Accepted Manuscript A coradiant based scalarization to characterize approximate solutions of vector optimization problems with variable ordering structures A. Sayadi-bander, R. Kasimbeyli, L. Pourkarimi PII: DOI: Reference:
S0167-6377(16)30296-6 http://dx.doi.org/10.1016/j.orl.2016.12.009 OPERES 6183
To appear in:
Operations Research Letters
Received date: 13 August 2016 Revised date: 26 December 2016 Accepted date: 26 December 2016 Please cite this article as: A. Sayadi-bander, R. Kasimbeyli, L. Pourkarimi, A coradiant based scalarization to characterize approximate solutions of vector optimization problems with variable ordering structures, Operations Research Letters (2016), http://dx.doi.org/10.1016/j.orl.2016.12.009 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Manuscript Click here to view linked References
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
A Coradiant Based Scalarization to Characterize Approximate Solutions of Vector Optimization Problems with Variable Ordering Structures A. Sayadi-bandera , R. Kasimbeyli∗b , L. Pourkarimic a Department
of Mathematics, Faculty of Mathematical Sciences and Computer, Shahid Chamran University of Ahvaz, Ahvaz, Iran of Industrial Engineering, Anadolu University, Iki Eylul Campus 26555, Eskisehir, Turkey c Department of Mathematics, Razi University, Kermanshah, Iran
b Department
Abstract This paper investigates some properties of approximate efficiency in variable ordering structures where the variable ordering structure is given by a special set valued map. We characterize ε-minimal and ε- nondominated elements as approximate solutions of a multiobjective optimization problem with a variable ordering structure and give necessary and sufficient conditions for these solutions, via scalarization. Keywords: Multiobjective optimization, ε-Efficiency, Variable ordering structures, Scalarization. 1. Introduction One of major problems of the optimization is that a decision maker should simultaneously obtain the optimal solution of several objectives which are in conflict. Such problems can be formulated in the form of multiobjective optimization problems. Applications of multiobjective optimization problems can be found in financial mathematics, economic theory, management science, design engineering and many other fields. In multiobjective optimization, it is usual to define the partial ordering by a fixed cone in the objective space. But sometimes decisions may depend on more that one objective and what is preferred may vary on the actual state. Already in 1974, Yu introduced the concept of variable ordering structure by using several different cones in vector optimization [25], and defined optimal solutions called nondominated solutions. In the literature, variable ordering structures usually were introduced by cone valued maps. These maps associate a cone to any objective vector. The variable ordering structure leads to various concepts of optimality in multiobjective optimization. In [3, 7, 8, 9, 25], different concepts of optimality and their properties have been studied. Some applications of the multiobjective optimization problem with variable ordering structures can be found in [23, 24]. ∗ Corresponding
author Email address:
[email protected] (R. Kasimbeyli∗ ) Preprint submitted to Operations Research Letters
In most cases, finding an exact optimal solution of a multiobjective optimization problem may be very hard. Different numerical algorithms are often applied to solve a given optimization problem. Numerical algorithms as usual provide approximate optimal solution. Hence, the study of approximate solutions is of interest. One of the first concepts of approximate solutions for multiobjective optimization problems is introduced by Kutateladze [19]. Later, different concepts of approximate solutions were introduced (see for instance [2, 4, 11, 20]). Gutierrez et al. using the concept of the coradiant set, introduced a new concept of approximate solutions and showed that many pervious concepts of approximate solutions are special cases of this new definition [12, 13]. In [1, 21], authors have generalized the concept of ε−efficiency (in the sense of Kutateladze) to a similar concept in variable ordering structures. They defined a set valued map C : Y ⇒ Y such that C(y) is a closed set and 0 ∈ ∂(C(y)). Using this map, they proposed the concept of k0 -nondominated points and k0 minimizers in vector optimization problem with variable ordering structures. In this work, we study approximate solutions of multiobjective problems by using a variable ordering structure which associates a coradiant set to any element of the objective space. By using the augmented dual cones given by Kasimbeyli in [16], we introduce the concepts of ε−dual coradiant set and augmented ε−dual coradiant set in this work. The elements of these sets are used to construct scalarizing functions which can be viewed December 26, 2016
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Definition 2.1. [13] The set C ⊆ R p is called a coradiant set if αC ⊆ C for all α ≥ 1.
