Skip to content

Advertisement

Open Access

Optimality Conditions for Approximate Solutions in Multiobjective Optimization Problems

Journal of Inequalities and Applications20102010:620928

https://doi.org/10.1155/2010/620928

Received: 18 July 2010

Accepted: 25 October 2010

Published: 27 October 2010

Abstract

We study first- and second-order necessary and sufficient optimality conditions for approximate (weakly, properly) efficient solutions of multiobjective optimization problems. Here, tangent cone, -normal cone, cones of feasible directions, second-order tangent set, asymptotic second-order cone, and Hadamard upper (lower) directional derivatives are used in the characterizations. The results are first presented in convex cases and then generalized to nonconvex cases by employing local concepts.

Keywords

Approximate SolutionEfficient SolutionVector OptimizationDirectional DerivativeTangent Cone

1. Introduction

The investigation of the optimality conditions is one of the most attractive topics of optimization theory. For vector optimization, the optimality solutions can be characterized with the help of different geometrical concepts. Miettinen and Mäkelä [1] and Huang and Liu [2] derived several optimality conditions for efficient, weakly efficient, and properly efficient solutions of vector optimization problems with the help of several kinds of cones. Engau and Wiecek [3] derived the cone characterizations for approximate solutions of vector optimization problems by using translated cones. In [4], Aghezzaf and Hachimi obtained second-order optimality conditions by means of a second-order tangent set which can be considered an extension of the tangent cone; Cambini et al. [5] and Penot [6] introduced a new second-order tangent set called asymptotic second-order cone. Later, second-order optimality conditions for vector optimization problems have been widely studied by using these second-order tangent sets; see [79].

During the past decades, researchers and practitioners in optimization had a keen interest in approximate solutions of optimization problems. There are several important reasons for considering this kind of solutions. One of them is that an approximate solution of an optimization problem can be computed by using iterative algorithms or heuristic methods. In vector optimization, the notion of approximate solution has been defined in several ways. The first concept was introduced by Kutateladze [10] and has been used to establish vector variational principle, approximate Kuhn-Tucker-type conditions and approximate duality theorems, and so forth, (see [1120]). Later, several authors have proposed other -efficiency concepts (see, e.g., White [21]; Helbig [22] and Tanaka [23]).

In this paper, we derive different characterizations for approximate solutions by treating convex case and nonconvex cases. Giving up convexity naturally means that we need local instead of global analysis. Some definitions and notations are given in Section 2. In Section 3, we derive some characterizations for global approximate solutions of multiobjective optimization problems by using tangent cone, the cone of feasible directions and -normal cone. Finally, in Section 3, we introduce some local approximate concepts and present some properties of these notions, and then, first and second-order sufficient conditions for local (properly) approximate efficient solutions of vector optimization problems are derived. These conditions are expressed by means of tangent cone, second-order tangent set and asymptotic second-order set. Finally, some sufficient conditions are given for local (weakly) approximate efficient solutions by using Hadamard upper (lower) directional derivatives.

2. Preliminaries

Let be the -dimensional Euclidean space and let be its nonnegative orthant. Let be a subset of , then, the cone generated by the set is defined as , and and referred to as the interior and the closure of the set , respectively. A set is said to be a cone if . We say that the cone is solid if , and pointed if . The cone is said to have a base if is convex, and . The positive polar cone and strict positive polar cone of are denoted by and , respectively.

Consider the following multiobjective optimization problem:
(2.1)
where is an arbitrary nonempty set, . As usual, the preference relation defined in by a closed convex pointed cone is used, which models the preferences used by the decision-maker:
(2.2)

We recall that is an efficient solution of (2.1) with respect to if . is a weakly efficient solution of (2.1) with respect to if (in this case, it is assumed that is solid). is a Benson properly efficient solution (see [24]) of (2.1) with respect to if . is a Henig' properly efficient solution (see [24]) of (2.1) with respect to if , for some convex cone with .

Definition 2.1 (see [18, 25]).

