On minimal representations by a family of sublinear functions

This paper is a continuation of Grzybowski et al. (J Glob Optim 46:589–601, 2010) and is motivated by the study of exhausters i.e. families of closed convex sets. By Minkowski duality closed convex sets correspond to sublinear functions. Here we study the criteria of reducing representations of pointwise infimum of an infinite family of sublinear functions. A family {fi}i∈I\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{f_i\}_{i\in I}$$\end{document} of sublinear functions is by definition an exhaustive family of upper convex approximations of its pointwise infimum infi∈Ifi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\inf _{i\in I}f_i$$\end{document}. A family of closed convex sets is by definition an exhauster of a pointwise infimum of a family of support functions of these convex sets. We establish codependence between infimum of a subset of Minkowski–Rådström–Hörmander cone and translation property of intersection of an exhauster (Sect. 4 ), between reducing an exhauster to single convex set and shadowing property of intersection of an exhauster (Theorem 5.1) and among all four of these properties (Theorem 6.1). In Grzybowski et al. (2010) the first example of two different minimal upper exhausters of the same function was presented. Here we give an example of infinitely many minimal exhausters of the same ph-function (Example 7.2). In Sect. 8, we give a criterion of minimality of an exhauster with the help of polars of sets belonging to the exhauster. We illustrate this criterion with two interesting examples (Example 8.3).


Introduction
The main object of this paper is to study reducibility and minimality of exhausters. The notion of upper/lower exhauster was introduced by Demyanov and Rubinov [10]. Exhausters were studied in a series of papers (see for example [3][4][5][6]28,29]).
Directional derivative of a function f : X −→ R at x 0 ∈ X is a positively homoge- If the directional derivative f (x 0 ; ·) is linear then f is differentiable at x 0 . If a function f is convex then f (x 0 ; ·) is sublinear. If the function f (x 0 ; ·) is sublinear then f is called subsmooth [27]. If a function f is a dc-function (difference of convex functions) then f (x 0 ; ·) a ds-function (difference of sublinear functions). If the function f (x 0 ; ·) is a ds-function then f is called quasi differentiable (tangentially dc-function) at x 0 ( [7,10,23]).
Let us notice that each ds-function f can be represented by a pair (A, B) of bounded closed convex sets, where f is equal to the difference of two support functions p A − p B . However, a much broader class of positively homogenous functions can be represented by a family of bounded closed convex sets as f = inf i∈I p A i . For example a very natural function f (x 1 , x 2 ) = √ |x 1 x 2 | is not a ds-function but it is a pointwise infimum of support functions of infinitely many rectangles [12].
The Minkowski duality is the correspondence between subdifferentials and support functions. Let Sublin(R n ) be the set of all sublinear functions and B(R n ) be the family of all nonempty bounded closed convex sets in R n . The set ∂ p| 0 = {x| x, · p} ∈ B(R n ) is called the subdifferential of p ∈ Sublin(R n ) at 0. The function p A = sup x∈A x, · ∈ Sublin(R n ) is called the support function of the set A ∈ B(R n ). The correspondence between (Sublin (R n ), +, ·, ) and (B(R n ), +, ·, ⊂) is an isomorphism of ordered abstract convex cones. The Minkowski duality can be extended to the family C(R n ) of all nonempty closed convex sets on one hand and to the set Sublin * (R n ) of all sublinear functions taking values in R = R ∪ {∞} on the other hand. Here, however, the addition A + B = {a + b|a ∈ a, b ∈ B} has to be replaced with A+B = A + B.
The following laws enable us to embed the abstract convex cone B(R n ) into the MRH space R n . Theorem 1.1 (Order law of cancellation) Let A, B, C ⊂ R n , B be bounded and C ∈ C(R n ).
The next law is a simple corollary.
The abstract convex cone C(R n ) cannot be embedded into a vector space. It can be embedded into the following ordered MRH cone R n c = C(R n ) × B(R n )/ ∼ (see [14]). For The upper (lower) exhauster is not unique. Pshenichnyi [23] introduced upper convex approximation with sublinear functions greater than h. Demyanov and Rubinov [8,9] studied exhaustive families of uca's.
Let us notice that all ds-functions are represented by upper exhausters. We have h = p A − p B = inf b∈B p A−b . Hence E * = {A − b} b∈B is an upper exhauster of h. Some functions represented with upper exhausters are not ds-functions. For example, the function h(x 1 , x 2 ) = √ |x 1 · x 2 |, which is not a ds-function and has an upper exhauster [12].

