim t ↘ 0 t − 1 ( F ( x 0 + t h ) − F ( x 0 ) ) , F x 0 ; h = lim t ↘ 0   t − 1 F x 0 + t h − F x 0 , F(x_(0);h)=lim_(t↘0)t^(-1)(F(x_(0)+th)-F(x_(0))),F\left(x_{0} ; h\right)=\lim _{t \searrow 0} t^{-1}\left(F\left(x_{0}+t h\right)-F\left(x_{0}\right)\right),F(x0;h)=limt↘0t−1(F(x0+th)−F(x0)),
if it exists.
Let x 0 x 0 x_(0)x_{0}x0 and h h hhh be fixed in X X XXX. If F F FFF is convex, then the operator
(*) ϕ ( t ) = t − 1 ( F ( x 0 + t h ) − F ( x 0 ) ) (*) ϕ ( t ) = t − 1 F x 0 + t h − F x 0 {:(*)phi(t)=t^(-1)(F(x_(0)+th)-F(x_(0))):}\begin{equation*} \phi(t)=t^{-1}\left(F\left(x_{0}+t h\right)-F\left(x_{0}\right)\right) \tag{*} \end{equation*}(*)ϕ(t)=t−1(F(x0+th)−F(x0))
is increasing on R ∖ { 0 } R ∖ { 0 } R\\{0}\mathbb{R} \backslash\{0\}R∖{0} (see for example [5] or [14]). Hence, when K K KKK is closed and F ( x 0 ; h ) F x 0 ; h F(x_(0);h)F\left(x_{0} ; h\right)F(x0;h) exists, ∇ F ( x 0 ; h ) ∇ F x 0 ; h grad F(x_(0);h)\nabla F\left(x_{0} ; h\right)∇F(x0;h) also exists (this follows from [12, Corollary II.3.2]).
Let L ( X , Y ) L ( X , Y ) L(X,Y)L(X, Y)L(X,Y) be the space of linear operators from X X XXX to Y Y YYY. The set
∂ F ( x 0 ) = { A ∈ L ( X , Y ) : A x ⩽ F ( x 0 + x ) − F ( x 0 ) for all x ∈ X } ∂ F x 0 = A ∈ L ( X , Y ) : A x ⩽ F x 0 + x − F x 0  for all  x ∈ X del F(x_(0))={A in L(X,Y):Ax <= F(x_(0)+x)-F(x_(0))" for all "x in X}\partial F\left(x_{0}\right)=\left\{A \in L(X, Y): A x \leqslant F\left(x_{0}+x\right)-F\left(x_{0}\right) \text { for all } x \in X\right\}∂F(x0)={A∈L(X,Y):Ax⩽F(x0+x)−F(x0) for all x∈X}
is called the subdifferential of F F FFF at x 0 x 0 x_(0)x_{0}x0. The elements of ∂ F ( x 0 ) ∂ F x 0 del F(x_(0))\partial F\left(x_{0}\right)∂F(x0) are called subgradients of F F FFF at x 0 x 0 x_(0)x_{0}x0.
For future reference we shall use Greek letters for alternatives in the following known result concerning the existence of directional minorants and directional derivatives.
Proposition 1. ( α α alpha\alphaα ) If ( Y , K Y , K Y,KY, KY,K ) is an ordered vector space with the monotone sequence property, then each convex operator with values in ( Y , K Y , K Y,KY, KY,K ) has a directional minorant in every direction at every point of its domain.
( β β beta\betaβ ) If ( Y , K Y , K Y,KY, KY,K ) is a sequentially regular ordered topological vector space, then each convex operator with values in ( Y , K Y , K Y,KY, KY,K ) has a directional derivative in every direction at every point of its domain.
The proof is straightforward. The alternative ( α α alpha\alphaα ) appears, for instance, in [14] and in [1, Proposition 3.7(a)].
Since ( β ) ( β ) (beta)(\beta)(β) was not explicitly stated in this form (usually the normality of K K KKK is also required-see for example [14, Theorem 5.1] or [1, Proposition 3.7(c)]) we give its proof.