as generalizations of the monotone sublinear functions introduced in [16] and [17] (see also [10]). Kasimbeyli has introduced the conic scalarization method and characterized efficient solutions of vector optimization problems without convexity and boundedness conditions [16, 17]. In this work, we introduce the scalarization approach to characterize approximate efficient elements of multiobjective optimization problems with variable ordering structure. The rest of the paper is organized as follows. Section 2 gives some notions and preliminaries. In Section 3, we present a special class of the variable ordering structures and introduce new concepts of approximate solutions for multiobjective optimization problem with variable ordering structure. Section 4 presents a scalarization method to characterize approximate solutions in variable ordering structures. Finally Section 5 draws some conclusions.
Throughout this paper, we assume that 0 < C and the coradiant set C is pointed if C ∩ (−C) = ∅. For given ε > 0, let C(ε) := {εc | c ∈ C }. Definition 2.2. [13] Let ε > 0. y¯ ∈ A is an ε−efficient element of A w.r.t. coradiant set C if (¯y − C(ε)) ∩ A\{¯y} = ∅. Let C be a coradiant set, and let ε > 0 be a given positive number. The sets C ∗ (ε) := {` ∈ R p | h`, yi ≥ 0 for all y ∈ C(ε) } , C # (ε) := {` ∈ R p | h`, yi > 0 for all y ∈ C(ε) } are called the ε−dual coradiant set of C and its quasiinterior, respectively. Let λ > 0 be a given positive number. For the coradiant set C, the augmented ε−dual coradiant set and its quasi-interior are denoted by C λ∗ (ε) and C λ# (ε), respectively, and are defined as follows: n C λ∗ (ε) := (`, α) ∈ C # (ε) × R+ | h`, yi − α kyk ≥ λ
2. Preliminaries
for all y ∈ C(ε)} . n # C (ε) := (`, α) ∈ C (ε) × R+ | h`, yi − α kyk > λ
Throughout the paper, int(G), cl(G), bd(G) and co(G) denote the interior, the closure, the boundary and the convex hull of a set G ⊆ R p , respectively. A set G ⊆ R p is called cone if αG = G for all α > 0. The set cone(G) = {αg | g ∈ G, α > 0 } denotes the cone generated by a set G. We say that a cone G is convex if G + G ⊆ G, G is solid if int(G) , ∅ and G is pointed if G ∩ (−G) ⊆ {0}, i.e. if G ∩ (−G) = {0} when 0 ∈ G and if G ∩ (−G) = ∅ when 0 < G. Let D be a convex, closed and pointed cone of R p , which defines a partial ordering on R p in the following form: y1 ≤ y2 ⇔ y2 − y1 ∈ D.
λ#
for all y ∈ C(ε)} .
3. Variable Ordering structure
In this paper the following multiobjective optimization problem is considered: min { f (x) | x ∈ S } ,
(1)
where f : S → R p and S ⊆ Rn . Let A := f (S ), we recall that y¯ ∈ A is an efficient element of (1) with respect to (w.r.t.) the cone D if (¯y − D) ∩ A ⊆ {¯y}. In recent decades, many researchers investigated approximate solutions of problem (1). Various concepts of approximate solutions have been introduced. A new notion for the approximate solutions, by using the concept of coradiant sets (instead of a cone in the classical definition) was given in [13]. Now we recall the definition of the coradiant set. 2
Variable ordering structure in vector optimization was firstly introduced by Yu in [25]. He suggested a variable cone instead of a fixed one in the evaluation process. After Yu, the multiobjective optimization problem with variable ordering structures was studied intensively. Motivated by these considerations, different concepts of optimality have been introduced in multiobjective optimization with variable ordering structure. In this section, we study approximate efficiency in multiobjective optimization with variable ordering structure on R p , which is defined by a set valued map C : A → 2A with C(y) ⊆ R p is a closed, convex and pointed coradiant set such that 0 < C(y) for every y ∈ A. For convenience, in the sequel we will use Cy := C(y). Let ε and λ be positive numbers. Then, the augmented ε−dual coradiant sets for a variable coradiant set Cy , can be defined in a similar way, which was used for a fixed coradiant set, as follows: n Cyλ∗ (ε) := (`, α) ∈ Cy# (ε) × R+ | h`, zi − α kzk ≥ λ o for all z ∈ Cy (ε) ,
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
n Cyλ# (ε) := (`, α) ∈ Cy# (ε) × R+ | h`, zi − α kzk > λ o for all z ∈ Cy (ε) .
