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Characterizing global maximizers of the difference of sub-topical functions
Accepted Manuscript Characterizing Global Maximizers of the Difference of Sub-Topical Functions
H. Bakhtiari, H. Mohebi
PII: DOI: Reference:
S0022-247X(16)30866-6 http://dx.doi.org/10.1016/j.jmaa.2016.12.070 YJMAA 21010
To appear in:
Journal of Mathematical Analysis and Applications
Received date:
15 September 2016
Please cite this article in press as: H. Bakhtiari, H. Mohebi, Characterizing Global Maximizers of the Difference of Sub-Topical Functions, J. Math. Anal. Appl. (2017), http://dx.doi.org/10.1016/j.jmaa.2016.12.070
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Characterizing Global Maximizers of the Difference of Sub-Topical Functions H. Bakhtiari1 and H. Mohebi1,2 1
2
Department of Mathematics, Shahid Bahonar University of Kerman, P.O. Box: 76169133, Postal Code: 7616914111, Kerman, Iran. e-mail: [email protected] e-mail: [email protected] Present Address: Department of Applied Mathematics, University of New South Wales, Sydney, NSW, 2052, Australia. e-mail: [email protected] Abstract. Applying for a certain class of elementary functions, which is a generalization of the class of min-type functions, the techniques from abstract convex analysis, we give various characterizations of maximal elements of support sets of sub-topical functions. As an application, by using this class of elementary functions, we characterize global maximizers of the difference of two strictly sub-topical functions. These results have many applications in various parts of applied mathematics, mathematical economics and game theory.
Key words: global optimization; DC-functions; topical function; sub-topical function; elementary function; support set; maximal element; abstract convexity. 2010 (AMS) Mathematics Subject Classification: 90C26; 90C46; 26B25; 26A48; 26A09.
1
Introduction
To explain the main idea behind abstract convex analysis, there are the following wellknown results from convex analysis: (i) each lower semi-continuous convex function f is the upper envelope of the set of all affine functions which are minorants of f ; (ii) A set C is closed and convex if and only if each point which does not belong to this set can be separated from it by a linear function [21]. In some cases the use of separation of a set from an outside point and the envelope representation of a function are very convenient, even when we consider separation by non-linear functions and the upper envelope of non-affine functions [13, 20]. Thus, the idea arises to investigate some of the main concepts of convex analysis in the case of non-convexity, to study abstract convex functions, that is, functions 1
which can be presented as upper envelopes of subsets of a given set of not necessarily linear ”elementary functions”. Special concepts and techniques for the study of abstract convex functions, such as, abstract support sets of functions, abstract subdifferentials of functions, various kinds of conjugacy and polarity, were developed in abstract convex analysis [8, 9, 10, 13, 14, 15, 16, 17, 18, 19, 20]. For applications of abstract convex analysis, we need to describe the main objects of this theory in concrete situations under consideration. Just as the set of linear functions leads to the theory of usual convex sets and convex functions [21], each class of elementary functions leads to a theory of abstract convex sets and abstract convex functions with respect to this class [13]. Abstract convexity and many special concepts, such as, support sets, subdifferentials, conjugacy and polarity of topical functions with respect to a certain class of elementary functions have been investigated [10, 11, 16, 19]. In fact, functions of this kind have arisen recently in several contexts [6], and the term ”topical function” is due to Gunawardena and Keane [7]. Topical functions were intensively studied (see [4, 5, 6] and the references therein) and they have many applications in various parts of applied mathematics, in particular, in the modelling of discrete event systems [4, 5, 6, 7]. Topical functions are also interesting from a different point of view, namely as a tool in the study of the so-called downward sets. Downward sets arise in the study of some problems of mathematical economics and game theory and also in the study of inequality systems involving increasing functions [16]. Besides topical functions, it has been studied the much more general class of increasing and plus-sub-homogeneous functions, which called sub-topical functions [7, 16, 19]. In [1, 16, 19], it has been shown that each sub-topical function is abstract convex with respect to the following set of L elementary functions. For each y ∈ X and each α ∈ R, consider the function η(y,α) : X −→ R is defined by η(y,α) (x) := sup{λ ∈ R : λ ≤ α, λ1 ≤ x + y}, ∀ x ∈ X, where L := {η(y,α) : y ∈ X, α ∈ R}
(1.1)
is called the class of elementary functions, which is a generalization of the class of mintype functions [16, 19]. Thus the question arises: to apply the main concepts and results of abstract convex analysis corresponding to the set of elementary functions defined by (1.1). We address this question in the present paper, we give various characterizations of maximal elements of support sets of sub-topical functions by using this class of elementary sub-topical functions. As an application, one of the most important global optimization problems [12], is that of minimizing a DC-functions (difference of two convex functions), that is, minimize h(x) subjet to x ∈ X, where h := g − f and f , g are convex functions. In a general case, DC-functions can be replaced by DAC-functions (difference of two abstract convex functions). In particular, 2
minimizing of the difference of two increasing and co-radiant (ICR) functions [3], and minimizing of the difference of two topical functions [2]. In this paper, we replace f and g by two strictly sub-topical (i.e., strictly increasing and sub-homogeneous) functions, and by using the class of elementary functions defined by (1.1), give necessary and sufficient conditions for global maximizers of the difference of two strictly sub-topical functions defined on a real normed linear space X. The layout of the paper is as follows: In Section 2, we collect definitions, notations and preliminary results relative to sub-topical and elementary functions. Characterizations of maximal elements of support sets of sub-topical functions are given in Section 3. In Section 4, as an application, we present necessary and sufficient conditions for global maximizers of the difference of two strictly sub-topical functions. Section 5, concludes with a discussion on conclusions and applications.
2
Preliminaries
Let X be a real normed linear space and S be a closed convex cone in X such that S ∩ {−S} = {0} and intS = ∅. We assume that X is equipped with the order relation induced by S : x ≤ y ⇐⇒ y − x ∈ S,
(x, y ∈ X).
Also, for x, y ∈ X, we say that x < y if and only if y − x ∈ S \ {0}. Moreover, we assume that S is normal in the sense that there exists a positive real number m > 0 such that x ≤ m y , whenever 0 ≤ x ≤ y with x, y ∈ X. Throughout the paper we assume that 1 ∈ intS. Let B := {x ∈ X : −1 ≤ x ≤ 1}. (2.1) It is well known and easy to check that B can be considered as the unit ball of a certain norm . 1 , which is equivalent to the initial norm . . ¯ := [−∞, +∞] is called increasing if x ≤ y =⇒ f (x) ≤ f (y), A function f : X −→ R (x, y ∈ X). ¯ is called plus-homogeneous if Recall [7, 16, 19] that a function f : X −→ R f (x + μ1) = f (x) + μ, ∀ x ∈ X, ∀ μ ∈ R. The function f is called topical if f is increasing and plus-homogeneous [16]. ¯ is called plus-sub-homogeneous if Recall [16, 19] that a function f : X −→ R f (x + μ1) ≤ f (x) + μ, ∀ x ∈ X, ∀ μ ≥ 0.
(2.2)
Functions satisfying (3.9) have been considered in [7], where they have been called subhomogeneous functions. 3
It is easy to check that f is plus-sub-homogeneous if f (x + μ1) ≥ f (x) + μ, ∀ x ∈ X, ∀ μ ≤ 0. The function f is called sub-topical if f is increasing and plus-sub-homogeneous [16, 19]. ¯ be a function. We say that f is strictly increasing if Definition 2.1. Let f : X −→ R x < y =⇒ f (x) < f (y), (x, y ∈ X). f is called strictly sub-topical if it is strictly increasing and plus-sub-homogeneous. Example 2.1. Let f, g : R2 −→ R be defined by 1 f (x, y) := x, ∀ x, y ∈ R, 2
and g(x, y) :=
1 1 4 x + 4 y, 1 3 4y − 2,
x ≤ y − 3, ∀ x, y ∈ R. x ≥ y − 3,
It is not difficult to check that f and g are strictly sub-topical functions. Moreover, it is clear that every topical function is sub-topical. Also, if f is a sub-linear function on X such that f (1) ≤ 1, then, f is sub-topical. ¯ the Remark 2.1. [16]. It is easy to see that for a sub-topical function f : X −→ R following assertions are true. (i) If there exists x ∈ X such that f (x) = +∞, then, f ≡ +∞. (ii) If there exists x ∈ X such that f (x) = −∞, then, f ≡ −∞. ¯ either dom f = X, or f ≡ +∞, where Hence, for any sub-topical function f : X −→ R, dom f := {x ∈ X : f (x) < +∞}. In the sequel, consider the function η : X × X × R −→ R is defined by η(x, y, α) := sup{λ ∈ R : λ ≤ α, λ1 ≤ x + y}, ∀ x, y ∈ X, ∀ α ∈ R.
(2.3)
It is worth noting that the function η is a generalization of the min-type function defined on Rn [16, 19]. In view of (2.1), it is easy to see that the set {λ ∈ R : λ ≤ α, λ1 ≤ x + y} is non-empty and bounded in R. Clearly, this set is also a closed subset of R. Therefore, one has sup = max, and η(x, y, α)1 ≤ x + y, ∀ x, y ∈ X, ∀ α ∈ R.
(2.4)
Also, in view of the definition, we have η(x, y, α) ≤ α, ∀ x, y ∈ X, ∀ α ∈ R. The following properties of the function η were obtained in [1, 11].
4
(2.5)
For every x, x , y, y ∈ X; γ, α, α ∈ R and μ ≥ 0, one has η(x + μ1, y, α) ≤ η(x, y, α) + μ,
(2.6)
η(x, y, α) = η(y, x, α),
(2.7)
x ≤ x =⇒ η(x, y, α) ≤ η(x , y, α),
(2.8)
y ≤ y =⇒ η(x, y, α) ≤ η(x, y , α),
(2.9)
η(x, −x + α1, α) = α,
(2.10)
α ≤ α =⇒ η(x, y, α) ≤ η(x, y, α ).