Let F F FFF be a convex operator from the vector space X X XXX to ( Y , K ) ( Y , K ) (Y,K)(Y, K)(Y,K). Let x 0 x 0 x_(0)x_{0}x0 and h h hhh be fixed in X X XXX. The operator ϕ ϕ phi\phiϕ defined by (*) is increasing on R ∖ { 0 } R ∖ { 0 } R\\{0}\mathbb{R} \backslash\{0\}R∖{0}. If we assume that lim t ↘ 0 ϕ ( t ) lim t ↘ 0   ϕ ( t ) lim_(t↘0)phi(t)\lim _{t \searrow 0} \phi(t)limt↘0ϕ(t) does not exist, then we get a neighbourhood U U UUU of 0 in Y Y YYY for which we can construct a decreasing sequence ( t n ) t n (t_(n))\left(t_{n}\right)(tn) of real numbers converging to 0 such that
ϕ ( t 2 k − 1 ) − ϕ ( t 2 k ) ∉ U for each k . ϕ t 2 k − 1 − ϕ t 2 k ∉ U  for each  k . phi(t_(2k-1))-phi(t_(2k))!in U quad" for each "k.\phi\left(t_{2 k-1}\right)-\phi\left(t_{2 k}\right) \notin U \quad \text { for each } k .ϕ(t2k−1)−ϕ(t2k)∉U for each k.
Now, since the sequence ( ϕ ( t n ) ) ϕ t n (phi(t_(n)))\left(\phi\left(t_{n}\right)\right)(ϕ(tn)) is decreasing and F ( x 0 ) − F ( x 0 − h ) F x 0 − F x 0 − h F(x_(0))-F(x_(0)-h)F\left(x_{0}\right)-F\left(x_{0}-h\right)F(x0)−F(x0−h) is a lower bound for it, we get a contradiction with the hypothesis.
Remark. For some ordered topological vector spaces, regularity implies normality. McArthur [7] has shown that every closed regular cone in a Fréchet space is normal. In the case of locally convex spaces he gave in [6] conditions in order that every closed normal cone be regular. There exist ordered normed spaces which are regular but lack normality [9].

2. Main results

We state first our principal results in the form of two theorems.
Theorem 1. Let ( Y , K ) ( Y , K ) (Y,K)(Y, K)(Y,K) be an ordered vector space and let X X XXX be a vector space of dimension greater than or equal to 1 . The following assertions are equivalent.
(i) ( Y , K ) ( Y , K ) (Y,K)(Y, K)(Y,K) has the monotone sequence property.
(ii) Each convex operator F : X → ( y , K ) F : X → ( y , K ) F:X rarr(y,K)F: X \rightarrow(y, K)F:X→(y,K) possesses a directional derivative in every direction at every point.
(iii) Each convex operator F : X → ( Y , K ) F : X → ( Y , K ) F:X rarr(Y,K)F: X \rightarrow(Y, K)F:X→(Y,K) possesses a directional minorant at 0 in some non-zero direction.
THEOREM 2. Let ( Y , K ) ( Y , K ) (Y,K)(Y, K)(Y,K) be an ordered topological vector space and let X X XXX be a vector space of dimension at least 1 . Then the following assertions are equivalent.
(i) ( Y , K ) ( Y , K ) (Y,K)(Y, K)(Y,K) is sequentially regular.
(ii) Each convex operator F : X → ( Y , K ) F : X → ( Y , K ) F:X rarr(Y,K)F: X \rightarrow(Y, K)F:X→(Y,K) possesses a directional derivative in every direction at every point.
(iii) Each convex operator F : X → ( Y , K ) F : X → ( Y , K ) F:X rarr(Y,K)F: X \rightarrow(Y, K)F:X→(Y,K) possesses a directional derivative at 0 in some non-zero direction.
We shall follow in the proofs a schema proposed by J. M. Borwein (which simplifies essentially our original version in [9]). Consider first some auxiliary results.
