R-convexity in R-vector spaces

In this paper, for every relation R on a vector space V, we consider the R-vector space (V,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(V,R)$\end{document} and define the notions of R-convexity, R-convex hull, and R-extreme point in this space. Some examples are provided to compare them with the reference cases. The effects of some operations on R-convex sets are investigated. In particular, it is shown that the R-interior of an R-convex set is also an R-convex set under some restrictions on R. Also, we give some equivalent conditions for R-extremeness. Moreover, the notions of R-convex and R-affine maps on R-vector spaces are defined, and some results that assert the relation between an R-convex map f and its R-epigraph under some limitations on R are considered. Several propositions, such as R-continuous maps preserve R-compact sets and R-affine maps preserve R-convex sets, are presented, and some results on the composition of R-convex and R-affine maps are considered. Finally, some applications of R-convexity are investigated in optimization. More precisely, we show that the extrema values of R-affine R-continuous maps are reached on R-extreme points. Moreover, local and global minimum points of an R-convex map f on R-convex set K are considered.


Introduction and preliminaries
Various generalizations of the classical concept of a convex function have been introduced, especially during the second half of the twentieth century. These generalizations have been explored in various fields, such as economics, engineering, statistics, and applied sciences, and they have provided interesting results in several branches related to mathematics such as convex analysis, nonlinear optimization, linear programming, geometric functional analysis, control theory, and dynamical systems; see for example [2,13,21], and the references therein. Recently, the extensions of convexity have been considered by many researchers. For example, Nikoufar et al. studied convexity in various branches of pure and applied mathematical areas [3,18,25]. Also, we refer the readers to η-convexity and coordinate convexity [9,27,37]; GA-convexity and GG-convexity [15,20,39]; s-convexity [1]; preinvexity [35]; strong convexity [29,30,38]; quasi-convexity [32]; Schur convexity [28,34]; and pseudo-convexity [24]. Also, see the following recent related references: [12,19,31], and [36].
Over the last forty years, another type of extension of convexity, in which the convex coefficients need not commute with each other, has been considered. Examples include C * -convexity [22,23], matrix convexity and operator convexity [8,33], and the extension of C * -convexity to * -rings [4][5][6], and [7]. The basic concepts of convex analysis can be seen in [26] and [14].
Recently, the notions of orthogonal metric spaces and metric spaces with relation have been considered by many researchers [10,16,17], and [11]. In [16], the authors introduced R-metric spaces and studied some of the properties of these spaces. We recall some notions and some notations as follows.
Suppose that (M, d) is a metric space and R is a relation on M. Then the triple (M, d, R) or in brief M is called an R-metric space. An R-sequence {x n } n∈N in an R-metric space M is a sequence {x n } n∈N such that x n R x n+k for each n, k ∈ N, and R-sequence {x n } n∈N is said to converge to x if, for every ε > 0, there is an integer N such that d(x n , x) < ε for every n ≥ N . In this case, we write x n R − → x, and the R-sequence {x n } n∈N in M is said to be an R-Cauchy sequence if, for every ε > 0, there exists an integer N such that d(x n , x m ) < ε for n ≥ N and m ≥ N . It is clear that x n R x m or x m R x n .
Also, the concepts of open and closed sets are defined in these spaces. For E ⊆ M, the element x ∈ M is called an R-limit point of E if there exists an R-sequence {x n } n∈N in E such that x n = x for all n ∈ N and x n R − → x. The set of all R-limit points of E is denoted by The paper is organized as follows. We continue this introductory section with a review of the basic definitions and notations of relative metric spaces, i.e., metric spaces equipped with relations, that are needed for the next sections.
In Sect. 2, we first define the notions of R-vector space, vector space equipped with relation, and R-convexity in these spaces. After giving some examples that distinct the notions of convexity and R-convexity in general, the effect of some operations on R-convex sets is investigated. More precisely, we show that the R-interior of an R-convex set is Rconvex under certain constraints on R.
Section 3 is devoted to studying R-extreme points, which are the relative extreme points of R-convex sets. After defining this notion and giving some examples, we prove that every extreme point is an R-extreme point, but the reverse is not necessarily true. Next, we define the R-convex hull of the sets and set some conditions that for R-convex set W , R -co(W ) = W . In the main theorem of this section, we give several equivalent conditions for R-extremeness, and in the last example of this section, we show that generally, the Krein-Milman type theorem does not hold. It seems that one can deduce a Krein-Milman type result for R-convex R-compact sets by putting additional restrictions on the relation R.
In Sect. 4, we introduce the notions of R-convex maps and R-affine maps on R-vector spaces. In classical convexity, f is a convex function if and only if the epigraph of f is a convex set. In this section, we prove such a result for R-convex maps, and then some corollaries of this theorem will be given. In continuation, by putting additional conditions on the relation R, we prove several propositions which assert that R-continuous maps take R-compact sets to R-compact sets, and R-affine maps preserve R-convexity. Also, the composition of an R-affine map and a preserving R-affine map is R-affine, and the composition of an increasing R-convex map and a preserving R-convex map is also an R-convex map.
The presented results in this manuscript make powerful tools for important applications in optimization theory. Finally, we concentrate on some applications of R-convexity in the optimization theory. More precisely, we show that the R-affine R-continuous maps take their extreme values on R-extreme points. Moreover, for an R-convex map f on R-convex set K , the set of all elements of K on which f takes its minimum is an R-convex set, and in R-vector metric space M, every local minimum x 0 of f is a global minimum of f on the set