(2.11)
η(x, y + γ1, α) = η(x, y, α − γ) + γ.
(2.12)
Now, for each y ∈ X and each α ∈ R, define the function η(y,α) : X −→ R by η(y,α) (x) := η(x, y, α) for all x ∈ X. In view of (2.6) and (2.8), we have the function η(y,α) is sub-topical for each y ∈ X and each α ∈ R. Let Ω := {η(y,α) : y ∈ X, α ∈ R}.
(2.13)
Ω is called the set of elementary functions. Lemma 2.1. The element (y, α) ∈ X ×R in the definition of η(y,α) is uniquely determined. That is, if (y1 , α1 ), (y2 , α2 ) ∈ X × R and η(y1 ,α1 ) (x) = η(y2 ,α2 ) (x), ∀ x ∈ X,
(2.14)
then, y1 = y2 and α1 = α2 . Proof. By putting x := −y1 + α1 1 and x := −y2 + α2 1 in (2.14), we obtain from (2.5) and (2.10) that α1 = α2 . Also, in view of (2.14), one has η(y1 ,α1 ) (x) ≥ η(y2 ,α2 ) (x), ∀ x ∈ X.
(2.15)
Put x := −y2 + α2 1 in (2.15), it follows from (2.10) and the fact that α1 = α2 , η(y1 ,α1 ) (−y2 + α2 1) = α1 . This together with (2.4) implies that α1 1 ≤ −y2 + α2 1 + y1 = −y2 + α1 1 + y1 . This implies that y2 ≤ y1 . Since, in view of (2.14), we have η(y1 ,α1 ) (x) ≤ η(y2 ,α2 ) (x), ∀ x ∈ X,
(2.16)
so, by putting x := −y1 + α1 1 in (2.16) and by using a similar argument as the above, we conclude that y1 ≤ y2 , and hence, y1 = y2 . Corollary 2.1. If we define the function ψ : X × R −→ Ω by ψ(y, α) := η(y,α) for all y ∈ X and all α ∈ R. Then, ψ is bijective (one-to-one and onto) and increasing. 5
Proof. This is an immediate consequence of Lemma 2.1, (2.9) and (2.11). ¯ the support set of f with In the following, recall [13, 20] that for a function f : X −→ R respect to Ω is defined by supp (f, Ω) := {η(y,α) ∈ Ω : η(y,α) (x) ≤ f (x), ∀ x ∈ X}. For an easy reference, we state the following results from [1]. ¯ be a sub-topical function. Then, Proposition 2.1. [1]. Let f : X −→ R supp (f, Ω) = {η(y,α) ∈ Ω : f (−y + α1) ≥ α}.
¯ be a function. Then, f is sub-topical if and only if Theorem 2.1. [1]. Let f : X −→ R there exists a set A ⊆ Ω such that f (x) =
sup η(y,α) (x), (x ∈ X).
η(y,α) ∈A
In this case, one can take A := supp (f, Ω).
3
Characterizations of Maximal Elements of Support Sets of Sub-Topical Functions
In this section, we give various characterizations of maximal elements of support sets of sub-topical functions with respect to a certain class of elementary functions. Let Ω be the set of elementary functions defined by (2.13). In the sequel, we assume that supp (f, Ω) = ∅, and supp (f, Ω) = Ω,
(3.1)
(see also, Corollary 2.1). First, we give a definition of a maximal element [13]. Definition 3.1. [13]. Let H be a set of real valued functions defined on a set Z. We assume that H is equipped with the natural (point-wise) order relation of functions. We say that a function f¯ ∈ H is a maximal element of the set H, if f ∈ H and f (x) ≥ f¯(x) for all x ∈ Z, then, f¯(x) = f (x) for all x ∈ Z. ¯ be a sub-topical function, and let η(y,α) ∈ Ω be a Proposition 3.1. Let f : X −→ R maximal element of supp (f, Ω). Then, f (−y + α1) = α.
6
Proof. Let η(y,α) ∈ Ω be a maximal element of supp (f, Ω). First, note that if f (−y +α1) = +∞, then, in view of Remark 2.1, one has f ≡ +∞, and so, supp (f, Ω) = Ω, which contradicts (3.1). Also, similarly, if f (−y + α1) = −∞, it follows from Remark 2.1 that supp (f, Ω) = ∅, which contradicts (3.1). So, f (−y + α1) ∈ R. Now, put y0 := −(−y + α1) + f (−y + α1)1 and α0 := f (−y + α1). Then, y0 ∈ X, α0 ∈ R, and in view of (2.10), we have η(y0 ,α0 ) (−y + α1) = η(−y + α1, −(−y + α1) + f (−y + α1)1, f (−y + α1)) = f (−y + α1).
(3.2)
¯ is a sub-topical function and η(y,α) ∈ supp (f, Ω), in view of Proposition Since f : X −→ R 2.1, f (−y +α1) ≥ α. This together with (3.2) implies that η(y0 ,α0 ) (−y +α1) ≥ α. Thus, by Proposition 2.1 and the fact that η(y0 ,α0 ) is sub-topical, we have η(y,α) ∈ supp (η(y0 ,α0 ) , Ω), and so, (3.3) η(y,α) (x) ≤ η(y0 ,α0 ) (x), ∀ x ∈ X. Also, we have f (−y0 + α0 1) = f (−y + α1 − f (−y + α1)1 + f (−y + α1)1) = f (−y + α1) = α0 . Therefore, it follows from Proposition 2.1 that η(y0 ,α0 ) ∈ supp (f, Ω). Since η(y,α) is a maximal element of supp (f, Ω), we conclude from (3.3) that η(y,α) (x) = η(y0 ,α0 ) (x), ∀ x ∈ X.
(3.4)
Put x := −y + α1 in (3.4). Hence, by using (2.10), one has α = η(y,α) (−y + α1) = η(y0 ,α0 ) (−y + α1) = η(−y + α1, −(−y + α1) + f (−y + α1)1, f (−y + α1)) = f (−y + α1), which completes the proof. Now, by the following example, we show that the converse statement to Proposition 3.1 is not valid. Example 3.1. Let f : R −→ R be defined by 0, x ≤ 0, f (x) := x, x > 0, 7
∀ x ∈ R.
It is easy to see that f is a sub-topical function, but not strictly. Moreover, for each α > 0, one has η(0,α) (x) = min{α, x}, ∀ x ∈ R. Then, η(0,α) (x) ≤ f (x) for all x ∈ R and all α > 0, and so, η(0,α) ∈ supp (f, Ω) for all α > 0. Moreover, f (−0 + α1) = f (α) = α for all α > 0. But, in view of Corollary 2.1, supp (f, Ω) does not have a maximal element. By using the following lemma, we shall show that the converse statement to Proposition 3.1 is valid. ¯ be a sub-topical function. Let (y, α) ∈ X × R be such Lemma 3.1. Let f : X −→ R that f (−y + α1) = α. Assume that there exists η(y0 ,α0 ) ∈ supp (f, Ω) such that η(y,α) (x) ≤ η(y0 ,α0 ) (x) for all x ∈ X. Then, α ≤ α0 , y ≤ y0 and f (−y0 + α0 1) = α0 . Proof. Suppose that there exists η(y0 ,α0 ) ∈ supp (f, Ω) such that η(y,α) (x) ≤ η(y0 ,α0 ) (x), ∀ x ∈ X.
(3.5)
Put x := −y + α1 in (3.5). So, by (2.10) and (2.5), we have α ≤ α0 . Moreover, since η(y0 ,α0 ) ∈ supp (f, Ω), by (2.10) one has α = η(y,α) (−y + α1) ≤ η(y0 ,α0 ) (−y + α1) ≤ f (−y + α1) = α. This implies that α = η(y0 ,α0 ) (−y + α1), and so, in view of (2.4), α1 ≤ −y + α1 + y0 . That is, y ≤ y0 . Now, we show that f (−y0 + α0 1) = α0 . Since y ≤ y0 , f is increasing and η(y0 ,α0 ) ∈ supp (f, Ω), it follows from Proposition 2.1 that f (−y + α0 1) ≥ f (−y0 + α0 1) ≥ α0 .
(3.6)
Let y1 := −(−y + α0 1) + f (−y + α0 1)1 and α1 := f (−y + α0 1). Clearly, y1 ∈ X and α1 ∈ R. By using (2.10) and (3.6), we deduce that η(y1 ,α1 ) (−y + α0 1) = f (−y + α0 1) ≥ α0 . So, since η(y1 ,α1 ) is a sub-topical function, in view of Proposition 2.1, one has η(y,α0 ) ∈ supp (η(y1 ,α1 ) , Ω). Then, η(y,α0 ) (x) ≤ η(y1 ,α1 ) (x), ∀ x ∈ X. 8
(3.7)
Also, we have f (−y1 + α1 1) = f (−y + α0 1 − f (−y + α0 1)1 + f (−y + α0 1)1) = f (−y + α0 1) = α1 . Thus, by Proposition 2.1, η(y1 ,α1 ) ∈ supp (f, Ω). This together with (2.11) and (3.7) and the fact that α ≤ α0 implies that η(y,α) (x) ≤ η(y,α0 ) (x) ≤ η(y1 ,α1 ) (x) ≤ f (x), ∀ x ∈ X.