Proposition 2. Let ( a n a n a_(n)a_{n}an ) and ( y n y n y_(n)y_{n}yn ) be sequences in R R R\mathbb{R}R and Y Y YYY, respectively with a n > a n + 1 a n > a n + 1 a_(n) > a_(n+1)a_{n}>a_{n+1}an>an+1 for each n ∈ N n ∈ N n inNn \in \mathbb{N}n∈N. In order that a convex operator f : R → Y f : R → Y f:Rrarr Yf: \mathbb{R} \rightarrow Yf:R→Y such that f ( a n ) = y n f a n = y n f(a_(n))=y_(n)f\left(a_{n}\right)=y_{n}f(an)=yn exists, it is necessary and sufficient that
Δ n = y n + 1 − y n a n + 1 − a n Δ n = y n + 1 − y n a n + 1 − a n Delta_(n)=(y_(n+1)-y_(n))/(a_(n+1)-a_(n))\Delta_{n}=\frac{y_{n+1}-y_{n}}{a_{n+1}-a_{n}}Δn=yn+1−ynan+1−an
is decreasing with respect to n n nnn.
Proof. We have Δ n ⩾ Δ n + 1 Δ n ⩾ Δ n + 1 Delta_(n) >= Delta_(n+1)\Delta_{n} \geqslant \Delta_{n+1}Δn⩾Δn+1 if and only if
y n + 1 ⩽ a n + 1 − a n + 2 a n − a n + 2 y n + a n − a n + 1 a n − a n + 2 y n + 2 . y n + 1 ⩽ a n + 1 − a n + 2 a n − a n + 2 y n + a n − a n + 1 a n − a n + 2 y n + 2 . y_(n+1) <= (a_(n+1)-a_(n+2))/(a_(n)-a_(n+2))y_(n)+(a_(n)-a_(n+1))/(a_(n)-a_(n+2))y_(n+2).y_{n+1} \leqslant \frac{a_{n+1}-a_{n+2}}{a_{n}-a_{n+2}} y_{n}+\frac{a_{n}-a_{n+1}}{a_{n}-a_{n+2}} y_{n+2} .yn+1⩽an+1−an+2an−an+2yn+an−an+1an−an+2yn+2.
This relation shows the necessity of the condition. To verify the sufficiency, define f n : R → Y f n : R → Y f_(n):Rrarr Yf_{n}: \mathbb{R} \rightarrow Yfn:R→Y by
f n ( t ) = ( t − a n ) Δ n + y n , f n ( t ) = t − a n Δ n + y n , f_(n)(t)=(t-a_(n))Delta_(n)+y_(n),f_{n}(t)=\left(t-a_{n}\right) \Delta_{n}+y_{n},fn(t)=(t−an)Δn+yn,
and observe that each f n f n f_(n)f_{n}fn is affine and
f n ( t ) − f n + 1 ( t ) = ( Δ n − Δ n + 1 ) ( t − a n + 1 ) . f n ( t ) − f n + 1 ( t ) = Δ n − Δ n + 1 t − a n + 1 . f_(n)(t)-f_(n+1)(t)=(Delta_(n)-Delta_(n+1))(t-a_(n+1)).f_{n}(t)-f_{n+1}(t)=\left(\Delta_{n}-\Delta_{n+1}\right)\left(t-a_{n+1}\right) .fn(t)−fn+1(t)=(Δn−Δn+1)(t−an+1).
Thus
f ( t ) = max k ∈ N f k ( t ) = f n ( t ) if t ∈ [ a n + 1 , a n ] f ( t ) = max k ∈ N   f k ( t ) = f n ( t )  if  t ∈ a n + 1 , a n f(t)=max_(k inN)f_(k)(t)=f_(n)(t)quad" if "t in[a_(n+1),a_(n)]f(t)=\max _{k \in \mathbb{N}} f_{k}(t)=f_{n}(t) \quad \text { if } t \in\left[a_{n+1}, a_{n}\right]f(t)=maxk∈Nfk(t)=fn(t) if t∈[an+1,an]
defines a convex operator with f ( a n ) = y n f a n = y n f(a_(n))=y_(n)f\left(a_{n}\right)=y_{n}f(an)=yn.
Proposition 3. Let ( u n ) u n (u_(n))\left(u_{n}\right)(un) be a sequence in K K KKK with u n + 1 ⩽ t n u n u n + 1 ⩽ t n u n u_(n+1) <= t_(n)u_(n)u_{n+1} \leqslant t_{n} u_{n}un+1⩽tnun for some t n t n t_(n)t_{n}tn in ( 0,1 ). Then one can select a sequence ( a n a n a_(n)a_{n}an ) in R R R\mathbb{R}R decreasing to zero such that the operator f f fff defined on { a n } a n {a_(n)}\left\{a_{n}\right\}{an} by f ( a n ) = a n u n f a n = a n u n f(a_(n))=a_(n)u_(n)f\left(a_{n}\right)=a_{n} u_{n}f(an)=anun has a convex extension to R R R\mathbb{R}R.