R-convex sets
In [10,16], and [11], the authors considered some spaces with relations to them and obtained important and interesting results. It seems that these properties are independent of the relation and this fact was not considered. This section is devoted to preliminaries of R-vector spaces that are needed to study the R-convexity property for sets. Some examples are considered to clarify the contents.

Definition 2.1
Let R be a relation on a vector space V . Then V (or the pair (V , R)) is called to be an R-vector space.
In [16], the authors introduced R-convex sets for R-metric space R k . We recall this notion for an R-vector space as follows.
Definition 2.2 A subset W of an R-vector space V is said to be R-convex if λw 1 + (1λ)w 2 ∈ W whenever w 1 , w 2 ∈ W , w 1 R w 2 , and 0 < λ < 1. In this case, the combination λw 1 + (1λ)w 2 is called an R-convex combination of two elements w 1 and w 2 .
The following remark and examples illustrate the relation between two notions "convexity" and "R-convexity". Remark 2.3 Every convex set W in an R-vector space V is an R-convex set. However, the reverse of the result is not true.
Example 2.4 Suppose that V = R and R is the equality relation on V , and W = N. Then N is an R-convex set, but it is not a convex set.
W is an R-convex set, but it is not convex.
Example 2.7 Let V be an R-vector space such that R is an equivalence relation. If there exists a v 0 ∈ V such that v 0 R v for all v ∈ V , then R = V × V and the notions of R-convexity and convexity are equivalent. Since for v 1 The union and intersection of sets preserve R-convexity property. In the next proposition, we investigate these subjects.

Proposition 2.8 Let V be an R-vector space. Then the following statements hold.
i

. The intersection of every family of R-convex sets in
Furthermore, not all properties of convex sets hold for R-convex sets, as is illustrated in the following two remarks.
Remark 2.9 The scalar multiplier of a convex set is convex. But it is not true for R-convex sets. Assume that E is an R-convex set and α ∈ C. Then the set αE is not necessarily Rconvex. For example, set E = (0, 1) ∪ (2, 3) and for x, y ∈ R, Remark 2.10 For convex sets E 1 and E 2 , the set E 1 + E 2 is also convex. But it is not valid for R-convex sets. To see this, let E 1 = (0, 2) ∪ (3, 5) and E 2 = {-1}, and for x, y ∈ R, xR y ⇐⇒ x, y ∈ 1 2 , 3 or x, y ∈ (3, 5) or x = y = -1.
It can be verified that E 1 and E 2 are R-convex, but the set E 1 + E 2 = (-1, 1) ∪ (2, 4) is not R-convex because for the numbers x = 3 4 and y = 2.5, xR y and some of their R-convex combinations are not in E 1 + E 2 .
The closure of any convex set is convex. This will be investigated in the following example using an R-convex set and its R-closure. Example 2.11 Let V = R and E = (0, 1) ∪ (4, 5), and The set of all interior points of a convex set is convex, but this is not true for R-convex sets. In other words, the R-interior points of any R-convex set are not necessarily an R-convex set. See the following example as a counterexample.
In the following theorem, we provide the conditions to preserve the R-convexity from E to R -int(E).