(3.8)
Put x := −y + α1 in (3.8). Then, by (2.10), α = η(y,α) (−y + α1) ≤ η(y,α0 ) (−y + α1) ≤ η(y1 ,α1 ) (−y + α1) ≤ f (−y + α1) = α. This implies that η(y,α0 ) (−y + α1) = η(y1 ,α1 ) (−y + α1), and hence, by (2.12), we conclude that η(y,α0 ) (−y + α1) = η(y1 ,α1 ) (−y + α1) = η(−y + α1, y + [f (−y + α0 1) − α0 ]1, f (−y + α0 1)) = η(−y + α1, y, f (−y + α0 1) − [f (−y + α0 1) − α0 ]) + f (−y + α0 1) − α0 = η(y,α0 ) (−y + α1) + f (−y + α0 1) − α0 . Therefore, it follows that f (−y+α0 1) = α0 , and so, in view of (3.6), we get f (−y0 +α0 1) = α0 , which completes the proof. ¯ we put In the sequel, for each y ∈ X and each function h : X −→ R, γy,h := max{α ∈ R : h(−y + α1) ≥ α}.
(3.9)
Now, we show that if f is a strictly sub-topical function, then the converse statement to Proposition 3.1 is valid. ¯ be a strictly sub-topical function. Then, η(y,α) ∈ Ω is a Theorem 3.1. Let f : X −→ R maximal element of supp (f, Ω) if and only if f (−y + α1) = α and α = γy,f , where γy,f defined by (3.9).
9
Proof. Suppose that η(y,α) ∈ Ω is a maximal element of supp (f, Ω). Since f is sub-topical, it follows from Proposition 3.1 that f (−y + α1) = α. Now, we show that α = γy,f . Let C := {α ∈ R : f (−y + α1) ≥ α}. Since f (−y + α1) = α, then, α ∈ C. Assume if possible that there exists γ0 ∈ C such that α < γ0 . Therefore, by (2.11), one has η(y,α) (x) ≤ η(y,γ0 ) (x) for all x ∈ X. Since by the hypothesis η(y,α) is a maximal element of supp (f, Ω), so, η(y,α) (x) = η(y,γ0 ) (x), ∀ x ∈ X, (3.10) (note that since γ0 ∈ C, hence, f (−y + γ0 1) ≥ γ0 . Therefore, in view of Proposition 2.1, η(y,γ0 ) ∈ supp (f, Ω)). Put x := −y + γ0 1 in (3.10). Hence, by using (2.5) and (2.10), α ≥ η(y,α) (−y + γ0 1) = η(y,γ0 ) (−y + γ0 1) = γ0 . So, α ≥ γ0 , which is a contradiction. Thus, α = γy,f . Conversely, suppose that f (−y + α1) = α and α = γy,f . Let η(y0 ,α0 ) ∈ supp (f, Ω) be such that η(y,α) (x) ≤ η(y0 ,α0 ) (x) for all x ∈ X. We prove that η(y,α) (x) = η(y0 ,α0 ) (x) ¯ is sub-topical, f (−y + α1) = α and for all x ∈ X. To this end, since f : X −→ R η(y0 ,α0 ) ∈ supp (f, Ω) is such that η(y,α) (x) ≤ η(y0 ,α0 ) (x) for all x ∈ X, it follows from Lemma 3.1 that y ≤ y0 , α ≤ α0 and f (−y0 + α0 1) = α0 . This together with f is increasing implies that f (−y + α0 1) ≥ α0 . Thus, α0 ∈ {α ∈ R : f (−y + α1) ≥ α}, and so, by (3.9) and the fact that α = γy,f , we have α ≥ α0 . Hence, α = α0 . Thus, one has f (−y0 + α1) = f (−y0 + α0 1) = α0 = α = f (−y + α1). This together with −y0 +α1 ≤ −y+α1 and f is strictly increasing implies that −y0 +α1 = −y + α1, and hence, y = y0 . Thus, η(y,α) (x) = η(y0 ,α0 ) (x) for all x ∈ X. So, η(y,α ) is a maximal element of supp (f, Ω) ( note that since f (−y + α1) = α, in view of Proposition 2.1, one has η(y,α) ∈ supp (f, Ω)). ¯ be a strictly sub-topical function. Let γy,f be as Proposition 3.2. Let f : X −→ R in (3.9). Then, for each η(y,α) ∈ supp (f, Ω) there exists a maximal element η(y0 ,α0 ) of supp (f, Ω) such that η(y,α) (x) ≤ η(y0 ,α0 ) (x) for all x ∈ X. Proof. Let η(y,α) ∈ supp (f, Ω) be arbitrary. Put y0 := −(−y + γy,f 1) + f (−y + γy,f 1)1 and α0 := f (−y + γy,f 1). Since f (−y + γy,f 1) ≥ γy,f , it follows that y ≤ y0 and γy,f ≤ α0 . Also, clearly, f (−y0 + α0 1) = α0 , and so, it follows from Proposition 2.1 that η(y0 ,α0 ) ∈ supp (f, Ω). Also, since η(y,α) ∈ supp (f, Ω) and f is sub-topical, by Proposition 2.1, one has f (−y + α1) ≥ α, and so, α ∈ {α ∈ R : f (−y + α1) ≥ α}. Hence, γy,f ≥ α. Thus, α0 ≥ γy,f ≥ α.
10
This together with (2.9) and (2.11) and the fact y ≤ y0 implies that η(y0 ,α0 ) (x) ≥ η(y,α0 ) (x) ≥ η(y,γy,f ) (x) ≥ η(y,α) (x), ∀ x ∈ X. Therefore, η(y,α) (x) ≤ η(y0 ,α0 ) (x), ∀ x ∈ X.
(3.11)
Now, we show that η(y0 ,α0 ) is a maximal element of supp (f, Ω). To this end, let δ ∈ R be such that f (−y0 + δ1) ≥ δ. Thus, since f (−y + γy,f 1) ≥ γy,f , we conclude that f (−y0 + δ1) ≥ δ ≥ δ − [f (−y + γy,f 1) − γy,f ] = δ − f (−y + γy,f 1) + γy,f . =⇒ f (−y0 + δ1) ≥ δ − f (−y + γy,f 1) + γy,f , =⇒ f (−y + γy,f 1 − f (−y + γy,f 1)1 + δ1) ≥ δ − f (−y + γy,f 1) + γy,f , =⇒ f (−y + [δ − f (−y + γy,f 1) + γy,f ]1) ≥ δ − f (−y + γy,f 1) + γy,f , =⇒ δ − f (−y + γy,f 1) + γy,f ∈ {α ∈ R : f (−y + α1) ≥ α}, =⇒ γy,f ≥ δ − f (−y + γy,f 1) + γy,f , =⇒ δ ≤ f (−y + γy,f 1) = α0 . This together with f (−y0 + α0 1) = α0 implies that α0 = γy0 ,f .
(3.12)
Therefore, the result follows from Theorem 3.1. Remark 3.1. It is worth noting that under the hypotheses of Proposition 3.2, α0 = γy0 ,f . The following theorem has a crucial role for characterizing maximizers of the difference of two strictly sub-topical functions. ¯ be strictly sub-topical functions. Let γy,f be as in (3.9). Theorem 3.2. Let f, g : X −→ R Then the following assertions are equivalent: (1) supp (f, Ω) ⊆ supp (g, Ω), (2) g(−y + γy,f 1) ≥ γy,f for each y ∈ X. Proof. (1) =⇒ (2). Suppose that (1) holds. Let y ∈ X be arbitrary. Then, in view of the hypothesis, f (−y + γy,f 1) ≥ γy,f , and so, by Proposition 2.1, we obtain η(y,γy,f ) ∈ supp (f, Ω). Thus, by the hypothesis (1), η(y,γy,f ) ∈ supp (g, Ω). Again, in view of Proposition 2.1, g(−y + γy,f 1) ≥ γy,f . That is, (2) holds. (2) =⇒ (1). Assume that (2) holds. Let η(y,α) ∈ supp (f, Ω) be arbitrary. Then, in view of 11
Proposition 3.2 and Remark 3.1, there exists a maximal element η(y0 ,γy0 ,f ) of supp (f, Ω) such that (3.13) η(y,α) (x) ≤ η(y0 ,γy0 ,f ) (x), ∀ x ∈ X. Now, set y1 := −(−y0 + γy0 ,f 1) + g(−y0 + γy0 ,f 1)1 and α1 := g(−y0 + γy0 ,f 1). Clearly, g(−y1 + α1 1) = α1 , and hence, by Proposition 2.1, we have η(y1 ,α1 ) ∈ supp (g, Ω). On the other hand, by the hypothesis (2), we have g(−y0 + γy0 ,f 1) ≥ γy0 ,f . Therefore, in view of (2.10), η(y1 ,α1 ) (−y0 + γy0 ,f 1) = g(−y0 + γy0 ,f 1) ≥ γy0 ,f .
(3.14)
Since η(y1 ,α1 ) is a sub-topical function, we conclude from (3.14) and Proposition 2.1 that η(y0 ,γy0 ,f ) ∈ supp (η(y1 ,α1 ) , Ω), and so, η(y0 ,γy0 ,f ) (x) ≤ η(y1 ,α1 ) (x), ∀ x ∈ X.
(3.15)
Since η(y1 ,α1 ) ∈ supp (g, Ω), it follows from (3.13) and (3.15) that η(y,α) ∈ supp (g, Ω). Hence, supp (f, Ω) ⊆ supp (g, Ω), which completes the proof.