Reducing finite exhausters
In this section we gather results from [15] on reducing finite upper exhausters to single set in the form of Theorem 3.1 In such a case we have an exhauster of (Clarke) regular function f = inf i∈I p A i . It means that Frechet and Clarke (Michel-Penot) subdifferentials of f coincide.
Theorem 3.1 (on reducing finite upper exhausters to single set) Let the set of indices I be finite, {A i } i∈I ⊂ B(R n ) and A = i∈I A i . Then the following statements are equivalent: Proof By Lemma 2.2 in [15] the set sup i∈I k∈I \{i} A k is a summand of i∈I A i if and only Since the supremum of convex sets sup i∈I k∈I \{i} A k is the convex hull i∈I k∈I \{i} A k and the infimum inf i∈I A i is the intersection i∈I A i , the equivalence (c) ⇔ (e) follows immediately. The equivalence (b) ⇔ (c) follows from Theorem 5.2 in [15]. It should be mentioned, however, that in [15] the property (c) is called a shadowing property.
The equivalence (e) ⇔ (f) is obvious. By summand, we understand a summand with respect to Minkowski addition. By the generalized Minkowski duality [20] we have inf i∈I p The translation property (d) is new and we will elaborate it in the next sections.

Translation property of intersection
The following property gives the equivalence between generalized properties (c) and (d) from Theorem 3.1.
Then the following statements are equivalent:

Corollary 4.2 Let
Then the following statements are equivalent:

Corollary 4.3
Let {A i } i∈I ⊂ C(R n ) and A = i∈I A i . Then the following statements are equivalent: .
The family {A t } t∈R (Fig. 1) is an exhauster of the function h defined by The function h is not continuous but it is upper semicontinuous. Let H be the closed lower halfplane

Shadowing property
The shadowing property, mentioned in Theorem 3.1 (b), is a generalization of separation of two sets by another (separating) set. A shadowing set shadows, in general, arbitrary number of sets or a family of sets. The notion of shadowing was introduced in [15]. Theorem 5.1 (shadowing property of intersection) Let {A i } i∈I ⊂ C(R n ) and A = i∈I A i . Then the following statements are equivalent: Proof (a) ⇒ (b) Let us assume that (b) does not hold. Then there exist {a i } i∈I ∈ i∈I A i and ε > 0 such that (A + εB) ∩ i∈I a i = ∅. Hence the convex sets (A + εB) and i∈I a i can be weakly separated by a hyperplane. Then A and i∈I a i can be strictly separated. There exist x ∈ R n and α, β ∈ R such that a, x α < β a i , x for all i ∈ I and all a ∈ A. Therefore, p A < inf i∈I p A i , and (a) does not hold. (

b) ⇒ (a) Let us assume that (a) does not hold. Then p A (x) α < β inf i∈I p A i (x)
for some x ∈ R n and α, β ∈ R. For each i ∈ I let us choose a i ∈ A i such that β a i , x . Then the sets A and i∈I a i are strictly separated by two parallel hyperlanes ·, x = α and ·, x = β. Hence the condition (b) does not hold.

Corollary 5.2
Let {A i } i∈I ⊂ B(R n ) and A = i∈I A i . Then the following statements are equivalent: (a) inf i∈I p A i = p A .
(b) For all {a i } ∈ i∈I A i we have A ∩ i∈I a i = ∅ (shadowing property).
Proof In this corollary the set A is compact and convex. Hence A does not intersect closed convex sets if and only if A can be strictly separated from that set. Then the corollary follows from Theorem 5.1.
The following example shows that the closedness of the set i∈I a i in the statement (b) of Corollary 5.2 is essential. (Fig. 2).

Example 5.3 Let us consider an upper exhauster
We

Reducing infinite exhausters to single sets
The ability of replacing a given exhauster with one convex set is applicable also to many cases of exhausters irreducible to one convex set. Indeed, if an exhauster {A i } i∈I , is a union of a number of exhausters {A i } i∈I λ , λ ∈ , where I = λ∈ I λ and if each of exhausters {A i } i∈I λ is reducible to one set B λ then the exhauster {A i } i∈I λ can be reduced to "smaller" exhauster {B λ } λ∈ . Also if for a given convex set C the exhauster {C . + A i } i∈I is reducible to one convex set B then the exhauster {A i } i∈I is reducible to the pair of sets (B, C) in the sense that inf i∈I p A i = p B − p C .
The following theorem connects reducibility to one set with the translation property of intersection.
Theorem 6.1 (on reducing infinite upper exhausters to one set) Let {A i } i∈I ⊂ C(R n ) and A = i∈I A i . Then the following statements are equivalent: Proof (a) ⇒ (b) Let us assume that (b) does not hold. Then there exist D ∈ C(R n ) and x ∈ i∈I (D+A i ) \ (D+A). Since x can be strictly separated from D+A, there exists y ∈ R n such that