Theorem 2.13 Let (M, d, R) be an R-metric vector space such that R is an equivalence relation on M, which has the following properties for every x, y ∈ M:
i. xR y ⇒ xR(λx The set E is R-convex, so xR y implies that z ∈ E. Using condition i, xR y implies that xR z, and hence yR z (since R is an equivalence relation).
On the other hand, since z n R − → z and xR z, by using condition ii, we conclude that xR z n , (∀n ≥ N 1 ), and hence yR z n (∀n ≥ N 1 ) for some N 1 ∈ N. For each m ∈ N, put Then, for each n ≥ N 1 , in view of condition i, we have xR x n,m and yR y n,m for all m ∈ N.
Since R is an equivalence relation on M, we can conclude from xR x n,m (∀m ∈ N) that implies that {y n,m } ∞ m=1 is an R-sequence in X, and y n,m R − → y as m → ∞. Thus, there are positive integers M 1 and M 2 such that x n,m ∈ E, ∀m ≥ M 1 , and y n,m ∈ E, ∀m ≥ M 2 . By and furthermore, we have Therefore, for all m ≥ M 0 , we conclude that , and the proof is completed.

R-extreme points
In this section, we define R-extreme point concept for an R-convex subset in R-vector spaces. Also, we define R-convex hull of the sets in R-vector spaces. The main results of this section are presented in Proposition 3.9 and Theorem 3.10, and some equivalent conditions for R-extremeness in the special R-vector spaces are obtained.

Definition 3.1
In an R-vector space V , an R-open line segment is a set of the following form: In the following, some examples are given to illustrate the concept of R-extreme points and the differences between the extreme points and R-extreme points.
It is well known that W is a convex set and also an R-convex set, and ext(W ) = ∅ but R -ext(W ) = {(x, 0); x ∈ R} because for every x ∈ R and 0 < λ < 1, if then 0 = λw + (1λ)v and 0 ≤ w < v, which is a contradiction, and hence (x, 0) cannot be written as an R-convex combination of elements of W . Note that if we replace '<' with '≤' in the relation R, then Clearly, W is a convex set, and so is R-convex. We know that ext If the relation is replaced with the following relation: then R -ext(W ) = ext(W ) = {(x, y) ∈ W ; x 2 + y 2 = 1}. Now, we define the concept of R-convex hull of a set, and then we appoint some limitations on the relation R, to obtain a necessary and sufficient condition for a set to be an R-convex set. Definition 3.8 Let W be a subset of an R-vector space V . The R-convex hull of W is denoted by R -co(W ) and is defined as follows: Moreover, every element of R -co(W ) is said to be an R-convex combination of elements of W . Proposition 3.9 Let V be an R-vector space such that the relation R has the following properties: i. vR v for all v ∈ V . ii. If vR v 1 and vRv 2 , then vR λv 1 + (1λ)v 2 for v, v 1 , v 2 ∈ V and every 0 < λ < 1.

Then every subset W of V is R-convex if and only if W = R -co(W ).
Proof Firstly, assume that W = R -co(W ), and v 1 , v 2 ∈ W such that v 1 R v 2 . Then, This shows that W ⊆ R -co(W ). Now, R -co(W ) ⊆ W is obtained by induction. 3 1-α 1 = 1, and v 2 R v 3 and by using R-convexity of W . Similarly, for every n ∈ N, we obtain R -co{v 1 , . . . , v n } ⊆ W . Thus R -co(W ) ⊆ W and the proof is complete.
Note that in Proposition 3.9 the given condition for R is necessary, and if this condition is omitted, then the result is not true. To see this, let V = R and R := ' < . It is clear that In the last theorem of this section, some equivalent conditions for an element to be an R-extreme point are given.