In the next section, we present characterization of the ε−nondominated and ε−minimal elements of A via scalarization.
Now we introduce two new concepts of approximate efficient elements in multiobjective optimization with a variable ordering structure. Definition 3.1. For a given ε > 0, y¯ ∈ A is called an ε−nondominated (weakly ε−nondominated) element of A w.r.t. the map C, if there is no y ∈ A such that y¯ ∈ {y} + Cy (ε) (¯y ∈ {y} + int(Cy (ε))).
4. Scalarization By using a nonlinear separation theorem for two cones, Kasimbeyli introduced the conic scalarization method to characterize efficient elements of problem (1) [16, 17]. In this scalarization, boundedness and convexity conditions are not required, while in most scalarization methods these conditions are essential (see e.g. [5, 14, 15, 18, 22]). By using nonlinear scalarization functionals, Eichfelder presented a complete characterization of nondominated and minimal elements in variable ordering structures [6, 7]. She also considered variable ordering structures which are defined by ordering maps with images being BP cones. This functional allows a complete characterization of the nondominated and also the minimal elements in variable ordering structures. In this section, we present a scalarization approach to characterize ε−nondominated and ε−minimal elements of a set A w.r.t. the variable ordering structure given by the map C, where we use elements of the augmented ε−dual coradiant sets and their quasi interior sets to construct scalarizing functions.
Definition 3.2. For given ε > 0, y¯ ∈ A is called an ε−minimal (weakly ε−minimal) element of A w.r.t. the map C if there is no y ∈ A such that y¯ ∈ {y} + Cy¯ (ε) (¯y ∈ {y} + int(Cy¯ (ε))).
Remark 3.3. From Definition 3.1 it can be concluded that if y¯ satisfies the following condition, y¯ − Cy (ε) ∩ A = ∅
for all y ∈ A,
(2)
then y¯ is an ε−nondominated element of A w.r.t. the map C. It should be noted that the relation (2) is a sufficient condition for y¯ to be an ε−nondominated element. Remark 3.4. From Definition 3.2 it can be concluded that y¯ is an ε−minimal element of A w.r.t. the map C if and only if y¯ − Cy¯ (ε) ∩ A = ∅. (3) The following example demonstrates that, the remarks 3.3 and 3.4 can be used to characterize the ε−minimal and ε-nondominated elements of A w.r.t. coradiant valued map C.
Example 3.5. Let ε > 0, C1 , C2 ⊆ R2 be two different closed, convex and pointed coradiant sets and C : R2 → R2 be a coradiant valued map as follows: ( C1 i f y1 ≥ y2 Cy = C2 i f y 1 < y2
Remark 4.1. In the remainder of this paper, we assume that ε and λ are positive real numbers and 0 < Cy (ε) for all y ∈ A. We also assume that for the given pair (`, α), the function T `,α (y) is defined by T `,α (y) := h`, yi + α kyk for every y ∈ A.
Assume that y¯ ∈ A. By Remark 3.4, if y¯ − Cy¯ (ε) ∩ A = ∅ then y¯ is an ε−minimal element of A w.r.t. the coradiant valued map C. In the other word, if ( (¯y − C1 (ε)) ∩ A = ∅ if y¯ 1 ≥ y¯ 2 , (4) (¯y − C2 (ε)) ∩ A = ∅ if y¯ 1 < y¯ 2 ,
Lemma 4.2. Assume that the variable ordering structure is given by a mapping C : A → 2A with coradiant set values, y1 , y2 , y¯ ∈ A, (`, α) ∈ Cyλ∗ ¯ (ε) ((`, α) ∈ λ# Cy¯ (ε)). If y1 − y2 ∈ Cy¯ (ε) then T `,α (y1 ) − T `,α (y2 ) ≥ λ (T `,α (y1 ) − T `,α (y2 ) > λ).
then y¯ ∈ A is an ε−minimal element of A w.r.t. coradiant valued map C. By Remark 3.3 if the conditions (5) and (6) satisfied then we conclude that y¯ is an ε−nondominated elements of A w.r.t. the coradiant valued map C. (¯y − C1 (ε)) ∩ A = ∅,
(5)
(¯y − C2 (ε)) ∩ A = ∅.