4
Necessary and Sufficient Conditions for Maximizers of the Difference of Two Sub-Topical Functions
In this section, we present necessary and sufficient conditions for global maximizers of the ¯ be sub-topical functions difference of two strictly sub-topical functions. Let f, g : X −→ R such that dom(f ) ⊆ dom(g). Let h := g − f , that is, g(x) − f (x), x ∈ dom (f ) , h(x) := (4.1) −∞, x∈ / dom (f ) . (with the convention (+∞) − (+∞) := −∞). ¯ be sub-topical functions. Then, Lemma 4.1. Let f, g : X −→ R f (x) ≤ g(x), ∀ x ∈ X ⇐⇒ supp (f, Ω) ⊆ supp (g, Ω).
(4.2)
Proof. Indeed, if f (x) ≤ g(x) for all x ∈ X, then it is clear that supp (f, Ω) ⊆ supp (g, Ω). Conversely, assume if possible that there exists x0 ∈ X such that f (x0 ) > g(x0 ). Since f is a sub-topical function, it follows from Theorem 2.1 that f (x0 ) =
sup η(y,α) ∈supp (f,Ω)
η(y,α) (x0 ).
This together with the fact that f (x0 ) > g(x0 ) implies that there exists η(y0 ,α0 ) ∈ supp (f, Ω) such that η(y0 ,α0 ) (x0 ) > g(x0 ). This implies that η(y0 ,α0 ) ∈ / supp (g, Ω), which is a contradiction. Hence, f (x) ≤ g(x) for all x ∈ X. 12
Now, in the following, we give characterizations of global maximizers of the difference of strictly sub-topical functions. ¯ be strictly sub-topical functions. Let γy,g be as in Proposition 4.1. Let f, g : X −→ R (3.9). Then, x0 ∈ X is a global maximizer of the function h (defined by (4.1)) if and only if f (−y + γy,g 1) + h(x0 ) ≥ γy,g for each y ∈ X. Proof. We have x0 is a global maximizer of the function h if and only if h(x) ≤ h(x0 ) for all x ∈ X, if and only if g(x) ≤ f (x) + h(x0 ) for all x ∈ X, if and only if g(x) ≤ f˜(x) := f (x) + h(x0 ) for all x ∈ X, in view of Lemma 4.1, if and only if supp(g, Ω) ⊆ supp(f˜, Ω), by Theorem 3.2, if and only if f˜(−y + γy,g 1) ≥ γy,g for each y ∈ X, if and only if f (−y + γy,g 1) + h(x0 ) ≥ γy,g for each y ∈ X (it is worth noting that the function f˜ is a strictly sub-topical function). ¯ be strictly sub-topical functions. Let γy,g be as in (3.9). Theorem 4.1. Let f, g : X −→ R Then, x0 ∈ X is a global maximizer of the function h if and only if h(x0 ) = sup {γy,g − f (−y + γy,g 1)}. y∈X
(4.3)
Proof. Suppose that (4.3) holds. Then, f (−y + γy,g 1) + h(x0 ) ≥ γy,g for each y ∈ X. Hence, in view of Proposition 4.1, x0 is a global maximizer of the function h. Conversely, assume that x0 ∈ X is a global maximizer of the function h. Then, by Proposition 4.1, one has f (−y + γy,g 1) + h(x0 ) ≥ γy,g for each y ∈ X, and so, h(x0 ) ≥ γy,g − f (−y + γy,g 1) for each y ∈ X. This implies that h(x0 ) ≥ z, where we define z by z := sup {γy,g − f (−y + γy,g 1)}. y∈X
We claim that h(x0 ) = z. If h(x0 ) = z, then, h(x0 ) > z. Consider z ∈ R such that h(x0 ) > z ≥ z. Since z ≥ γy,g − f (−y + γy,g 1) for each y ∈ X, hence, z ≥ γy,g − f (−y + γy,g 1) for each y ∈ X. This implies that f (−y + γy,g 1) + z ≥ γy,g for each y ∈ X, and so, by Theorem 3.2 (the implication (2) =⇒ (1)), we conclude that supp(g, Ω) ⊆ supp(f¯, Ω), where f¯(x) := f (x) + z for all x ∈ X. Thus, in view of Lemma 4.1, we deduce that g(x) ≤ f¯(x) = f (x) + z for all x ∈ X. Therefore, in particular, h(x0 ) ≤ z , which is a contradiction (note that f¯ is a strictly sub-topical function). Example 4.1. Let X := {f : [−1, 1] −→ R : f is a continuos f unction} be the Banach space of all continuous real valued functions defined on [−1, 1], and let S := {f ∈ X : f (t) ≥ 0, ∀ t ∈ [−1, 1]}. It is clear that S is a closed convex pointed and normal cone in X, which contains the function 1 (1(t) := 1 for all t ∈ [−1, 1]). Fix, t0 ∈ [−1, 1], and consider functions ϕ, ψ : X −→ R are defined by 1 ϕ(f ) := f (t0 ), ∀ f ∈ X, 2 13
and ψ(f ) :=
1 inf f (t), ∀ f ∈ X. 2 −1≤t≤1
Clearly, ϕ and ψ are strictly sub-topical functions. Let h := ψ − ϕ. Also, one has γf,ψ = max{δ ∈ R : ψ(−f + δ1) ≥ δ} 1 = max{δ ∈ R : inf (−f (t) + δ) ≥ δ} 2 −1≤t≤1 1 1 = max{δ ∈ R : δ ≤ inf (−f )(t)} 2 2 −1≤t≤1 = max{δ ∈ R : δ ≤ inf (−f )(t)} −1≤t≤1
=
inf (−f )(t), ∀ f ∈ X.
−1≤t≤1
Now, consider sup {γf,ψ − ϕ(−f + γf,ψ 1)} = sup { inf (−f )(t) − ϕ(−f + [ inf (−f )(t)]1)} f ∈X −1≤t≤1
f ∈X
−1≤t≤1
1 = sup { inf (−f )(t) − [(−f )(t0 ) + inf (−f )(t)]} −1≤t≤1 2 f ∈X −1≤t≤1 1 1 inf (−f )(t)} = sup { inf (−f )(t) + f (t0 ) − 2 2 −1≤t≤1 f ∈X −1≤t≤1 1 1 inf (−f )(t) + f (t0 )} = sup { −1≤t≤1 2 2 f ∈X 1 1 (4.4) max f (t))}. = sup { f (t0 ) − 2 2 −1≤t≤1 f ∈X Since f (t0 ) ≤ max−1≤t≤1 f (t) for all f ∈ X, then, 1 1 f (t0 ) − max f (t) ≤ 0, ∀ f ∈ X. 2 2 −1≤t≤1 So, 1 1 max f (t))} ≤ 0. sup { f (t0 ) − 2 −1≤t≤1 f ∈X 2
(4.5)
But, for the constant function g ∈ X, one has 1 1 g(t0 ) − max g(t) = 0. 2 2 −1≤t≤1
(4.6)
Therefore, in view of 4.4), (4.5) and (4.6), we have sup {γf,ψ − ϕ(−f + γf,ψ 1)} = 0.
f ∈X
14
(4.7)
Now, let f0 ∈ X. Thus, h(f0 ) = 0, 1 1 ⇐⇒ inf f0 (t) − f0 (t0 ) = 0, 2 −1≤t≤1 2 ⇐⇒ f0 (t0 ) = inf f0 (t), −1≤t≤1
⇐⇒ f0 (t0 ) = min f0 (t).
(4.8)
−1≤t≤1
Hence, in view of Theorem 4.1, (4.7) and (4.8), one has f0 ∈ X is a global maximizer of the function h if and only if h(f0 ) = 0, if and only if t0 is a global minimizer of the function f0 over [−1, 1].
5
Conclusions
We studied a new class of elementary sub-topical functions, which is a generalization of the class of min-type functions. By using this class, we obtained various characterizations of maximal elements of support sets of sub-topical functions. As an application, by using the obtained results, we presented necessary and sufficient conditions for global maximizers of the difference of two strictly sub-topical functions. These results have many applications in various parts of applied mathematics, mathematical economics and game theory, in particular, in the modelling of discrete event systems [4, 5, 6, 7, 16]. Acknowledgment: The authors are very grateful to the anonymous referee for his/her useful suggestions on an earlier version of this paper. The comments of the referee were very fruitful and these comments have enabled the authors to improve the paper significantly.
References [1] A.R. Doagooei, Sub-topical functions and plus-co-radiant dx.doi.org/10.1080/02331934.2014.994629, (2014).