Theorem 3.10 Let V be an R-vector space such that R is reflexive and for
vR v 1 and vR v 2 , then vR λv 1 + (1λ)v 2 for all 0 < λ < 1. Then the following statements are equivalent for every R-convex subset W of V : where v i ∈ W for i = 1, . . . , n and n ∈ N, then there exists Proof i − → ii. By definition of R-extreme point, it is clear.
ii − → iii. If λ = 1 2 , without loss of generality, we assume that 1 2 < λ < 1. Then the following equality is obtained: and v 1 Ry by the assumption. Therefore, the part iii is valid. iii By induction, the properties of R, and Proposition 3.9, we have n i=2 This is a contradiction, and so λv 1 proper R-open line segment containing v. Then v = λv 1 + (1λ)v 2 for some 0 < λ < 1. It is known that v 1 = v 2 , then v = v 1 and v = v 2 . Also, W \ {v} is R-convex, and v 1 , v 2 ∈ W \ {v}, and so v ∈ W \ {v}. But it is a contradiction, and hence v is an R-extreme point for W .
One of the most important subjects is considering Krein-Milman theorem for R-vector spaces. In the following example, we see that this theorem is not valid for an R-compact R-convex set in R-vector spaces generally.

R-convex functions
An important part of subjections in mathematics is studying the properties of a type of map between two spaces. One type of the map is a convex map. This section introduces R-convex maps and relative concepts and considers their properties with respect to relation R. i. Assume that R 1 is another relation on V . The map f is called to be R-convex with respect to R 1 if, for each 0 < λ < 1 and v 1 , v 2 ∈ V such that v 1 R v 2 , the following relation holds: ii. The map f : V → R is called to be R-convex if, for each 0 < λ < 1 and v 1 , v 2 ∈ V such that v 1 R v 2 , the following relation holds: iii. The map f : V → V (also the function f : V → R) is called to be R-affine if, for each 0 < λ < 1 and v 1 , v 2 ∈ V such that v 1 R v 2 , the following equation holds: The following example illustrates that every R-convex map is not necessarily a convex map. Then the map f : R − → R defined by is an R-convex map on R, but it is not a convex map. Because for α = 1 2 , v 1 = -1, and v 2 = 1, In the classical convexity, there is a straight relation between the convex functions and their epigraphs. In the following theorem and its corollaries, by giving some conditions, we obtain similar results for R-convex maps.

Theorem 4.4
Let R 1 and R 2 be two relations on a vector space V , and let f : V − → V be a map. Also, assume that R 2 is transitive and reflexive with the following property: For Moreover, suppose that S is a relation on V × V with two properties as follows: i.
Proof Suppose that f is an R 1 -convex map on V with respect to R 2 . For (v 1 , w 1 ) and (v 2 , Hence, by the property of R 2 , for each 0 < λ < 1, we conclude that By ii, we have v 1 R 1 v 2 , and by the R 1 -convexity of f , Now, since R 2 is transitive, we deduce f (λv 1 + (1λ)v 2 )R 2 (λw 1 + (1λ)w 2 ). Therefore, Conversely, let R 2 -epi(f ) be an S-convex set. Suppose that 0 < λ < 1 and v 1 and v 2 in V such that v 1 R 1 v 2 . By the reflexivity of R 2 and the property i, the following statements hold: Then the S-convexity of R 2 -epi(f ) concludes and hence, So, and f is R 1 -convex on V with respect to R 2 .
The special cases of the above theorem for a real vector space with different relations are concluded in the following corollaries. It is well known that for f : R − → R, the epigraph of f is {(x, y); f (x) ≤ y}. Corollary 4.5 Let R be a relation on vector space R, f be a map on R, and S be a relation on R × R such that Proof Since the relation ' ≤ is reflexive and transitive on R, so it is a straightforward conclusion of Theorem 4.4.