Hence (a) does not hold. (b) ⇒ (a) Let us assume that (a) does not hold. Then p A (x) α < β inf i∈I p A i (x)
for some x ∈ R n and α, β ∈ R. By H α we denote the halfspace {y ∈ R n | y, x α}. Proof By Theorem 6.1 the condition (b) follows from (a). Let us assume that (a) does not hold. Then p A (x) α < β inf i∈I p A i (x) for some x ∈ R n and α, β ∈ R. Hence there exists {a i } i∈I ∈ i∈I A i such that a i , x = β. For all y ∈ i∈I a i we have y, x = β. Hence A and i∈I a i do not intersect. Denote D = − i∈I a i . The vector 0 does not belong to A + D. On the other hand 0 ∈ D + A i for all i ∈ I . Then D + A i∈I (D + A i ), and since i∈I a i ⊂ i∈I A i is bounded, the condition (b) does not hold.

Minimality of upper exhausters
We say that an upper exhauster E * (h) = {A i } i∈I ⊂ B(R n ) is inclusion-minimal (or minimal in the sense of Demyanov [3]) if inf i∈I \{k} p A i = inf i∈I p A i = h for all k ∈ I . It means that if we drop any element of the exhauster we obtain an exhauster of different function h.
An exhauster E * (h) is I -minimal if inf( p A k , inf i∈I \{k} p A i ) = inf i∈I p A i = h for all k ∈ I and all B(R n ) A k A k . It means that if we replace any element of the exhauster with its proper subset we obtain an exhauster of a different function h.
We call an upper exhauster E * (h) = {A i } i∈I minimal if it is inclusion-minimal and I -minimal. For everyC ∈Ẽ * (h), there exists C ∈ E * (h) such thatC ⊂ C [28]. In other words the exhausterẼ * (h) is finer than E * (h).

minimal in the sense of Demyanov-Roshchina if and only if it is inclusion-minimal and I -minimal.
Proof If an exhauster E * (h) is minimal in the sense of Demyanov-Roshchina, then obviously it is minimal by inclusion and I -minimal.
Conversely, letẼ Then the familyÊ * = {A i } i∈I \{k} ∪ {A l } is also an upper exhauster of h, i.e.Ê * =Ê * (h). Since the exhauster E * (h) is I -minimal, we haveÊ * = E * (h). Hence A k = A l , and Example 7. 2 We present different minimal exhausters of the same function h. Consider two triangles Moreover the upper exhauster E * (h) is minimal. On the other hand, the upper exhauster E * 1 (h) = {B 1 , B 2 } is also minimal. Then the uniqueness of minimal exhausters does not hold (Fig. 3). These two exhausters were already presented in [15].
one of a continuum of three elements minimal exhausters (see Fig. 4).
Let   Fig. 6 An infinite minimal exhauster Example 7.3 In this example, we present a minimal infinite exhauster E * (h) = {A n } n∈N , where A n = cos i n , sin i n ∨ − cos i n , − sin i n ⊂ R 2 and h = inf n∈N p A n (see Fig. 6).
Notice that for any k ∈ N we have Hence our exhauster E * (h) is inclusion-minimal. On the other hand for any k ∈ N if we replace the segment A k with its proper subsegment A k then there exists a small real number x (positive or negative) such that Hence the exhauster E * (h) is I (= N)-minimal.

Reducing exhausters with the help of polars of convex sets
Let C 0 (R n ) be a family of sets from C(R n ) containing 0. Let A ∈ C 0 (R n ). By A • we denote the set {x ∈ R n | p A (x) 1} and call it the polar set or polar of A. Let R be the family of all radiant subsets B of R n , that is [0, 1]· B ⊂ B and the intersection of B and any ray L with initial point 0 is closed. We call a family is not a convex covering of B for all k ∈ I and all C ∈ C(R n ) such that B k C. We call the covering {B i } i∈I minimal if it is inclusion-minimal and I -minimal.    In Fig. 10 the set G is a union of polars of all maximal convex subsets of B. By the way a convex hull of G is equal to Clarke subdifferential [2] of the function h at origin.
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