sets,
Optim.,
[2] A.R. Doagooei and H. Mohebi, Optimization of the difference of topical functions, J. Global Optim., 57 (2013), 1349-1358. [3] A.R. Doagooei and H. Mohebi, Optimization of the difference of ICR functions, Nonlinear Anal., 71 (2009), 4493-4499. [4] S. Gaubert and J. Gunawardena, A non-linear hierarchy for discrete event dynamical systems, In: Proceedings of the 4th workshop on discrete event systems, Calgiari, Technical Report HPL-BRIMS-98-20, Hewlett-Packard Labs., Cambridge University Press, Cambridge, (1998). 15
[5] J. Gunawardena, An introduction to idempotency, Cambridge University Press, Cambridge, (1998). [6] J. Gunawardena, From max-plus algebra to non-expansive mappings: A non-linear theory for discrete event systems. Theoretical Computer Science, Technical Report HPL-BRIMS-99-07, Hewlett-Packard Labs., Cambridge University Press, Cambridge, (1999). [7] J. Gunawardena and M. Keane, On the existence of cycle times for some nonexpansive maps, Technical Report HPL-BRIMS-95-003, Hewlett-Packard Labs., Cambridge University Press, Cambridge, (1995). [8] S. Mirzadeh and H. Mohebi, Increasing co-radiant and quasi-concave functions with applications in mathematical economic, J. Optim. Theory Appl., 169 (2016), No. 2, 443-472. [9] H. Mohebi and H. Sadeghi, Monotonic analysis over ordered topological vector spaces: I, Optim., 56 (2007), 305-321. [10] H. Mohebi, Topical functions and their properties in a class of ordered Banach spaces, Appl. Optim., Springer, 99 (2005), 343-361. [11] H. Mohebi and M. Samet, Abstract convexity of topical functions, J. Global Optim., 58 (2014), 365-375. [12] D. Pallaschke and S. Rolewicz, Foundations of mathematical optimization, Kluwer Academic Publishers, Boston, Dordrecht, London, 1997. [13] A.M. Rubinov, Abstract convexity and global optimization, Kluwer Academic Publishers, Boston, Dordrecht, London, 2000. [14] A.M. Rubinov and M.Y. Andramonov, Minimizing increasing star-shaped functions based on abstract convexity, J. Global Optim., 15 (1999), No. 1, 19-39. [15] A.M. Rubinov and B.M. Glover, Increasing convex-along-rays functions with applications to global optimization, J. Optim. Theory Appl., 102 (1999), No. 3, 615-542. [16] A.M. Rubinov and I. Singer, Topical and sub-topical functions, downward sets and abstract convexity, Optim., 50 (2001), 307-351. [17] A.M. Rubinov and Uderzo, On global optimality conditions via separation functions, J. Optim. Theory Appl., 109 (2001), No. 2, 345-370. [18] A.M. Rubinov and A. Vladimirov, Convex-along-rays functions and star-shaped sets, Numer. Funct. Anal. Optim., 19 (1998), Nos. 5 & 6, 593-613. [19] I. Singer, On radiant sets, downward sets, topical functions and sub-topical functions in lattice ordered groups, Optim., 53 (2004), 393-428.
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[20] I. Singer, Abstract convex analysis, Wiley-Interscience, New York, 1997. [21] C. Zalinescu, Convex analysis in general vector spaces, World Scientific, London, 2002.
17
H. Bakhtiari, H. Mohebi
PII: DOI: Reference:
S0022-247X(16)30866-6 http://dx.doi.org/10.1016/j.jmaa.2016.12.070 YJMAA 21010
To appear in:
Journal of Mathematical Analysis and Applications
Received date:
15 September 2016
Please cite this article in press as: H. Bakhtiari, H. Mohebi, Characterizing Global Maximizers of the Difference of Sub-Topical Functions, J. Math. Anal. Appl. (2017), http://dx.doi.org/10.1016/j.jmaa.2016.12.070
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Characterizing Global Maximizers of the Difference of Sub-Topical Functions H. Bakhtiari1 and H. Mohebi1,2 1
2
Department of Mathematics, Shahid Bahonar University of Kerman, P.O. Box: 76169133, Postal Code: 7616914111, Kerman, Iran. e-mail: [email protected] e-mail: [email protected] Present Address: Department of Applied Mathematics, University of New South Wales, Sydney, NSW, 2052, Australia. e-mail: [email protected] Abstract. Applying for a certain class of elementary functions, which is a generalization of the class of min-type functions, the techniques from abstract convex analysis, we give various characterizations of maximal elements of support sets of sub-topical functions. As an application, by using this class of elementary functions, we characterize global maximizers of the difference of two strictly sub-topical functions. These results have many applications in various parts of applied mathematics, mathematical economics and game theory.
Key words: global optimization; DC-functions; topical function; sub-topical function; elementary function; support set; maximal element; abstract convexity. 2010 (AMS) Mathematics Subject Classification: 90C26; 90C46; 26B25; 26A48; 26A09.
1
Introduction
To explain the main idea behind abstract convex analysis, there are the following wellknown results from convex analysis: (i) each lower semi-continuous convex function f is the upper envelope of the set of all affine functions which are minorants of f ; (ii) A set C is closed and convex if and only if each point which does not belong to this set can be separated from it by a linear function [21]. In some cases the use of separation of a set from an outside point and the envelope representation of a function are very convenient, even when we consider separation by non-linear functions and the upper envelope of non-affine functions [13, 20]. Thus, the idea arises to investigate some of the main concepts of convex analysis in the case of non-convexity, to study abstract convex functions, that is, functions 1
which can be presented as upper envelopes of subsets of a given set of not necessarily linear ”elementary functions”. Special concepts and techniques for the study of abstract convex functions, such as, abstract support sets of functions, abstract subdifferentials of functions, various kinds of conjugacy and polarity, were developed in abstract convex analysis [8, 9, 10, 13, 14, 15, 16, 17, 18, 19, 20]. For applications of abstract convex analysis, we need to describe the main objects of this theory in concrete situations under consideration. Just as the set of linear functions leads to the theory of usual convex sets and convex functions [21], each class of elementary functions leads to a theory of abstract convex sets and abstract convex functions with respect to this class [13]. Abstract convexity and many special concepts, such as, support sets, subdifferentials, conjugacy and polarity of topical functions with respect to a certain class of elementary functions have been investigated [10, 11, 16, 19]. In fact, functions of this kind have arisen recently in several contexts [6], and the term ”topical function” is due to Gunawardena and Keane [7]. Topical functions were intensively studied (see [4, 5, 6] and the references therein) and they have many applications in various parts of applied mathematics, in particular, in the modelling of discrete event systems [4, 5, 6, 7]. Topical functions are also interesting from a different point of view, namely as a tool in the study of the so-called downward sets. Downward sets arise in the study of some problems of mathematical economics and game theory and also in the study of inequality systems involving increasing functions [16]. Besides topical functions, it has been studied the much more general class of increasing and plus-sub-homogeneous functions, which called sub-topical functions [7, 16, 19]. In [1, 16, 19], it has been shown that each sub-topical function is abstract convex with respect to the following set of L elementary functions. For each y ∈ X and each α ∈ R, consider the function η(y,α) : X −→ R is defined by η(y,α) (x) := sup{λ ∈ R : λ ≤ α, λ1 ≤ x + y}, ∀ x ∈ X, where L := {η(y,α) : y ∈ X, α ∈ R}
(1.1)
is called the class of elementary functions, which is a generalization of the class of mintype functions [16, 19]. Thus the question arises: to apply the main concepts and results of abstract convex analysis corresponding to the set of elementary functions defined by (1.1). We address this question in the present paper, we give various characterizations of maximal elements of support sets of sub-topical functions by using this class of elementary sub-topical functions. As an application, one of the most important global optimization problems [12], is that of minimizing a DC-functions (difference of two convex functions), that is, minimize h(x) subjet to x ∈ X, where h := g − f and f , g are convex functions. In a general case, DC-functions can be replaced by DAC-functions (difference of two abstract convex functions). In particular, 2
minimizing of the difference of two increasing and co-radiant (ICR) functions [3], and minimizing of the difference of two topical functions [2]. In this paper, we replace f and g by two strictly sub-topical (i.e., strictly increasing and sub-homogeneous) functions, and by using the class of elementary functions defined by (1.1), give necessary and sufficient conditions for global maximizers of the difference of two strictly sub-topical functions defined on a real normed linear space X. The layout of the paper is as follows: In Section 2, we collect definitions, notations and preliminary results relative to sub-topical and elementary functions. Characterizations of maximal elements of support sets of sub-topical functions are given in Section 3. In Section 4, as an application, we present necessary and sufficient conditions for global maximizers of the difference of two strictly sub-topical functions. Section 5, concludes with a discussion on conclusions and applications.
2
Preliminaries
Let X be a real normed linear space and S be a closed convex cone in X such that S ∩ {−S} = {0} and intS = ∅. We assume that X is equipped with the order relation induced by S : x ≤ y ⇐⇒ y − x ∈ S,
(x, y ∈ X).
Also, for x, y ∈ X, we say that x < y if and only if y − x ∈ S \ {0}. Moreover, we assume that S is normal in the sense that there exists a positive real number m > 0 such that x ≤ m y , whenever 0 ≤ x ≤ y with x, y ∈ X. Throughout the paper we assume that 1 ∈ intS. Let B := {x ∈ X : −1 ≤ x ≤ 1}. (2.1) It is well known and easy to check that B can be considered as the unit ball of a certain norm . 1 , which is equivalent to the initial norm . . ¯ := [−∞, +∞] is called increasing if x ≤ y =⇒ f (x) ≤ f (y), A function f : X −→ R (x, y ∈ X). ¯ is called plus-homogeneous if Recall [7, 16, 19] that a function f : X −→ R f (x + μ1) = f (x) + μ, ∀ x ∈ X, ∀ μ ∈ R. The function f is called topical if f is increasing and plus-homogeneous [16]. ¯ is called plus-sub-homogeneous if Recall [16, 19] that a function f : X −→ R f (x + μ1) ≤ f (x) + μ, ∀ x ∈ X, ∀ μ ≥ 0.
(2.2)
Functions satisfying (3.9) have been considered in [7], where they have been called subhomogeneous functions. 3
It is easy to check that f is plus-sub-homogeneous if f (x + μ1) ≥ f (x) + μ, ∀ x ∈ X, ∀ μ ≤ 0. The function f is called sub-topical if f is increasing and plus-sub-homogeneous [16, 19]. ¯ be a function. We say that f is strictly increasing if Definition 2.1. Let f : X −→ R x < y =⇒ f (x) < f (y), (x, y ∈ X). f is called strictly sub-topical if it is strictly increasing and plus-sub-homogeneous. Example 2.1. Let f, g : R2 −→ R be defined by 1 f (x, y) := x, ∀ x, y ∈ R, 2
and g(x, y) :=
1 1 4 x + 4 y, 1 3 4y − 2,
x ≤ y − 3, ∀ x, y ∈ R. x ≥ y − 3,
It is not difficult to check that f and g are strictly sub-topical functions. Moreover, it is clear that every topical function is sub-topical. Also, if f is a sub-linear function on X such that f (1) ≤ 1, then, f is sub-topical. ¯ the Remark 2.1. [16]. It is easy to see that for a sub-topical function f : X −→ R following assertions are true. (i) If there exists x ∈ X such that f (x) = +∞, then, f ≡ +∞. (ii) If there exists x ∈ X such that f (x) = −∞, then, f ≡ −∞. ¯ either dom f = X, or f ≡ +∞, where Hence, for any sub-topical function f : X −→ R, dom f := {x ∈ X : f (x) < +∞}. In the sequel, consider the function η : X × X × R −→ R is defined by η(x, y, α) := sup{λ ∈ R : λ ≤ α, λ1 ≤ x + y}, ∀ x, y ∈ X, ∀ α ∈ R.