(6) 3
Proof. By definition, n # Cyλ∗ ¯ (ε) = (`, α) ∈ C y¯ (ε) × R+ | h`, zi − α kzk ≥ λ o for all z ∈ Cy¯ (ε) ,
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Since (`, α) ∈ Cyλ∗ ¯ (ε) and y1 − y2 ∈ C y¯ (ε), we have: h`, y1 − y2 i − α ky1 − y2 k ≥ λ.
(7)
Theorem 4.6. Assume that the variable ordering structure is given by a mapping C : A → 2A with coraS diant set values. Let D := Cy , r ∈ D(ε) and
On the other hand, since ky2 k − ky1 k ≤ ky1 − y2 k , it follows from (7) that
y∈A
h`, y1 i + α ky1 k −(h`, y2 i + α ky2 k)
(`, α) ∈ Dλ∗ (ε). If yˆ ∈ A is an optimal solution of the problem min T `,α (y) | y ∈ A , then any point of the set yˆ + θr θ ∈ (0, λ ) ∩ A is an ε−nondominated ele-
≥ h`, y1 − y2 i − α ky1 − y2 k ≥ λ,
and the proof is completed.
T `,α (r)
ment of A.
Remark 4.3. The above lemma introduces a property for T `,α on the coradiant sets Cy (ε). If (`, α) ∈ Cyλ∗ (ε) and yˆ ∈ Cy (ε), then according to this property we conclude T `,α (ˆy) = T `,α (ˆy) − T `,α (0) ≥ λ > 0 because yˆ − 0 ∈ Cy (ε) and T `,α (0) = 0.
Proof. Since r ∈ D(ε) and (`, α) ∈ Dλ∗ (ε), it follows from Remark 4.3 that T `,α (r) ≥ λ > 0 or equivalently 0 < T`,αλ(r) ≤ 1. Therefore, the term θ ∈ (0, T`,αλ(r) ) is well defined. On the contrary, assume that θˆ ∈ (0, T`,αλ(r) ) and y1 := ˆ ∈ A but y1 is not an ε−nondominated element of yˆ + θr A w.r.t. the map C. Then there exists y2 ∈ A\{y1 } such that y1 − y2 ∈ Cy2 (ε).
Lemma 4.4. If Cy1 ⊆ Cy2 , then Cyλ∗2 (ε) ⊆ Cyλ∗1 (ε) and Cyλ#2 (ε) ⊆ Cyλ#1 (ε).
Since Cy2 ⊆ D, we have Dλ∗ (ε) ⊆ Cyλ∗2 (ε). Therefore (`, α) ∈ Cyλ∗2 (ε). Hence, by Lemma 4.2 it can be concluded that T `,α (y1 ) − T `,α (y2 ) ≥ λ or equivalently
Proof. The proof is straightforward the definitions of Cyλ∗ (ε) and Cyλ# (ε).
ˆ − T `,α (y2 ) ≥ λ. T `,α (ˆy + θr) (8)
ˆ , it holds T `,α (ˆy) + θT ˆ `,α (r) ≥ Since kˆyk + θˆ krk ≥ yˆ + θr ˆ T `,α (ˆy + θr). Then from (8) it follows that T `,α (ˆy) + ˆ `,α (r) − T `,α (y2 ) ≥ λ or equivalently θT
The following two theorems give necessary and sufficient conditions for ε−nondominated elements.
ˆ `,α (r) T `,α (ˆy) − T `,α (y2 ) ≥ λ − θT
Theorem 4.5. Assume that the variable ordering structure is given by a mapping C : A → 2A with coradiant S set values. Let y¯ ∈ A, and let D := Cy .