Let be a fixed element, and .

(i) is said to be a weakly -efficient solution of problem (2.1) if (in this case it is assumed that is solid).

(ii) is said to be a efficient -solution of problem (2.1) if .

(iii) is said to be a properly -efficient solution of problem (2.1), if .

The sets of -efficient solutions, weakly -efficient solutions, and properly -efficient solutions of problem (2.1) are denoted by , , and , respectively.

Remark 2.2.

If , then -efficient solution, weakly -efficient solution, and properly -efficient solution reduce to efficient solution, weakly efficient solution and properly efficient solution of problem (2.1).

Definition 2.3.

Let be a nonempty convex set.

The contingent cone of at is defined as
(2.3)
The cone of feasible directions of at is defined as
(2.4)
Let , the -normal set of at is defined as
(2.5)

Lemma 2.4 (see [26]).

Let be closed convex cones such that . Suppose that is pointed and locally compact, or , then, .

3. Cone Characterizations of Approximate Solutions: Convex Case

In this section, we assume that is a convex set.

Theorem 3.1.

Let and . If
(3.1)

then .

Proof.

Suppose, on the contrary, that , then, there exist and such that . That is, . Therefore, , which is a contradiction to . This completes the proof.

Theorem 3.2.

Let .

(i)If , then .

(ii)Let , and is solid set and . If , then .

Proof.
  1. (i)

    Suppose, on the contrary, that , then, there exists such that . Hence, there exist , and such that . Since , there exists such that .

     

Since is convex set, . Hence, . From Lemma 2.4, there exists such that , for all .

On the other hand, from , we have . Therefore, there exists such that , and so , which deduces a contradiction, and the proof is completed.
  1. (ii)
    Now, we let . From , we have
    (3.2)
     

In fact, if there exists such that , then, from and , there exists such that . Hence, , which is a contradiction to the assumption.

Since is a convex set, . Hence,
(3.3)

By using the convex separation theorem, there exists such that , for all and , for all . It is easy to get that , for all . Hence, , for all .

Suppose, on the contrary, that , then, there exists such that
(3.4)
and there exist , for all such that . That is, there exist and , for all such that , for all . Since , there exists such that , for all . From , and , for all , we have , for all . Therefore,
(3.5)

Which implies . On the other hand, from , we have , which yields a contradiction. This completes the proof.

Remark 3.3.

If , then the conditions of Theorems 3.1 and 3.2 are also necessary(see [2]). But for , these are not necessary conditions, see the following example.

Example 3.4.

Let , , , , , and , then, and . But . Hence, and .

Theorem 3.5.

Let , , be a solid set and . If there exists such that and , then . Conversely, if , then there exists such that and .

Proof.

Assume that, there exists such that and . Suppose, on the contrary, that , then, there exist and such that . From and , we have . Hence,
(3.6)

On the other hand, from , we have , which is a contradiction to the above inequality. Hence, .

Conversely, let , then, . Since is convex and is a convex cone, there exists such that , for all . Since , there exists such that and , for all . Therefore, , for all , which implies . This completes the proof.

Theorem 3.6.

Let and . If there exists such that and , then . Conversely, assume that is a locally compact set, if , then there exists such that and .

Proof.

Assume that, there exists such that and . Suppose, on the contrary, that , then, there exists such that
(3.7)

and there exists , for all such that . From and , we have . Hence, there exists such that , for all . From , for all , there exist , , and such that , for all . Therefore, , for all , which combing with and yields , for all , which is a contradiction to . Hence, .

Conversely, let , then,
(3.8)

Since is a convex set, is a closed convex cone. From Lemma 2.4, there exists such that . Since , and are cone, there exists such that and .

Now, we prove that . That is, , for all .

From , we have
(3.9)
Since and , we have
(3.10)

Which implies . This completes the proof.

Example 3.7.

Let , , , , , and , then, and . Let , then and .

Remark 3.8.
  1. (i)

    If and , then Theorems 3.1 and 3.5 reduce to the corresponding results in [1].