(2.3)
It is worth noting that the function η is a generalization of the min-type function defined on Rn [16, 19]. In view of (2.1), it is easy to see that the set {λ ∈ R : λ ≤ α, λ1 ≤ x + y} is non-empty and bounded in R. Clearly, this set is also a closed subset of R. Therefore, one has sup = max, and η(x, y, α)1 ≤ x + y, ∀ x, y ∈ X, ∀ α ∈ R.
(2.4)
Also, in view of the definition, we have η(x, y, α) ≤ α, ∀ x, y ∈ X, ∀ α ∈ R. The following properties of the function η were obtained in [1, 11].
4
(2.5)
For every x, x , y, y ∈ X; γ, α, α ∈ R and μ ≥ 0, one has η(x + μ1, y, α) ≤ η(x, y, α) + μ,
(2.6)
η(x, y, α) = η(y, x, α),
(2.7)
x ≤ x =⇒ η(x, y, α) ≤ η(x , y, α),
(2.8)
y ≤ y =⇒ η(x, y, α) ≤ η(x, y , α),
(2.9)
η(x, −x + α1, α) = α,
(2.10)
α ≤ α =⇒ η(x, y, α) ≤ η(x, y, α ).
(2.11)
η(x, y + γ1, α) = η(x, y, α − γ) + γ.
(2.12)
Now, for each y ∈ X and each α ∈ R, define the function η(y,α) : X −→ R by η(y,α) (x) := η(x, y, α) for all x ∈ X. In view of (2.6) and (2.8), we have the function η(y,α) is sub-topical for each y ∈ X and each α ∈ R. Let Ω := {η(y,α) : y ∈ X, α ∈ R}.
(2.13)
Ω is called the set of elementary functions. Lemma 2.1. The element (y, α) ∈ X ×R in the definition of η(y,α) is uniquely determined. That is, if (y1 , α1 ), (y2 , α2 ) ∈ X × R and η(y1 ,α1 ) (x) = η(y2 ,α2 ) (x), ∀ x ∈ X,
(2.14)
then, y1 = y2 and α1 = α2 . Proof. By putting x := −y1 + α1 1 and x := −y2 + α2 1 in (2.14), we obtain from (2.5) and (2.10) that α1 = α2 . Also, in view of (2.14), one has η(y1 ,α1 ) (x) ≥ η(y2 ,α2 ) (x), ∀ x ∈ X.
(2.15)
Put x := −y2 + α2 1 in (2.15), it follows from (2.10) and the fact that α1 = α2 , η(y1 ,α1 ) (−y2 + α2 1) = α1 . This together with (2.4) implies that α1 1 ≤ −y2 + α2 1 + y1 = −y2 + α1 1 + y1 . This implies that y2 ≤ y1 . Since, in view of (2.14), we have η(y1 ,α1 ) (x) ≤ η(y2 ,α2 ) (x), ∀ x ∈ X,
(2.16)
so, by putting x := −y1 + α1 1 in (2.16) and by using a similar argument as the above, we conclude that y1 ≤ y2 , and hence, y1 = y2 . Corollary 2.1. If we define the function ψ : X × R −→ Ω by ψ(y, α) := η(y,α) for all y ∈ X and all α ∈ R. Then, ψ is bijective (one-to-one and onto) and increasing. 5
Proof. This is an immediate consequence of Lemma 2.1, (2.9) and (2.11). ¯ the support set of f with In the following, recall [13, 20] that for a function f : X −→ R respect to Ω is defined by supp (f, Ω) := {η(y,α) ∈ Ω : η(y,α) (x) ≤ f (x), ∀ x ∈ X}. For an easy reference, we state the following results from [1]. ¯ be a sub-topical function. Then, Proposition 2.1. [1]. Let f : X −→ R supp (f, Ω) = {η(y,α) ∈ Ω : f (−y + α1) ≥ α}.
¯ be a function. Then, f is sub-topical if and only if Theorem 2.1. [1]. Let f : X −→ R there exists a set A ⊆ Ω such that f (x) =
sup η(y,α) (x), (x ∈ X).
η(y,α) ∈A
In this case, one can take A := supp (f, Ω).
3
Characterizations of Maximal Elements of Support Sets of Sub-Topical Functions
In this section, we give various characterizations of maximal elements of support sets of sub-topical functions with respect to a certain class of elementary functions. Let Ω be the set of elementary functions defined by (2.13). In the sequel, we assume that supp (f, Ω) = ∅, and supp (f, Ω) = Ω,
(3.1)
(see also, Corollary 2.1). First, we give a definition of a maximal element [13]. Definition 3.1. [13]. Let H be a set of real valued functions defined on a set Z. We assume that H is equipped with the natural (point-wise) order relation of functions. We say that a function f¯ ∈ H is a maximal element of the set H, if f ∈ H and f (x) ≥ f¯(x) for all x ∈ Z, then, f¯(x) = f (x) for all x ∈ Z. ¯ be a sub-topical function, and let η(y,α) ∈ Ω be a Proposition 3.1. Let f : X −→ R maximal element of supp (f, Ω). Then, f (−y + α1) = α.
6
Proof. Let η(y,α) ∈ Ω be a maximal element of supp (f, Ω). First, note that if f (−y +α1) = +∞, then, in view of Remark 2.1, one has f ≡ +∞, and so, supp (f, Ω) = Ω, which contradicts (3.1). Also, similarly, if f (−y + α1) = −∞, it follows from Remark 2.1 that supp (f, Ω) = ∅, which contradicts (3.1). So, f (−y + α1) ∈ R. Now, put y0 := −(−y + α1) + f (−y + α1)1 and α0 := f (−y + α1). Then, y0 ∈ X, α0 ∈ R, and in view of (2.10), we have η(y0 ,α0 ) (−y + α1) = η(−y + α1, −(−y + α1) + f (−y + α1)1, f (−y + α1)) = f (−y + α1).
(3.2)
¯ is a sub-topical function and η(y,α) ∈ supp (f, Ω), in view of Proposition Since f : X −→ R 2.1, f (−y +α1) ≥ α. This together with (3.2) implies that η(y0 ,α0 ) (−y +α1) ≥ α. Thus, by Proposition 2.1 and the fact that η(y0 ,α0 ) is sub-topical, we have η(y,α) ∈ supp (η(y0 ,α0 ) , Ω), and so, (3.3) η(y,α) (x) ≤ η(y0 ,α0 ) (x), ∀ x ∈ X. Also, we have f (−y0 + α0 1) = f (−y + α1 − f (−y + α1)1 + f (−y + α1)1) = f (−y + α1) = α0 . Therefore, it follows from Proposition 2.1 that η(y0 ,α0 ) ∈ supp (f, Ω). Since η(y,α) is a maximal element of supp (f, Ω), we conclude from (3.3) that η(y,α) (x) = η(y0 ,α0 ) (x), ∀ x ∈ X.
(3.4)
Put x := −y + α1 in (3.4). Hence, by using (2.10), one has α = η(y,α) (−y + α1) = η(y0 ,α0 ) (−y + α1) = η(−y + α1, −(−y + α1) + f (−y + α1)1, f (−y + α1)) = f (−y + α1), which completes the proof. Now, by the following example, we show that the converse statement to Proposition 3.1 is not valid. Example 3.1. Let f : R −→ R be defined by 0, x ≤ 0, f (x) := x, x > 0, 7
∀ x ∈ R.
It is easy to see that f is a sub-topical function, but not strictly. Moreover, for each α > 0, one has η(0,α) (x) = min{α, x}, ∀ x ∈ R. Then, η(0,α) (x) ≤ f (x) for all x ∈ R and all α > 0, and so, η(0,α) ∈ supp (f, Ω) for all α > 0. Moreover, f (−0 + α1) = f (α) = α for all α > 0. But, in view of Corollary 2.1, supp (f, Ω) does not have a maximal element. By using the following lemma, we shall show that the converse statement to Proposition 3.1 is valid. ¯ be a sub-topical function. Let (y, α) ∈ X × R be such Lemma 3.1. Let f : X −→ R that f (−y + α1) = α. Assume that there exists η(y0 ,α0 ) ∈ supp (f, Ω) such that η(y,α) (x) ≤ η(y0 ,α0 ) (x) for all x ∈ X. Then, α ≤ α0 , y ≤ y0 and f (−y0 + α0 1) = α0 . Proof. Suppose that there exists η(y0 ,α0 ) ∈ supp (f, Ω) such that η(y,α) (x) ≤ η(y0 ,α0 ) (x), ∀ x ∈ X.
(3.5)
Put x := −y + α1 in (3.5). So, by (2.10) and (2.5), we have α ≤ α0 . Moreover, since η(y0 ,α0 ) ∈ supp (f, Ω), by (2.10) one has α = η(y,α) (−y + α1) ≤ η(y0 ,α0 ) (−y + α1) ≤ f (−y + α1) = α. This implies that α = η(y0 ,α0 ) (−y + α1), and so, in view of (2.4), α1 ≤ −y + α1 + y0 . That is, y ≤ y0 . Now, we show that f (−y0 + α0 1) = α0 . Since y ≤ y0 , f is increasing and η(y0 ,α0 ) ∈ supp (f, Ω), it follows from Proposition 2.1 that f (−y + α0 1) ≥ f (−y0 + α0 1) ≥ α0 .