(9)
ˆ `,α (r) > 0. From Since θˆ ∈ (0, T`,αλ(r) ), it holds λ − θT (9) it follows that T `,α (ˆy) − T `,α (y2 ) > 0 or T `,α (y2 ) < T `,α (ˆy), but this contradicts the optimality of yˆ for the ˆ is an problem min T `,α (y) | y ∈ A . Therefore, yˆ + θr ε−nondominated element of A w.r.t. C.
y∈A
a. If (`, α) ∈ D (ε) and T `,α (¯y) − T `,α (y) < λ for all y ∈ A\{¯y}, then y¯ is an ε−nondominated elements of A. b. If (`, α) ∈ Dλ# (ε) and T `,α (¯y) − T `,α (y) ≤ λ for all y ∈ A\{¯y}, then y¯ is an ε−nondominated elements of A. λ∗
The following example illustrates Theorem 4.6.
Proof. a. On the contrary, assume that y¯ is not an ε−nondominated element of A w.r.t. the map C, then there exists y ∈ A\{¯y} such that y¯ − y ∈ Cy (ε). Since (`, α) ∈ Dλ∗ (ε) and Cy ⊆ D, it follows from Lemma 4.4 that (`, α) ∈ Cyλ∗ (ε). Therefore, it can be concluded that T `,α (¯y) − T `,α (y) ≥ λ by Lemma 4.2 and this is a contradiction. b. The proof of (b) is similar to the proof of (a), and we only mention briefly that in Lemma 4.2, if (`, α) ∈ Cyλ# ¯ (ε) then T `,α (y1 ) − T `,α (y2 ) > λ.
Example 4.7. Let A ⊆ R2 and C1 and C2 be two coradiant sets as follows: n o A := (y1 , y2 ) ∈ R2 (y1 − 2)2 + (y2 − 2)2 ≤ 10 , n o C1 := (y1 , y2 ) ∈ R2 | y2 ≥ 1, y2 ≥ 3y1 , n o C2 := (y1 , y2 ) ∈ R2 | y1 + y2 ≥ 1, y2 ≤ 2y1 , y1 ≤ 2y2 .
4
Consider the variable ordering structures C : A → 2A such that ( C1 i f y1 > y2 Cy := C2 i f y1 ≤ y2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Proof. From Remark 4.3, since r ∈ Cy¯ (ε) and (`, α) ∈ Cyλ∗ ¯ (ε), we have T `,α (r) ≥ λ > 0 or equivalently 0 < λ λ T `,α (r) ≤ 1. Therefore, the term θ ∈ (0, T `,α (r) ) is well defined. On the contrary, assume that θˆ ∈ (0, T`,αλ(r) ) and ˆ ∈ A but y1 is not an ε−minimal element of y1 := yˆ + θr A w.r.t. C. Then there exists y2 ∈ A\{y1 } such that
Suppose D := C1 ∪ C2 , ε = 0.25 and λ = 0.5. It easily can be seen that r := (0.25, 0.75) ∈ D(ε). Choosing (`, α) = ((3, 4), 1) ∈ Dλ∗ (ε), the point yˆ = (3.0093, 2.5430) becomes an optimal solution of the problem min T `,α (y) | y ∈ A and T`,αλ(r) = 0.1101. The line segment L with ( ) λ L := yˆ + θr | 0 ≤ θ < T (r)
y1 − y2 ∈ Cy1 (ε).
= {(3.0093, 2.5430) + θ(0.25, 0.75) | θ ∈ [0, 0.1101) } .
Under the hypotheses of theorem, Cy1 ⊆ Cy¯ , then we have y1 − y2 ∈ Cy¯ (ε) and since (`, α) ∈ Cyλ∗ ¯ (ε), then By Lemma 4.2 it concluded that T `,α (y1 ) − T `,α (y2 ) ≥ λ or equivalently
is a subset of ε−nondominated elements of A w.r.t. the map C.
ˆ − T `,α (y2 ) ≥ λ. T `,α (ˆy + θr)
(10)
The following two theorems give necessary and sufficient conditions for ε−minimal elements.
The rest of the proof similar to the proof of Theorem 4.6.
Theorem 4.8. Assume that the variable ordering structure is given by a mapping C : A → 2A with coradiant set values. Let y¯ ∈ A.
In the sequel, we characterize other sufficient conditions for existence the ε−nondominated and ε−minimal elements of A w.r.t. the map C. Suppose that (`y , αy ) ∈ Cyλ∗ (ε) for all y ∈ A and y¯ ∈ A. We introduce the functional T `,α,¯y : A → R as follows:
a. If (`, α) ∈ Cyλ∗ y) − T `,α (y) < λ for all ¯ (ε) and T `,α (¯ y ∈ A\{¯y}, then y¯ is an ε−minimal element of A w.r.t. the map C. b. If (`, α) ∈ Cyλ# y) − T `,α (y) ≤ λ for all ¯ (ε) and T `,α (¯ y ∈ A\{¯y}, then y¯ is an ε−minimal element of A w.r.t. the map C.
