Generalised subdifferentials

Analogously to the subdifferentials defined for standard abstract convexity notions we now want to define subdifferentials for the generalised abstract convexity of the previous section. It is not difficult to imagine how this would look like given that subdifferentials are defined for a point x0∈ X of a function f ∈ R^X and that in the above generalisation we have f as the pointwise supremum of functions from the sets W_x⊆ R^X.

Definition 5.18. Let X be a set, E = R^X, W_x⊆ R^X for all x ∈ X and W_X = {W_x}_x∈X. Further let f ∈ R^X and x0 ∈ X with f (x₀) ∈ R. Then wx0 ∈ W_x0 is called the WX -subgradient of f at x0 if

f (x) − f (x₀) ≥ w_x0(x) − w_x0(x₀)

for all x ∈ X. Further we denote the subdifferential of f by the set ∂WXf (x0) as the sets of all W_X-subgradients w_x0 of f at x₀, i.e.

∂_WXf (x0) = {wx0 ∈ W_x0 : f (x) − f (x0) ≥ wx0(x) − wx0(x0) for all x ∈ X}.

Analogously we define the sets VWx as the closure of each Wx under vertical shifts, so that for each x ∈ X we have V_Wx = {h : h(·) = w_x(·) − c, w_x ∈ W_x, c ∈ R}. Then yet again we obtain:

Proposition 5.19. Let f ∈ R^X and x0 ∈ X such that f (x₀) ∈ R. Then the WX -subdifferential ∂WXf (x0) of f at x0 is nonempty if and only if

f (x₀) = max{h(x₀) : h ≤ f and h ∈ V_Wx0}.

The proof can be transferred one to one from the original setting of Proposition 5.7 and will be omitted.

5.2.3. Dualities and Legendre-Fenchel transforms

In the corresponding section for original abstract convexity we discussed conjugations c between the sets R^X and R^W where X and W are two sets and in particular when W ⊆ R^X. It is obvious that we now immediately have a family of conjugations at our disposal, namely c_X = {cx}_x∈X where cx is a conjugation between R^X and R^Wx for each x ∈ X. By Theorem 5.8 each cx defines or is defined by a coupling function ϕ_x : X × W_x → R such that f^cx(w_x) = sup_y∈X{ϕ_x(y, w_x) u

−f (y)} for all f ∈ R^X and wx ∈ R^Wx. Furthermore, each of the conjugations cx has a dual conjugation c⁰_x and we define c⁰_X = {c⁰_x}_x∈X as the family of all dual conjugations. Accordingly, for h ∈ R^Wx we have that

h^c0x(y) = sup

wx∈W_x

{ϕ_c⁰

x(wx, x) u −h(w_x)} = sup

wx∈W_x

{ϕ_c(x, wx) u −h(w_x)}

Then for f ∈ R^X we call f^cxc0x the biconjugate of f at x. In the case of W_x ⊆ R^X for all x ∈ X it is now no longer appropriate to use the notation f^∗∗for the biconjugate since it depends on x and would otherwise be ambiguous. More implications of this dependence will become apparent in the next section.

5.2.4. Biconjugates and abstract convex hulls

In this section we only consider the case when W_x⊆ R^X for all x ∈ X. From the previous section we have the biconjugates f^cxc0x for each x ∈ X. Taking these we now define a new function f^cXc0X that we will be able to relate back to the WX-laminate operator from Section 5.2.1.

Definition 5.20. Let X be a set and W_X = {W_x}_x∈X with W_x ⊆ R^X. Further let cX = {cx}_x∈X be the family of conjugations associated to the natural coupling function ϕ_nat: X × W_x→ R, ϕnat(x, w) = w(x) for each x ∈ X and c^∗_X = {c⁰_x}_x∈X the family of their respective dual conjugations. Then we call the function f^cXc∗X defined by

f^cXc∗X(x) = f^cxc0x(x) the X-biconjugate of f .

Now let W_X = {W_x}_x∈X with W_x ⊆ R^X for all x ∈ X. Then by an abuse of notation we denote by WX + R the family {Wx+ R}x∈X, i.e. the closure of Wx under vertical shifts for each x ∈ X. Then we obtain the following relation between the X-biconjugate f^cXc∗X of f and the (W_X + R)-laminate (WX+ R)-lam(f ) of f .

Theorem 5.21. With the same assumption as in Definition 5.20 we have that f^cXc∗X = (W_X + R)-lam(f ).

Proof. By [55, Thm. 8.5] it holds in general that f^cxc0x = sup{ϕcx(·, wx) u r : wx ∈ W_x, r ∈ R, ϕcx(·, w_x) u r ≤ f } = sup{ϕ_cx(·, w_x) + r : w_x∈ W_x, r ∈ R, ϕcx(·, w_x) + r ≤ f }.

Specifically for ϕcx = ϕnat for all x ∈ X we obtain at x

f^cxc0x(x) = sup{wx(x) u r : wx∈ W_x, r ∈ R} = sup{wx(x) + r : wx∈ W_x, r ∈ R}

= ((WX + R)-lam f )(x)

and thus, since the above holds for all x ∈ X, we deduce f^cXc∗X = (W_X + R)-lam f .