(3.6)
Let y1 := −(−y + α0 1) + f (−y + α0 1)1 and α1 := f (−y + α0 1). Clearly, y1 ∈ X and α1 ∈ R. By using (2.10) and (3.6), we deduce that η(y1 ,α1 ) (−y + α0 1) = f (−y + α0 1) ≥ α0 . So, since η(y1 ,α1 ) is a sub-topical function, in view of Proposition 2.1, one has η(y,α0 ) ∈ supp (η(y1 ,α1 ) , Ω). Then, η(y,α0 ) (x) ≤ η(y1 ,α1 ) (x), ∀ x ∈ X. 8
(3.7)
Also, we have f (−y1 + α1 1) = f (−y + α0 1 − f (−y + α0 1)1 + f (−y + α0 1)1) = f (−y + α0 1) = α1 . Thus, by Proposition 2.1, η(y1 ,α1 ) ∈ supp (f, Ω). This together with (2.11) and (3.7) and the fact that α ≤ α0 implies that η(y,α) (x) ≤ η(y,α0 ) (x) ≤ η(y1 ,α1 ) (x) ≤ f (x), ∀ x ∈ X.
(3.8)
Put x := −y + α1 in (3.8). Then, by (2.10), α = η(y,α) (−y + α1) ≤ η(y,α0 ) (−y + α1) ≤ η(y1 ,α1 ) (−y + α1) ≤ f (−y + α1) = α. This implies that η(y,α0 ) (−y + α1) = η(y1 ,α1 ) (−y + α1), and hence, by (2.12), we conclude that η(y,α0 ) (−y + α1) = η(y1 ,α1 ) (−y + α1) = η(−y + α1, y + [f (−y + α0 1) − α0 ]1, f (−y + α0 1)) = η(−y + α1, y, f (−y + α0 1) − [f (−y + α0 1) − α0 ]) + f (−y + α0 1) − α0 = η(y,α0 ) (−y + α1) + f (−y + α0 1) − α0 . Therefore, it follows that f (−y+α0 1) = α0 , and so, in view of (3.6), we get f (−y0 +α0 1) = α0 , which completes the proof. ¯ we put In the sequel, for each y ∈ X and each function h : X −→ R, γy,h := max{α ∈ R : h(−y + α1) ≥ α}.
(3.9)
Now, we show that if f is a strictly sub-topical function, then the converse statement to Proposition 3.1 is valid. ¯ be a strictly sub-topical function. Then, η(y,α) ∈ Ω is a Theorem 3.1. Let f : X −→ R maximal element of supp (f, Ω) if and only if f (−y + α1) = α and α = γy,f , where γy,f defined by (3.9).
9
Proof. Suppose that η(y,α) ∈ Ω is a maximal element of supp (f, Ω). Since f is sub-topical, it follows from Proposition 3.1 that f (−y + α1) = α. Now, we show that α = γy,f . Let C := {α ∈ R : f (−y + α1) ≥ α}. Since f (−y + α1) = α, then, α ∈ C. Assume if possible that there exists γ0 ∈ C such that α < γ0 . Therefore, by (2.11), one has η(y,α) (x) ≤ η(y,γ0 ) (x) for all x ∈ X. Since by the hypothesis η(y,α) is a maximal element of supp (f, Ω), so, η(y,α) (x) = η(y,γ0 ) (x), ∀ x ∈ X, (3.10) (note that since γ0 ∈ C, hence, f (−y + γ0 1) ≥ γ0 . Therefore, in view of Proposition 2.1, η(y,γ0 ) ∈ supp (f, Ω)). Put x := −y + γ0 1 in (3.10). Hence, by using (2.5) and (2.10), α ≥ η(y,α) (−y + γ0 1) = η(y,γ0 ) (−y + γ0 1) = γ0 . So, α ≥ γ0 , which is a contradiction. Thus, α = γy,f . Conversely, suppose that f (−y + α1) = α and α = γy,f . Let η(y0 ,α0 ) ∈ supp (f, Ω) be such that η(y,α) (x) ≤ η(y0 ,α0 ) (x) for all x ∈ X. We prove that η(y,α) (x) = η(y0 ,α0 ) (x) ¯ is sub-topical, f (−y + α1) = α and for all x ∈ X. To this end, since f : X −→ R η(y0 ,α0 ) ∈ supp (f, Ω) is such that η(y,α) (x) ≤ η(y0 ,α0 ) (x) for all x ∈ X, it follows from Lemma 3.1 that y ≤ y0 , α ≤ α0 and f (−y0 + α0 1) = α0 . This together with f is increasing implies that f (−y + α0 1) ≥ α0 . Thus, α0 ∈ {α ∈ R : f (−y + α1) ≥ α}, and so, by (3.9) and the fact that α = γy,f , we have α ≥ α0 . Hence, α = α0 . Thus, one has f (−y0 + α1) = f (−y0 + α0 1) = α0 = α = f (−y + α1). This together with −y0 +α1 ≤ −y+α1 and f is strictly increasing implies that −y0 +α1 = −y + α1, and hence, y = y0 . Thus, η(y,α) (x) = η(y0 ,α0 ) (x) for all x ∈ X. So, η(y,α ) is a maximal element of supp (f, Ω) ( note that since f (−y + α1) = α, in view of Proposition 2.1, one has η(y,α) ∈ supp (f, Ω)). ¯ be a strictly sub-topical function. Let γy,f be as Proposition 3.2. Let f : X −→ R in (3.9). Then, for each η(y,α) ∈ supp (f, Ω) there exists a maximal element η(y0 ,α0 ) of supp (f, Ω) such that η(y,α) (x) ≤ η(y0 ,α0 ) (x) for all x ∈ X. Proof. Let η(y,α) ∈ supp (f, Ω) be arbitrary. Put y0 := −(−y + γy,f 1) + f (−y + γy,f 1)1 and α0 := f (−y + γy,f 1). Since f (−y + γy,f 1) ≥ γy,f , it follows that y ≤ y0 and γy,f ≤ α0 . Also, clearly, f (−y0 + α0 1) = α0 , and so, it follows from Proposition 2.1 that η(y0 ,α0 ) ∈ supp (f, Ω). Also, since η(y,α) ∈ supp (f, Ω) and f is sub-topical, by Proposition 2.1, one has f (−y + α1) ≥ α, and so, α ∈ {α ∈ R : f (−y + α1) ≥ α}. Hence, γy,f ≥ α. Thus, α0 ≥ γy,f ≥ α.
10
This together with (2.9) and (2.11) and the fact y ≤ y0 implies that η(y0 ,α0 ) (x) ≥ η(y,α0 ) (x) ≥ η(y,γy,f ) (x) ≥ η(y,α) (x), ∀ x ∈ X. Therefore, η(y,α) (x) ≤ η(y0 ,α0 ) (x), ∀ x ∈ X.
(3.11)
Now, we show that η(y0 ,α0 ) is a maximal element of supp (f, Ω). To this end, let δ ∈ R be such that f (−y0 + δ1) ≥ δ. Thus, since f (−y + γy,f 1) ≥ γy,f , we conclude that f (−y0 + δ1) ≥ δ ≥ δ − [f (−y + γy,f 1) − γy,f ] = δ − f (−y + γy,f 1) + γy,f . =⇒ f (−y0 + δ1) ≥ δ − f (−y + γy,f 1) + γy,f , =⇒ f (−y + γy,f 1 − f (−y + γy,f 1)1 + δ1) ≥ δ − f (−y + γy,f 1) + γy,f , =⇒ f (−y + [δ − f (−y + γy,f 1) + γy,f ]1) ≥ δ − f (−y + γy,f 1) + γy,f , =⇒ δ − f (−y + γy,f 1) + γy,f ∈ {α ∈ R : f (−y + α1) ≥ α}, =⇒ γy,f ≥ δ − f (−y + γy,f 1) + γy,f , =⇒ δ ≤ f (−y + γy,f 1) = α0 . This together with f (−y0 + α0 1) = α0 implies that α0 = γy0 ,f .
(3.12)
Therefore, the result follows from Theorem 3.1. Remark 3.1. It is worth noting that under the hypotheses of Proposition 3.2, α0 = γy0 ,f . The following theorem has a crucial role for characterizing maximizers of the difference of two strictly sub-topical functions. ¯ be strictly sub-topical functions. Let γy,f be as in (3.9). Theorem 3.2. Let f, g : X −→ R Then the following assertions are equivalent: (1) supp (f, Ω) ⊆ supp (g, Ω), (2) g(−y + γy,f 1) ≥ γy,f for each y ∈ X. Proof. (1) =⇒ (2). Suppose that (1) holds. Let y ∈ X be arbitrary. Then, in view of the hypothesis, f (−y + γy,f 1) ≥ γy,f , and so, by Proposition 2.1, we obtain η(y,γy,f ) ∈ supp (f, Ω). Thus, by the hypothesis (1), η(y,γy,f ) ∈ supp (g, Ω). Again, in view of Proposition 2.1, g(−y + γy,f 1) ≥ γy,f . That is, (2) holds. (2) =⇒ (1). Assume that (2) holds. Let η(y,α) ∈ supp (f, Ω) be arbitrary. Then, in view of 11
Proposition 3.2 and Remark 3.1, there exists a maximal element η(y0 ,γy0 ,f ) of supp (f, Ω) such that (3.13) η(y,α) (x) ≤ η(y0 ,γy0 ,f ) (x), ∀ x ∈ X. Now, set y1 := −(−y0 + γy0 ,f 1) + g(−y0 + γy0 ,f 1)1 and α1 := g(−y0 + γy0 ,f 1). Clearly, g(−y1 + α1 1) = α1 , and hence, by Proposition 2.1, we have η(y1 ,α1 ) ∈ supp (g, Ω). On the other hand, by the hypothesis (2), we have g(−y0 + γy0 ,f 1) ≥ γy0 ,f . Therefore, in view of (2.10), η(y1 ,α1 ) (−y0 + γy0 ,f 1) = g(−y0 + γy0 ,f 1) ≥ γy0 ,f .