Corollary 4.6
Assume that R is a relation on the vector space R, and f is a map on R. Let S be the induced relation of R on R × R as follows: Then the function f : Proof It is concluded by Corollary 4.5.

Corollary 4.7 Let V be an R-vector space and f : V − → R be a function. Also, S is a relation on V × V with the following properties:
i

Then f is R-convex if and only if epi(f ) is an S-convex set.
Proof It is a consequence of Theorem 4.4, since the relation ' ≤ is reflexive and transitive.
In the classical convexity, every convex function is a continuous function. But there exist some R-convex functions which are not R-continuous. Then the function is an R-convex function on the R-convex set R, and it is not R-continuous. This is because by setting x n = 1 n for all n ∈ N, {x n } n∈N is an R-sequence converging to zero, and It is known that every continuous map preserves compact sets. In the following proposition, we show that every R-continuous map, by an additional condition, preserves Rcompact sets.
The goal of the following proposition is to show the preservation of R-convex sets under the special R-affine maps.

Proposition 4.3
In an R-metric vector space, the following statements are valid: i. Summation, subtraction, and scalar multiplication of R-affine maps are also R-affine. ii. If f and g are R-affine maps and g is an R-preserving map, then fog is also R-affine.

Proposition 4.4 Let f be an increasing R-convex function on the R-metric vector space M, and let g be an R-preserving R-convex map on M. Then fog is also an R-convex map.
Proof Let x, y ∈ M such that xR y. Then g(x)R g(y). For 0 < α < 1, This shows that fog is R-convex.

Some applications in optimization
An optimization problem considers minimizing or maximizing a given real function on a subset of its domain. In other words, in an optimization problem, one obtains the best available values for some functions that have different types corresponding to objective functions and types of their domains. The optimization theory and its techniques are useful and very important in a large area of applied mathematics. In this section, we study some results in optimization theory. More precisely, we study important results about the extreme values of some R-convex maps on R-convex sets. In the first theorem, we show that every R-continuous R-affine function attains its extrema at R-extreme points. Proof Let f take its maximum on B at x 0 ∈ B. Then there exists an R-sequence {x n } n∈N ⊂ R -co(Rext(K)) such that x n R − → x 0 . Notice that x n = N n i=1 λ n,i y n,i where N n ∈ N and y n,i ∈ R -ext(K), (1 ≤ i ≤ N n ) and λ n,i ∈ (0, 1] such that N n i=1 λ n,i = 1. Thus, On the other hand, f (x 0 ) is maximum of f on B, so f (x 0 ) = f (y 0 ), and f attains its maximum on B at y 0 . Similarly, we can prove the theorem for the minimum case.
In the succeeding proposition, we show that the set of all elements on which an R-convex function takes its minimum is an R-convex set.
Proposition 5.2 Let V be an R-vector space, K be an R-convex subset of V , and f : K − → R be an R-convex function on K . Then, the set B = {x ∈ V ; f (x) = min y∈K f (y)} is R-convex.
The following theorem asserts that every local minimum is a global minimum for Rconvex functions. Proof Suppose that f takes its minimum at x 0 on the neighborhood N of x 0 , and x ∈ [x 0 ] R ∩ K . Then, for sufficiently small λ > 0, we have and hence λ(f (x)f (x 0 )) ≥ 0, which implies that f (x 0 ) ≤ f (x), and the proof is completed. In addition, if x 0 R x for all x ∈ K , then x 0 is a global minimum of f on K since f (x 0 ) ≤ f (x) for all x ∈ K . Proof Since f is strictly R-convex on K , as the proof of the previous theorem, we obtain f (x 0 ) < f (x) for all x ∈ [x 0 ] R ∩ K where x = x 0 . Theorem 5.5 Let (V , R) be an R-vector space such that R is an equivalence relation on V with the following property: aRb ⇒ aR λa + (1λ)b , ∀λ ∈ (0, 1).
If K is an R-convex subset of V and f : K − → R is an R-convex function which has a global maximum at x 0 , then f is constant on [x 0 ] R ∩ K .