     
  2. (ii)

    In [1], the cone characterizations of Henig' properly efficient solution were derived. We know that Henig' properly efficient solution equivalent to Benson properly efficient solution, when is a closed convex pointed cone(see [24]). Therefore, if and , Theorems 3.2 and 3.6 reduce to the corresponding results in [1].

     

4. Cone Characterizations of Approximate Solutions: Nonconvex Case

In this section, is no longer assumed to be convex. In nonconvex case, the corresponding local concepts are defined as follows.

Definition 4.1.

Let be a fixed element and .

(i) is said to be a local weakly -efficient solution of problem (2.1), if there exists a neighborhood of such that (in this case, it is assumed that is solid).

(ii) is said to be a local -efficient solution of problem (2.1), if there exists a neighborhood of such that .

(iii) is said to be a local properly -efficient solution of problem (2.1), if there exists a neighborhood of such that .

The sets of local -efficient solutions, local weakly -efficient solutions and local properly -efficient solutions of problem (2.1) are denoted by , and , respectively.

If , then, (i), (ii), and (iii) reduce to the definitions of local weakly efficient solution, local efficient solution and local properly efficient solution, respectively, and the sets of local (weakly, properly) efficient solutions of problem (2.1) are denoted by , , respectively.

Definition 4.2 (see [4, 5]).

Let and .

(i)The second-order tangent set to at is defined as
(4.1)
(ii)The asymptotic second-order tangent cone to at is defined as
(4.2)

In [49], some properties of second-order tangent sets have been derived, see the following Lemma.

Lemma 4.3.

Let and , then,

(i) and are closed sets contained in , and is a cone.

(ii)If , then . If , then . If , then , and .

(iii)Let is convex. If and , then .

Definition 4.4 (see [27]).

Let and be a nonsmooth function. The Hadamard upper directional derivative and the Hadamard lower directional derivative derivative of at in the direction are given by
(4.3)

Lemma 4.5 (see [7]).

Let be a finite-dimensional space and . If the sequence converges to , then there exists a subsequence (denoted the same) such that converges to some nonnull vector , where , and either converges to some vector or there exists a sequence such that and converges to some vector , where denotes the orthogonal subspace to .

In the following theorem, we derive several properties of local (weakly, properly) approximate efficient solutions.

Theorem 4.6.
  1. (i)
    Let , then, for any fixed ,
    (4.4)
     
Conversely, if , and there exists a neighborhood of such that , for all , that is, , for all , then .
  1. (ii)

    For any fixed , . Conversely, if and there exists a neighborhood of such that for any fixed and , , then .

     
  2. (iii)

    For any fixed , . Conversely, if and there exists a neighborhood of such that for any fixed and , is a closed set, and , then .

     
Proof.
  1. (i)
    Let , then, there exists a neighborhood of such that . From , we have
    (4.5)
     

Which implies .

Conversely, we assume that there exists a neighborhood of such that , for all . Suppose, on the contrary, that , then, for any neighborhood of . Take , then, there exist and such that . Therefore, if is sufficiently small, we have , which is a contradiction to , for all . This completes the proof.
  1. (ii)

    It is easy to see that .

     
Conversely, we assume that there exists a neighborhood of such that for any fixed and , . Suppose, on the contrary, that , then, for any neighborhood of , we have . Take , then, there exist and such that . Take and , then, , which is a contradiction to the assumption. This completes the proof.
  1. (iii)

    It is easy to see that .

     

Conversely, we assume that there exists a neighborhood of such that for any fixed and , is a closed set, and . Suppose, on the contrary, that , then, for any neighborhood of , we have . Take , then, there exist , , and such that . Take and , similar to the proof of (ii) we can complete the proof.

Theorem 4.7.

Let be a continuous function on , , and .

(i)If , then .

(ii)If , and for each
(4.6)

then .

Proof.
  1. (i)

    Let . Suppose, on the contrary, that , then, there exists and such that , for all . Since is a continuous function and is a pointed cone, , for all and . Therefore, .