Therefore we can deduce analogously:

Corollary 5.22. Let X be a set and W_X be as above. Then the following statements are equivalent

(i) f ∈ C(W_X u R), (ii) f ∈ C(W_X + R), (iii) f = f^cXc∗X,

(iv) f (x) = sup_wx∈Wx{w_x(x) u −f^cx(wx)} for all x ∈ X.

Note that in all the above we have used the notion c^∗_X instead of c⁰_X. This is due to the observation that c^∗_X is not the dual of c_X and therefore c⁰_X could lead to confusion.

The fact that c^∗_X is in general not the dual of c_X can be easily acknowledged with the equivalence to the WX-laminate operator, which does not in general equal the WX -convex hull operator and only converges to it by potentially infinite repeated application.

However, this raises the question whether there exists a suitable target space F and a conjugation c : R^X → F such that the biconjugate f^cc0 does equal the WX-convex hull of f , where c⁰: F → R^X is the dual conjugate of c. In the following we will show that there is always a trivial choice of such a conjugation.

Proposition 5.23. Let X be a set C(W_X) be the convexity system defined by W_X = {W_x}_x∈X for Wx ⊆ R^X for all x ∈ X and where Wx is closed under vertical shifts for all x ∈ X. Then the map c₀ : R^X → C(W_X) → R
defined such that

(c0(f ))(g) = sup

x∈X

{g(x) u −f (x)} (5.5)

is a conjugation and its biconjugate satisfies

c⁰₀c0f = WX-co f,

i.e. the biconjugate c⁰₀c₀ is corresponds to the W_X-convex hull operator.

Proof. For brevity we denote the space C(W_X) → R
by F . Note that F is not the dual space of C(W_X) as we do not require the maps from C(W_X) into R to be linear or continuous. In (5.5) we could replace the term g(x) by ϕ(x, g) if ϕ : X × C(WX) → R is the coupling function defined by ϕ(x, g) = g(x). Thus, by Theorem 8.2 of [55] it follows that c₀ is a conjugation.

We now show that its dual c⁰₀ : F → R^X is such that c⁰_Xc_X : R^X → R^X is the W_X-convex hull operator, i.e. c⁰₀c0= WX-co. As usual c⁰₀ is defined as

c⁰₀G = inf{h ∈ R^X : c₀h ≤ G}

= inf{h ∈ R^X : sup

x∈X

{g(x) u −h(x)} ≤ G(g) for all g ∈ C(W_X)} (5.6)

for G ∈ F = C(W_X) → R
. Let f ∈ R^X. First we show that c⁰₀c₀f ≤ W_X-co f by proving that c0(W_X-co f ) ≤ c0f , making W_X-co f a candidate for the attaining the infimum in (5.6) with G = c0f . Thus we must show that c0(WX-co f )(g) ≤ (c0f )(g) for all g ∈ C(W_X), which is equivalent to

sup

x∈X

{g(x) u − W_X-co f (x)} ≤ sup

x∈X

{g(x) u −f (x)}.

We may assume that for g ∈ C(W_X) the right hand side is finite and that the supremum is attained for x ∈ X, i.e. we have that

g(x) − f (x) ≤ g(x) − f (x)

for all x ∈ X (we can replace u by + since the right hand is finite). Rearranging the inequality we find that g(x) − g(x) + f (x) ≤ f (x) for all x ∈ X. Thus, since C(W_X) is closed under vertical shift the function g − g(x) + f (x) is W_X-convex and minorises f . Because WX-co f is the largest function in C(WX) that minorises f it also follows that g − g(x) + f (x) minorises W_X-co f , i.e.

g(x) − g(x) + f (x) ≤ W_X-co f (x)

for all x ∈ X. Again, by rearranging we find that sup_x∈X{g(x) u − W_X-co f (x)} ≤ g(x) − f (x) = sup_x∈X{g(x) u −f (x)}. Therefore, we have c_XW_X-co f ≤ c_Xf and hence c⁰_Xc_Xf ≤ W_X-co f .

To show the reverse inequality, let h = c⁰_XcXf . We now simply test cXh ≤ cXf with W_X-co f ∈ C(W_X). Clearly, (c_Xf )(W_X-co f ) = sup_x∈X{W_X-co f (x) u −f (x) ≤ 0} and hence we must have that sup_x∈X{W_X-co f (x) u −h(x)} ≤ 0. This is only possible if h ≥ WX-co f .

Although this proposition proves that c⁰₀c0 = WX-co it does not provide any advantages.

This is because the space F = C(W_X) → R
is extremely large. On the other hand, in classical abstract convexity it is known that a much smaller space can be used,

in particular c_W : R^X → F with F = (W → R) and cW defined analogously to c_W also satisfies c⁰_WcW = WX-co and W can be much smaller than C(W ) (e.g. affine functions as opposed to convex functions). Therefore, a natural question is whether another conjugation c_X : R^X → F with a smaller target space F can be defined such that c⁰_XcX = WX-co for a convexity system C(WX) in our generalised abstract convexity theory.