Proof. Inductively, suppose that a 1 , … , a n a 1 , … , a n a_(1),dots,a_(n)a_{1}, \ldots, a_{n}a1,…,an have been selected. By Proposition 2 it is sufficient to find r r rrr small enough so that
Δ n − 1 = a n − 1 u n − 1 − a n u n a n − 1 − a n ⩾ a n u n − r u n + 1 a n − r = Δ n ( r ) . Δ n − 1 = a n − 1 u n − 1 − a n u n a n − 1 − a n ⩾ a n u n − r u n + 1 a n − r = Δ n ( r ) . Delta_(n-1)=(a_(n-1)u_(n-1)-a_(n)u_(n))/(a_(n-1)-a_(n)) >= (a_(n)u_(n)-ru_(n+1))/(a_(n)-r)=Delta_(n)(r).\Delta_{n-1}=\frac{a_{n-1} u_{n-1}-a_{n} u_{n}}{a_{n-1}-a_{n}} \geqslant \frac{a_{n} u_{n}-r u_{n+1}}{a_{n}-r}=\Delta_{n}(r) .Δn−1=an−1un−1−anunan−1−an⩾anun−run+1an−r=Δn(r).
From the hypothesis, we can find δ n > 0 δ n > 0 delta_(n) > 0\delta_{n}>0δn>0 such that
Δ n − 1 = a n − 1 a n − 1 − a n ( u n − 1 − u n ) + u n ⩾ ( 1 + δ n ) u n Δ n − 1 = a n − 1 a n − 1 − a n u n − 1 − u n + u n ⩾ 1 + δ n u n Delta_(n-1)=(a_(n-1))/(a_(n-1)-a_(n))(u_(n-1)-u_(n))+u_(n) >= (1+delta_(n))u_(n)\Delta_{n-1}=\frac{a_{n-1}}{a_{n-1}-a_{n}}\left(u_{n-1}-u_{n}\right)+u_{n} \geqslant\left(1+\delta_{n}\right) u_{n}Δn−1=an−1an−1−an(un−1−un)+un⩾(1+δn)un
while, since u n + 1 ⩾ 0 u n + 1 ⩾ 0 u_(n+1) >= 0u_{n+1} \geqslant 0un+1⩾0, for every r r rrr with 0 < r < a n 0 < r < a n 0 < r < a_(n)0<r<a_{n}0<r<an we obtain
Δ n ( r ) = r a n − r ( u n − u n + 1 ) + u n ⩽ ( 1 + r a n − r ) u n . Δ n ( r ) = r a n − r u n − u n + 1 + u n ⩽ 1 + r a n − r u n . Delta_(n)(r)=(r)/(a_(n)-r)(u_(n)-u_(n+1))+u_(n) <= (1+(r)/(a_(n)-r))u_(n).\Delta_{n}(r)=\frac{r}{a_{n}-r}\left(u_{n}-u_{n+1}\right)+u_{n} \leqslant\left(1+\frac{r}{a_{n}-r}\right) u_{n} .Δn(r)=ran−r(un−un+1)+un⩽(1+ran−r)un.
Take r r rrr sufficiently small in order to have r / ( a n − r ) ⩽ δ n r / a n − r ⩽ δ n r//(a_(n)-r) <= delta_(n)r /\left(a_{n}-r\right) \leqslant \delta_{n}r/(an−r)⩽δn as well; then Δ n − 1 ⩽ Δ n ( r ) Δ n − 1 ⩽ Δ n ( r ) Delta_(n-1) <= Delta_(n)(r)\Delta_{n-1} \leqslant \Delta_{n}(r)Δn−1⩽Δn(r) and 0 < a n + 1 < a n 0 < a n + 1 < a n 0 < a_(n+1) < a_(n)0<a_{n+1}<a_{n}0<an+1<an, as desired.