(3.14)
Since η(y1 ,α1 ) is a sub-topical function, we conclude from (3.14) and Proposition 2.1 that η(y0 ,γy0 ,f ) ∈ supp (η(y1 ,α1 ) , Ω), and so, η(y0 ,γy0 ,f ) (x) ≤ η(y1 ,α1 ) (x), ∀ x ∈ X.
(3.15)
Since η(y1 ,α1 ) ∈ supp (g, Ω), it follows from (3.13) and (3.15) that η(y,α) ∈ supp (g, Ω). Hence, supp (f, Ω) ⊆ supp (g, Ω), which completes the proof.
4
Necessary and Sufficient Conditions for Maximizers of the Difference of Two Sub-Topical Functions
In this section, we present necessary and sufficient conditions for global maximizers of the ¯ be sub-topical functions difference of two strictly sub-topical functions. Let f, g : X −→ R such that dom(f ) ⊆ dom(g). Let h := g − f , that is, g(x) − f (x), x ∈ dom (f ) , h(x) := (4.1) −∞, x∈ / dom (f ) . (with the convention (+∞) − (+∞) := −∞). ¯ be sub-topical functions. Then, Lemma 4.1. Let f, g : X −→ R f (x) ≤ g(x), ∀ x ∈ X ⇐⇒ supp (f, Ω) ⊆ supp (g, Ω).
(4.2)
Proof. Indeed, if f (x) ≤ g(x) for all x ∈ X, then it is clear that supp (f, Ω) ⊆ supp (g, Ω). Conversely, assume if possible that there exists x0 ∈ X such that f (x0 ) > g(x0 ). Since f is a sub-topical function, it follows from Theorem 2.1 that f (x0 ) =
sup η(y,α) ∈supp (f,Ω)
η(y,α) (x0 ).
This together with the fact that f (x0 ) > g(x0 ) implies that there exists η(y0 ,α0 ) ∈ supp (f, Ω) such that η(y0 ,α0 ) (x0 ) > g(x0 ). This implies that η(y0 ,α0 ) ∈ / supp (g, Ω), which is a contradiction. Hence, f (x) ≤ g(x) for all x ∈ X. 12
Now, in the following, we give characterizations of global maximizers of the difference of strictly sub-topical functions. ¯ be strictly sub-topical functions. Let γy,g be as in Proposition 4.1. Let f, g : X −→ R (3.9). Then, x0 ∈ X is a global maximizer of the function h (defined by (4.1)) if and only if f (−y + γy,g 1) + h(x0 ) ≥ γy,g for each y ∈ X. Proof. We have x0 is a global maximizer of the function h if and only if h(x) ≤ h(x0 ) for all x ∈ X, if and only if g(x) ≤ f (x) + h(x0 ) for all x ∈ X, if and only if g(x) ≤ f˜(x) := f (x) + h(x0 ) for all x ∈ X, in view of Lemma 4.1, if and only if supp(g, Ω) ⊆ supp(f˜, Ω), by Theorem 3.2, if and only if f˜(−y + γy,g 1) ≥ γy,g for each y ∈ X, if and only if f (−y + γy,g 1) + h(x0 ) ≥ γy,g for each y ∈ X (it is worth noting that the function f˜ is a strictly sub-topical function). ¯ be strictly sub-topical functions. Let γy,g be as in (3.9). Theorem 4.1. Let f, g : X −→ R Then, x0 ∈ X is a global maximizer of the function h if and only if h(x0 ) = sup {γy,g − f (−y + γy,g 1)}. y∈X
(4.3)
Proof. Suppose that (4.3) holds. Then, f (−y + γy,g 1) + h(x0 ) ≥ γy,g for each y ∈ X. Hence, in view of Proposition 4.1, x0 is a global maximizer of the function h. Conversely, assume that x0 ∈ X is a global maximizer of the function h. Then, by Proposition 4.1, one has f (−y + γy,g 1) + h(x0 ) ≥ γy,g for each y ∈ X, and so, h(x0 ) ≥ γy,g − f (−y + γy,g 1) for each y ∈ X. This implies that h(x0 ) ≥ z, where we define z by z := sup {γy,g − f (−y + γy,g 1)}. y∈X
We claim that h(x0 ) = z. If h(x0 ) = z, then, h(x0 ) > z. Consider z ∈ R such that h(x0 ) > z ≥ z. Since z ≥ γy,g − f (−y + γy,g 1) for each y ∈ X, hence, z ≥ γy,g − f (−y + γy,g 1) for each y ∈ X. This implies that f (−y + γy,g 1) + z ≥ γy,g for each y ∈ X, and so, by Theorem 3.2 (the implication (2) =⇒ (1)), we conclude that supp(g, Ω) ⊆ supp(f¯, Ω), where f¯(x) := f (x) + z for all x ∈ X. Thus, in view of Lemma 4.1, we deduce that g(x) ≤ f¯(x) = f (x) + z for all x ∈ X. Therefore, in particular, h(x0 ) ≤ z , which is a contradiction (note that f¯ is a strictly sub-topical function). Example 4.1. Let X := {f : [−1, 1] −→ R : f is a continuos f unction} be the Banach space of all continuous real valued functions defined on [−1, 1], and let S := {f ∈ X : f (t) ≥ 0, ∀ t ∈ [−1, 1]}. It is clear that S is a closed convex pointed and normal cone in X, which contains the function 1 (1(t) := 1 for all t ∈ [−1, 1]). Fix, t0 ∈ [−1, 1], and consider functions ϕ, ψ : X −→ R are defined by 1 ϕ(f ) := f (t0 ), ∀ f ∈ X, 2 13
and ψ(f ) :=
1 inf f (t), ∀ f ∈ X. 2 −1≤t≤1
Clearly, ϕ and ψ are strictly sub-topical functions. Let h := ψ − ϕ. Also, one has γf,ψ = max{δ ∈ R : ψ(−f + δ1) ≥ δ} 1 = max{δ ∈ R : inf (−f (t) + δ) ≥ δ} 2 −1≤t≤1 1 1 = max{δ ∈ R : δ ≤ inf (−f )(t)} 2 2 −1≤t≤1 = max{δ ∈ R : δ ≤ inf (−f )(t)} −1≤t≤1
=
inf (−f )(t), ∀ f ∈ X.
−1≤t≤1
Now, consider sup {γf,ψ − ϕ(−f + γf,ψ 1)} = sup { inf (−f )(t) − ϕ(−f + [ inf (−f )(t)]1)} f ∈X −1≤t≤1
f ∈X
−1≤t≤1
1 = sup { inf (−f )(t) − [(−f )(t0 ) + inf (−f )(t)]} −1≤t≤1 2 f ∈X −1≤t≤1 1 1 inf (−f )(t)} = sup { inf (−f )(t) + f (t0 ) − 2 2 −1≤t≤1 f ∈X −1≤t≤1 1 1 inf (−f )(t) + f (t0 )} = sup { −1≤t≤1 2 2 f ∈X 1 1 (4.4) max f (t))}. = sup { f (t0 ) − 2 2 −1≤t≤1 f ∈X Since f (t0 ) ≤ max−1≤t≤1 f (t) for all f ∈ X, then, 1 1 f (t0 ) − max f (t) ≤ 0, ∀ f ∈ X. 2 2 −1≤t≤1 So, 1 1 max f (t))} ≤ 0. sup { f (t0 ) − 2 −1≤t≤1 f ∈X 2
(4.5)
But, for the constant function g ∈ X, one has 1 1 g(t0 ) − max g(t) = 0. 2 2 −1≤t≤1
(4.6)
Therefore, in view of 4.4), (4.5) and (4.6), we have sup {γf,ψ − ϕ(−f + γf,ψ 1)} = 0.
f ∈X
14
(4.7)
Now, let f0 ∈ X. Thus, h(f0 ) = 0, 1 1 ⇐⇒ inf f0 (t) − f0 (t0 ) = 0, 2 −1≤t≤1 2 ⇐⇒ f0 (t0 ) = inf f0 (t), −1≤t≤1
⇐⇒ f0 (t0 ) = min f0 (t).
(4.8)
−1≤t≤1
Hence, in view of Theorem 4.1, (4.7) and (4.8), one has f0 ∈ X is a global maximizer of the function h if and only if h(f0 ) = 0, if and only if t0 is a global minimizer of the function f0 over [−1, 1].
5
Conclusions
We studied a new class of elementary sub-topical functions, which is a generalization of the class of min-type functions. By using this class, we obtained various characterizations of maximal elements of support sets of sub-topical functions. As an application, by using the obtained results, we presented necessary and sufficient conditions for global maximizers of the difference of two strictly sub-topical functions. These results have many applications in various parts of applied mathematics, mathematical economics and game theory, in particular, in the modelling of discrete event systems [4, 5, 6, 7, 16]. Acknowledgment: The authors are very grateful to the anonymous referee for his/her useful suggestions on an earlier version of this paper. The comments of the referee were very fruitful and these comments have enabled the authors to improve the paper significantly.
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