T `,α,¯y (y) := h`y , y − y¯ i + αy ky − y¯ k . It should be noted that in the the functional T `,α,¯y , for any y ∈ A we choose (`y , αy ) ∈ Cyλ∗ (ε). In the other word, by changing element y ∈ A, the pair (`y , αy ) may also change.
Proof. a. By the contrary, assume that y¯ is not an ε−minimal element of A w.r.t. the map C, then there exists yˆ ∈ A\{¯y} such that y¯ − yˆ ∈ Cy¯ (ε). On the other hand, since (`, α) ∈ Cyλ∗ ¯ (ε), it follows from Lemma 4.2 that T `,α (¯y) − T `,α (ˆy) ≥ λ which is a contradiction. b. The proof of (b) is similar to the proof of (a), and we only mention briefly that in Lemma 4.2, if (`, α) ∈ Cyλ# ¯ (ε) then it can be concluded that T `,α (y1 ) − T `,α (y2 ) > λ. Theorem 4.9. Assume that the variable ordering structure is given by a mapping C : A → 2A with coradiant (ε) and yˆ is set values. Let y¯ ∈ A, r ∈ Cy¯ (ε), (`, α) ∈ Cyλ∗ ¯ an optimal solution of the problem min T `,α (y) | y ∈ A . If Cyˆ+θr ⊆ Cy¯ for θ ∈ (0, T`,αλ(r) ), then any point of set yˆ + θr θ ∈ (0, T`,αλ(r) ) ∩ A is an ε−minimal element of A w.r.t. map C.
Theorem 4.10. Let y¯ ∈ A. If T `,α,¯y (y) > −λ for all y ∈ A then y¯ is an ε−nondominated element of A w.r.t. the map C. Proof. On the contrary, assume that y¯ is not an ε−nondominated element of A w.r.t. the map C, then there exists yˆ ∈ A\{¯y} such that y¯ − yˆ ∈ Cyˆ (ε). Let λ∗ (`yˆ , αyˆ ) ∈ Cyλ∗ ˆ (ε), then from the definition of C yˆ (ε) we have h`yˆ , yˆ −¯yi−αyˆ kˆy − y¯ k ≥ λ or −h`yˆ , y¯ −ˆyi−αyˆ k¯y − yˆ k ≥ λ, in other words, h`yˆ , y¯ − yˆ i + αyˆ k¯y − yˆ k ≤ −λ. This means that T `,α,¯y (ˆy) ≤ −λ which contradicts the hypothesis.
5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
¯ α) Theorem 4.11. Let y¯ ∈ A and (`, ¯ ∈ Cyλ∗ ¯ (ε). If T `,¯ α,¯ (y) > −λ for all y ∈ A then y ¯ is an ε−minimal ¯y element of A w.r.t. the map C.
[15] N. Kasimbeyli, T. Sarac and R. Kasimbeyli, A two-objective mathematical model without cutting patterns for one-dimensional assortment problems, Journal of Computational and Applied Mathematics 235 (2011) 4663–4674. [16] R. Kasimbeyli, A nonlinear cone separation theorem and scalarization in nonconvex vector optimization. SIAM J. Optim. 20 (2010) 1591–1619. [17] R. Kasimbeyli, A conic scalarization method in multi-objective optimization. Journal of Global Optimization 56 (2013) 279–297. [18] R. Kasimbeyli, Z.K. Ozturk, N. Kasimbeyli, G.D. Yalcin, B. Icmen, Conic Scalarization Method in Multiobjective Optimization and Relations with Other Scalarization Methods, Proceedings of the 3rd International Conference on Modelling, Computation and Optimization in Information Systems and Management SciencesMCO 2015-Part I, In: L.T.H. An, P.D. Tao, N. N. Thanh (Eds.), Volume of Series: Advances in Intelligent Systems and Computing, Springer 359 (2015) 319–329 . [19] S.S. Kutateladze, Convex ε-programming. Soviet Math. Dokl. 20 (1979) 391–393. [20] P. Loridan, ε-solutions in vector minimization problems, J Optim Theory Appl. 43 (1984) 265–276. [21] B. Soleimani, Characterization of approximate solutions of vector optimization problems with variable order structure based on scalarization, J. Optim. Theory Appl. 162 (2014) 605-632. [22] O. Ustun and R. Kasimbeyli, Combined forecasts in portfolio optimization: a generalized approach, Computers & Operations Research 39 (2012) 805–819. [23] M. Wacker, Multikriterielle Optimierung bei Registrierung medizinischer Daten, Diplomarbeit, Univ. Erlangen-Nurnberg, Germany (2008). [24] M. Wacker, F. Deinzer, Automatic robust medical image registration using a new democratic vector optimization approach with multiple measures. In: Medical Image Computing and ComputerAssisted Intervention MICCAI 2009, Eds. G.-Z. Yang et al., (2009) 590–597. [25] P.L. Yu, Cone convexity, cone extreme points, and nondominated solutions in decision problems with multiobjectives, J. Optim. Theory Appl. 14 (1974) 319–377.