Lemma. If each convex operator f : X → ( Y , K ) f : X → ( Y , K ) f:X rarr(Y,K)f: X \rightarrow(Y, K)f:X→(Y,K) possesses a directional minorant at 0 in some non-zero direction, then ( Y , K Y , K Y,KY, KY,K ) is Archimedian.
Proof. We shall show that if ( Y , K ) ( Y , K ) (Y,K)(Y, K)(Y,K) is not Archimedian, then there exists a convex operator from R R R\mathbb{R}R to ( Y , K Y , K Y,K\mathrm{Y}, \mathrm{K}Y,K ) without directional minorant at 0 ∈ R 0 ∈ R 0inR0 \in \mathbb{R}0∈R in the direction 1 ∈ R 1 ∈ R 1inR1 \in \mathbb{R}1∈R. Observe first that ( Y , K Y , K Y,KY, KY,K ) is Archimedian if and only if each set of the
form { t x : t > 0 } { t x : t > 0 } {tx:t > 0}\{t x: t>0\}{tx:t>0} with x ∈ K x ∈ K x in Kx \in Kx∈K has an infimum. Indeed, suppose that v v vvv is an infimum of the set { t x : t > 0 } { t x : t > 0 } {tx:t > 0}\{t x: t>0\}{tx:t>0}, where x ∈ K x ∈ K x in Kx \in Kx∈K. Then v ⩾ 0 v ⩾ 0 v >= 0v \geqslant 0v⩾0 and s x ⩾ r v s x ⩾ r v sx >= rvs x \geqslant r vsx⩾rv for arbitrary s > 0 s > 0 s > 0s>0s>0 and r > 0 r > 0 r > 0r>0r>0. If we fix r = 2 r = 2 r=2r=2r=2, then it follows that 2 v 2 v 2v2 v2v is a lower bound for the set { t x : t > 0 } { t x : t > 0 } {tx:t > 0}\{t x: t>0\}{tx:t>0}, thus v ⩾ 2 v v ⩾ 2 v v >= 2vv \geqslant 2 vv⩾2v, that is, v ⩽ 0 v ⩽ 0 v <= 0v \leqslant 0v⩽0. Hence if every set of this kind has an infimum, it must be 0 . But then, if t x ⩾ y t x ⩾ y tx >= yt x \geqslant ytx⩾y for some x ∈ K x ∈ K x in Kx \in Kx∈K and every t > 0 t > 0 t > 0t>0t>0, it follows that y ⩽ 0 y ⩽ 0 y <= 0y \leqslant 0y⩽0, that is, ( Y , K ) ( Y , K ) (Y,K)(Y, K)(Y,K) is Archimedian. The converse is immediate. Now assuming that ( Y , K Y , K Y,KY, KY,K ) is not Archimedian and considering a set { t x : t > 0 } { t x : t > 0 } {tx:t > 0}\{t x: t>0\}{tx:t>0}, where x ∈ K x ∈ K x in Kx \in Kx∈K, without infimum, we define f : R → ( Y , K ) f : R → ( Y , K ) f:Rrarr(Y,K)f: \mathbb{R} \rightarrow(Y, K)f:R→(Y,K) by putting f ( t ) = 0 f ( t ) = 0 f(t)=0f(t)=0f(t)=0 for t ⩽ 0 t ⩽ 0 t <= 0t \leqslant 0t⩽0 and f ( t ) = t 2 x f ( t ) = t 2 x f(t)=t^(2)xf(t)=t^{2} xf(t)=t2x for t > 0 t > 0 t > 0t>0t>0. Then f f fff is obviously convex and the set { t − 1 ( f ( t ) − f ( 0 ) ) = t x : t > 0 } t − 1 ( f ( t ) − f ( 0 ) ) = t x : t > 0 {t^(-1)(f(t)-f(0))=tx:t > 0}\left\{t^{-1}(f(t)-f(0))=t x: t>0\right\}{t−1(f(t)−f(0))=tx:t>0} has no infimum, that is, ∇ f ( 0 ; 1 ) ∇ f ( 0 ; 1 ) grad f(0;1)\nabla f(0 ; 1)∇f(0;1) does not exist.