A natural candidate would be to consider c_X with the target space F =

X → R^W· whereby we mean that an element g ∈ F is a function on X and each g(x) is in turn a function from Wx into R. In particular, we could define (cXf )(x) = cxf ∈ (Wx → R).

Note that this is simply a different notation to writing c_X = {c_x}_x∈X as in Definition 5.20.

In fact, c_X is a conjugation since it satisfies (5.2) and (5.3). We formalise this in the following proposition.

Proposition 5.24. Let X and WX be as in Theorem 5.21. Then cX is a conjugation.

Proof. We need to verify (5.2) and (5.3). Let I be an index set and {fi}_i∈I ⊆ R^X a family of functions. Then, since each cx is a conjugation we have

c_X with the constant function in various spaces. We then have

c_X(f u d)(x) = cx(f u d) = cx(f ) u −d = c_X(f )(x) u −d(x) = (c_X(f ) u −d) (x), and therefore cX(f u d) = cX(f ) u −d.

We now consider the dual conjugation c⁰_X of cX. By the theory of duality it holds that c_Xc⁰_Xc_X = c_X and c⁰_Xc_Xc⁰_X = c⁰_X. Thus, c_Xc⁰_X is a hull operator. We therefore suspect that c⁰_X 6= c^∗_X in general as discussed before. We did not define the operator c^∗_X in our new notation, but in the context of F =



X → R^W·



the following is consistent. For g ∈

c_x(f )^c0x(x) = f^cxc0x(x). Recall, however, that the dual c⁰_X is defined via

c⁰_X(g) = inf{f ∈ R^X : c_X(f ) ≤ g} = inf{f ∈ R^X : c_X(f )(x) ≤ g(x) for all x ∈ X}

= inf{f ∈ R^X : f^cx ≤ g(x) for all x ∈ X}

= inf{f ∈ R^X : f^cx(w_x) ≤ g(x)(w_x) for all x ∈ X and w_x ∈ W_x},

where the last equality follows from f^cx, g(x) ∈ R^Wx where R^Wx is a lattice with the partial ordering ‘≤’ as a pointwise comparison, i.e. for f₁, f₂ ∈ R^Wx we have f₁ ≤ f₂ if and only if f1(wx) ≤ f2(wx) for all wx ∈ W_x. The following example will show that c⁰_XcXf does not always yield the desired result of being equal to W_X-co f . Note, however, that duality theory requires that c⁰_Xc_X is a hull operator, i.e. it must hold that c⁰_Xc_X(c⁰_Xc_X) = c⁰_Xc_X. Example 5.25. Let X = R^2×2 and let WX = {WF}_{F ∈X} with WF = {wF : X → R : w_F is 1-polyaffine at F }. Define the function f : R^2×2→ R such that

f (F ) =







0, F 6= 0 1, F = 0.

Then it holds that W_X-co f ≡ 0 while c⁰_Xc_Xf = f .

Proof. Let h = c⁰_Xc_Xf . Since f trivially satisfies c_Xf ≤ c_Xf it follows that h ≤ f . This in turn implies that cXh ≥ cXf , but h also satisfies cXh ≤ cXf by definition and hence c_Xh = c_Xf . In particular this implies that (c_Xh)(I) = (c_Xf )(I), or equivalently (cIh)(wI) = (cIf )(wI) for all wI ∈ W_I, where I ∈ R^2×2 is the identity matrix. Note that f is 1-polyaffine at F = I, i.e. f ∈ WI. Thus we must have that (cIh)(f ) = (cIf )(f ) = sup_{F ∈R}2×2{f (F ) − f (F )} = 0. It holds that 0 = (c_Ih)(f ) = sup_{F ∈R}2×2{f (F ) − h(F )} = max{sup_{F 6=0}{−h(F )}, 1 − h(0)}, which implies that both sup_{F 6=0}{−h(F )} ≤ 0 and 1 − h(0) ≤ 0 must be true. These two conditions imply that h ≥ f , which together with the inequality h ≤ f as observed above, implies h = f , as claimed.

Although W_X-co f 6= c⁰_Xc_Xf for this example it is still true that c⁰_Xc_X is a hull operator, albeit not a very useful one, as in the particular case we have that c⁰_Xc_X(c⁰_Xc_Xf ) = c⁰_XcXf = f . Furthermore, if instead we choose WF as the set of polyaffine functions from R^2×2→ R for all F ∈ R^2×2 we do obtain that W_R2×2-co f = c⁰_Xc_Xf = 0 as desired.

The definition of the conjugation c_X : R^X →

X → R^W·

was an attempt to choose a smaller target space than C(W_X) → R
for c₀, but it does not seem to be suitable for WX-convexity. It is currently not clear whether there exists a better definition of a conjugation c_X and its target space such that c⁰_Xc_X = W_X-co or whether the operator

c^∗_Xc_X that corresponds to the laminate operator W_X-lam is already the best possible choice.