Proof. On the contrary, assume that y¯ is not an ε−minimal element of A w.r.t. the map C, then there exists yˆ ∈ A\{¯y} such that y¯ − yˆ ∈ Cy¯ (ε). Since ¯ α) (`, ¯ ∈ Cyλ∗ ¯ (ε), ¯ yˆ − y¯ i − α¯ kˆy − y¯ k ≥ λ or − h`, ¯ y¯ − yˆ i − α¯ k¯y − yˆ k ≥ λ, h`, ¯ y¯ − yˆ i + α¯ k¯y − yˆ k ≤ −λ. This in other words,, h`, means that T `,¯ α,¯ y) ≤ −λ which contradicts the hypoth¯ y (ˆ esis.
References [1] T. Q. Bao, G. Eichfelder, B. Soleimani, C. Tammer, Ekeland’s Variational Principle for Vector Optimization with Variable Ordering Structure, J. Convex Analysis, vol 24. [2] C. Bazgan, F. Jamain, D. Vanderpooten, Approximate Pareto sets of minimal size for multi-objective optimization problems, Operations Research Letters 43 (2015) 1–6. [3] G.Y. Chen, X.Q. Yang, Characterizations of variable domination structures via nonlinear scalarization, J. Optim. Theory Appl. 112 (2002) 97–110. [4] J. Dutta, V. Vetrivel, On approximate minima in vector optimization. Numer Funct Anal Optim. 22 (2001) 845–859. [5] M. Ehrgott, C. Waters, R. Kasimbeyli and O. Ustun, Multiobjective Programming and Multiattribute Utility Functions in Portfolio Optimization, INFOR: Information Systems and Operational Research 47 (2009) 31–42. [6] G. Eichfelder. Variable Ordering Structures in Vector Optimization. Habilitation Thesis, Friedrich-Alexander-Universitat Erlangen-Nurnberg (2001). [7] G. Eichfelder, Variable ordering structures in vector optimization. Springer Berlin Heidelberg (2014). [8] G. Eichfelder, R. Kasimbeyli, Properly optimal elements in vector optimization with variable ordering structures. Journal of Global Optimization 60 (2014) 689–712. [9] G. Eichfelder, T. Gerlach, Characterization of properly optimal elements with variable ordering structures. Optimization 65 (2016) 571-588. [10] R. N. Gasimov, G. Ozturk, Separation via polyhedral conic functions. Optimization Methods and Software 21 (2006) 527–540. [11] B.L. Gorissen, D. Den Hertog, Approximating the pareto set of multiobjective linear programs via robust optimization, Operations Research Letters 40 (2012) 319–324. [12] C. Gutierrez, B. Jimenez, V. Novo, A unified approach and optimality conditions for approximate solutions of vector optimization problems, SIAM J. Optim. 17 (2006) 688–710. [13] C. Gutierrez, B. Jimenez, V. Novo, On approximate efficiency in multiobjective programming, Math. Methods Oper. Res. 64 (2006) 165–185. [14] E. K. Karasakal, W. Michalowski, Incorporating wealth information into a multiple criteria decision making model. European Journal of Operational Research, 150 (2003) 204–219.
6