Proof of Theorem 1. Clearly (ii) implies (iii), while (i) implies (ii) by Proposition 1 ( α ) 1 ( α ) 1(alpha)1(\alpha)1(α). To show that (iii) implies (i) we argue as follows. Let ( v n ) v n (v_(n))\left(v_{n}\right)(vn) be a decreasing sequence in K K KKK. Let u n = v n ( n + 1 ) / n u n = v n ( n + 1 ) / n u_(n)=v_(n)(n+1)//nu_{n}=v_{n}(n+1) / nun=vn(n+1)/n. Then 0 ⩽ u n ⩽ ( 1 − 1 / n 2 ) u n − 1 0 ⩽ u n ⩽ 1 − 1 / n 2 u n − 1 0 <= u_(n) <= (1-1//n^(2))u_(n-1)0 \leqslant u_{n} \leqslant\left(1-1 / n^{2}\right) u_{n-1}0⩽un⩽(1−1/n2)un−1 for each n n nnn, and Proposition 3 applies. Let F ( t ) = f ( | t | ) F ( t ) = f ( | t | ) F(t)=f(|t|)F(t)=f(|t|)F(t)=f(|t|) with f f fff constructed as in Proposition 3. Then d = ∇ F ( 0 ; 1 ) d = ∇ F ( 0 ; 1 ) d=grad F(0;1)d=\nabla F(0 ; 1)d=∇F(0;1) exists by hypothesis and
d = inf n ∈ N a n − 1 ( F ( a n ) − F ( 0 ) ) = inf n ∈ N u n . d = inf n ∈ N   a n − 1 F a n − F ( 0 ) = inf n ∈ N   u n . d=i n f_(n inN)a_(n)^(-1)(F(a_(n))-F(0))=i n f_(n inN)u_(n).d=\inf _{n \in \mathbb{N}} a_{n}^{-1}\left(F\left(a_{n}\right)-F(0)\right)=\inf _{n \in \mathbb{N}} u_{n} .d=infn∈Nan−1(F(an)−F(0))=infn∈Nun.
Since ( Y , K Y , K Y,KY, KY,K ) must be Archimedian by the preceding lemma, d d ddd also will be an infimum of ( v n ) v n (v_(n))\left(v_{n}\right)(vn). Indeed, we have d ⩽ ( 1 + 1 / n ) v m d ⩽ ( 1 + 1 / n ) v m d <= (1+1//n)v_(m)d \leqslant(1+1 / n) v_{m}d⩽(1+1/n)vm for all n n nnn and m m mmm. Fix m m mmm for the moment. Then d − v m ⩽ v m / n d − v m ⩽ v m / n d-v_(m) <= v_(m)//nd-v_{m} \leqslant v_{m} / nd−vm⩽vm/n for all n n nnn and hence d ⩽ v m d ⩽ v m d <= v_(m)d \leqslant v_{m}d⩽vm, that is, d d ddd is a lower bound for ( v m ) v m (v_(m))\left(v_{m}\right)(vm). Every other lower bound v v vvv of ( v n ) v n (v_(n))\left(v_{n}\right)(vn) will be a lower bound for ( u n ) u n (u_(n))\left(u_{n}\right)(un) too, and hence v ⩽ d v ⩽ d v <= dv \leqslant dv⩽d. This completes the proof.
Proof of Theorem 2. Clearly (ii) implies (iii), while (i) implies (ii) by Proposition 1 ( β ) 1 ( β ) 1(beta)1(\beta)1(β). For the proof that (iii) implies (i) we consider a decreasing sequence ( v n ) v n (v_(n))\left(v_{n}\right)(vn) in K K KKK. Then, since u n = v n ( n + 1 ) / n u n = v n ( n + 1 ) / n u_(n)=v_(n)(n+1)//nu_{n}=v_{n}(n+1) / nun=vn(n+1)/n, we can construct by Proposition 3 a convex function F : R → Y F : R → Y F:R rarr YF: R \rightarrow YF:R→Y such that F ( 0 ) = 0 F ( 0 ) = 0 F(0)=0F(0)=0F(0)=0 and F ( a n ) = a n u n F a n = a n u n F(a_(n))=a_(n)u_(n)F\left(a_{n}\right)=a_{n} u_{n}F(an)=anun with some sequence ( a n ) a n (a_(n))\left(a_{n}\right)(an) decreasing to zero in R R R\mathbb{R}R. If we consider now
lim n → ∞ a n − 1 ( F ( a n ) − F ( 0 ) ) , lim n → ∞   a n − 1 F a n − F ( 0 ) , lim_(n rarr oo)a_(n)^(-1)(F(a_(n))-F(0)),\lim _{n \rightarrow \infty} a_{n}^{-1}\left(F\left(a_{n}\right)-F(0)\right),limn→∞an−1(F(an)−F(0)),
which certainly exists, we deduce that lim n → ∞ u n lim n → ∞   u n lim_(n rarr oo)u_(n)\lim _{n \rightarrow \infty} u_{n}limn→∞un exists. But v n = ( n / ( n + 1 ) ) u n v n = ( n / ( n + 1 ) ) u n v_(n)=(n//(n+1))u_(n)v_{n}=(n /(n+1)) u_{n}vn=(n/(n+1))un has obviously the same limit, and so Theorem 2 is proved.

3. Application to subdifferentiability

Let X X XXX be a vector space and let ( Y , K Y , K Y,KY, KY,K ) be an ordered (topological) vector space. Suppose that F : X → ( Y , K ) F : X → ( Y , K ) F:X rarr(Y,K)F: X \rightarrow(Y, K)F:X→(Y,K) is a convex operator. We shall say that F F FFF is fully subdifferentiable at x 0 ∈ X x 0 ∈ X x_(0)in Xx_{0} \in Xx0∈X if (i) ∇ F ( x 0 ; ⋅ ) ∇ F x 0 ; ⋅ grad F(x_(0);*)\nabla F\left(x_{0} ; \cdot\right)∇F(x0;⋅) (or F ( x 0 ; ⋅ ) F x 0 ; ⋅ F(x_(0);*)F\left(x_{0} ; \cdot\right)F(x0;⋅) ) is defined on X X XXX, (ii) ∂ F ( x 0 ) ∂ F x 0 del F(x_(0))\partial F\left(x_{0}\right)∂F(x0) is non-empty, and (iii) for every h h hhh in X X XXX one has the relation ∇ F ( x 0 ; h ) = max { A h : A ∈ ∂ F ( x 0 ) } ( ∇ F x 0 ; h = max A h : A ∈ ∂ F x 0 grad F(x_(0);h)=max{Ah:A in del F(x_(0))}(:}\nabla F\left(x_{0} ; h\right)=\max \left\{A h: A \in \partial F\left(x_{0}\right)\right\}\left(\right.∇F(x0;h)=max{Ah:A∈∂F(x0)}( or F ( x 0 ; h ) = max { A h : A ∈ ∂ F ( x 0 ) } ) F x 0 ; h = max A h : A ∈ ∂ F x 0 {:F(x_(0);h)=max{Ah:A in del F(x_(0))})\left.F\left(x_{0} ; h\right)=\max \left\{A h: A \in \partial F\left(x_{0}\right)\right\}\right)F(x0;h)=max{Ah:A∈∂F(x0)}).
As a consequence of a result of Fel'dman [4] (see also [ 8 , 2 ] [ 8 , 2 ] [8,2][8,2][8,2] ) combined with some continuity properties of convex operators in [ 3 , 1 ] [ 3 , 1 ] [3,1][\mathbf{3}, \mathbf{1}][3,1] we have the following.
Proposition 4. Let F F FFF be a convex operator from the vector space X X XXX to the ordered vector space ( Y , K Y , K Y,KY, KY,K ) with the monotone net property or to the ordered topological vector space ( Y , K Y , K Y,KY, KY,K ) which is Daniell. Then F F FFF is fully subdifferentiable at each point of X X XXX.
If X X XXX is a topological vector space, ( Y , K Y , K Y,KY, KY,K ) is an ordered normal topological vector space which is Daniell, and if F F FFF is continuous at x 0 x 0 x_(0)x_{0}x0, then it is fully subdifferentiable at x 0 x 0 x_(0)x_{0}x0 and all the maps in the definition of the full subdifferentiability are continuous.
By using Theorems 1 and 2 we can give the following characterization of the ordered vector spaces in which all the convex operators are fully subdifferentiable.
′ ′ ^(^(')){ }^{'}′ Theorem 3. Let ( Y , K Y , K Y,KY, KY,K ) be an ordered vector space (respectively an ordered topological vector space with closed positive cone) which admits an isotone functional. Then the following assertions are equivalent.
(i) ( Y , K Y , K Y,KY, KY,K ) has the monotone sequence property (respectively is countable Daniell).
(ii) Every convex operator F F FFF from a vector space X X XXX to ( Y , K ) ( Y , K ) (Y,K)(Y, K)(Y,K) has directional minorants (respectively directional derivatives) in every direction at every point in X X XXX.
(iii) Every convex operator F F FFF from a vector space X X XXX to ( Y , K ) ( Y , K ) (Y,K)(Y, K)(Y,K) is fully subdifferentiable at each point of X X XXX.
Proof. Condition (i) implies condition (ii) by Proposition 1. To verify that (ii) implies (iii) we observe that in the presence of an isotone functional the monotone sequence property (respectively the countable Daniell property) implies the monotone net property (respectively the Daniell property), the reasoning being similar to that in [13, Proposition II.4.9]. Hence Proposition 4 can be used. Finally (iii) implies (i) according Theorem 1 (respectively Theorem 2).
Remark. In [ 4 , 2 ] [ 4 , 2 ] [4,2][4,2][4,2] it was shown that the condition ∂ F ( x ) ≠ ∅ ∂ F ( x ) ≠ ∅ del F(x)!=O/\partial F(x) \neq \varnothing∂F(x)≠∅ for each convex operator F F FFF with values in ( Y , K Y , K Y,KY, KY,K ), called the subgradient property of ( Y , K Y , K Y,KY, KY,K ), does not imply that the space has the monotone sequence property. In the given counterexamples ( Y , K Y , K Y,KY, KY,K ) is not Archimedian. Our results say nothing about the relationship between the monotone sequence property and either the subgradient or the Archimedian properties. (For other problems concerning the subgradient property see [10].)

References

  1. J. M. Borwein, 'Continuity and differentiability properties of convex operators', Proc. London Math. Soc. (3) 44 (1982) 420-444.
  2. J. M. Borwein, 'Subgradients of convex operators', Math. Operationsforsch. Statist. Ser. Optim. 15 (1984) 179-191.
  3. W. W. Breckner and G. Orbán, 'On the continuity of convex mappings, Rev. Anal. Numér. Théor. Approx. 6 (1977) 117-123.
  4. M. M. Fel'dman, 'About the sufficient conditions of the existence of supporting operators to sublinear mappings', Sibirsk. Mat. Ž. 16 (1975) 132-138 (Russian).
  5. V. L. LEVIN, 'The subdifferentials of convex mappings and composed functionals', Sibirsk. Mat. Ž. 13 (1972) 1295-1303 (Russian).
  6. C. W. McArthur, 'In what spaces is every closed normal cone regular?', Proc. Edinburgh Math. Soc. (2) 17 (1970) 121 − 125 121 − 125 121-125121-125121−125.
  7. C. W. McArthur, 'Convergence of monotone nets in ordered topological vector spaces', Studia Math. 34 (1970) 1-16.
  8. A. B. Németh, 'Some differential properties of convex mappings', Mathematica (Cluj) 22 (45) (1980) 107-114.
  9. A. B. Németh, 'Sequential regularity and directional differentiability of convex operators are equivalent', preprint, Babes-Bolyai University 1984.
  10. A. B. Németh, 'On some universal subdifferentiability properties of ordered vector spaces', preprint, Babes-Bolyai University 1985.
  11. N. S. Papagforgiu, 'Nonsmooth analysis on partially ordered vector spaces: part 1-convex case', Pacific J. Math 107 (1983) 403-485.
  12. A. L. Peressina, Ordered topological vector spaces (Harper & Row, New York 1967).
  13. H. Schaffer, Banach lattices and positive operators (Springer, Berlin 1974).
  14. M. Valadier, 'Sous-differentiabilité de fonctions convexes à valuers dans un espace vectoriel ordonné', Math. Scand. 30 (1972) 65-74.
  15. J. ZowE, 'Subdifferentiability of convex functions with values in ordered topological vector spaces', Math. Scand. 34 (1974) 69-83.
Institutul de Matematica CP 68
3400 Cluj-Napoca
Romania
  1. Received 21 October 1985.
    1980 Mathematics Subject Classification 47B55.
1986

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