     
On the other hand, for any , we have
(4.7)
Since and , there exists such that
(4.8)
Hence, , which is a contradiction to the assumption. This completes the proof.
  1. (ii)
    Suppose, on the contrary, that . Similar to the proof of (i), we have there exists such that
    (4.9)
     

Let and , for all . Similar to the proof of Lemma 4.3, we have there exists such that or , which is a contradiction to the assumptions. This completes the proof.

Corollary 4.8.

Let be a continuous function on , and .

(i)If , then is a local efficient solution of problem (2.1).

(ii)If , and for each
(4.10)

then is a local efficient solution of problem (2.1).

Proof.

The proof is similar to Theorem 4.7.

Remark 4.9.

If is convex, then the condition (ii) of Theorem 4.7 is equivalent to the following condition

, and for each

(4.11)

since by Lemma 4.3(iii).

Theorem 4.10.

Let be continuous on , , and .

(i)Assume that has a compact base , for and , and there exists such that . If , then .

(ii)Assume that , and there exists such that for each the following conditions hold
(4.12)

then , where, denotes the closed unit ball of .

Proof.
  1. (i)
    Let , then, , for all . The assumptions and the separation result [28, page 9] implies that for any there exists a neighborhood of 0 such that
    (4.13)
     
Suppose, on the contrary, that , then, for any neighborhood of 0, we have
(4.14)
Therefore,
(4.15)
That is, for any there exist , and so, for any there exists , and such that and . Since , there exists such that , for all . By , we have
(4.16)
Let for and , then,
(4.17)
Let , then, , since is a convex set, and so,
(4.18)
On the other hand, from , we have when and . Since is a continuous function, when and , which combining with the assumption yields there exist and such that
(4.19)
From , there exists such that , for all . Take , and let . Since is an arbitrary set, it follows that
(4.20)
Which is a contradiction to (4.13). This completes the proof.
  1. (ii)
    Suppose, on the contrary, that , then, for any and , we have
    (4.21)
     
Let . Similar to the proof of (i), we have for any there exist , , and such that and . It is obvious that . Otherwise, , which is a contradiction to the assumption that is a pointed cone. Since and , there exists such that and , for all . From , we have , when and . Since is a continuous function and , it is easy to see that . From , we have for sufficiently large
(4.22)
On the other hand, we have
(4.23)
for sufficiently large . In fact, for sufficiently large
(4.24)
Hence,
(4.25)
when and sufficiently large enough. Since is arbitrary,
(4.26)

Let and . Similar to the proof of Lemma 4.3, we have there exists such that or , which is a contradiction to the assumptions. This completes the proof.

Remark 4.11.

If is convex, then the conditions (i) and (ii) of Theorem 4.10 are equivalent to and , respectively.

has a compact base , for some , , and .

(ii) , and there exists such that for each
(4.27)

Remark 4.12.

The conditions of Theorem 4.7, Corollary 4.8 and Theorem 4.10 are not necessary conditions, see Examples 4.14 and 4.15.

Now, we give some examples to verify the results of Theorem 4.7, Theorem 4.10 and Corollary 4.8.

Example 4.13.

Let , , , , , and . We consider . It is easy to see that and . That is, the condition (i) of Theorem 4.10 is valid, and , for all .

If we let and , then, . But the condition (ii) of Theorem 4.10 is valid. Hence, .

Let , then, . But for all , the condition (ii) of Corollary 4.8 satisfies (see Example 3.7 in [7]), and is an efficient solution of this problem, since is a convex set. But for any , it is easy to check that there exists such that . In fact, for any , take , then, and . Hence, the condition (ii) of Theorem 4.10 is false, and is not a properly efficient solution of this problem.

Example 4.14.

Let , , , and . Take , then, it is easy to see that there exists such that and , for all . Hence, , for all . But is not a global properly efficient solution, where, is closed unit ball of .

We let and , then, , for all , and (ii) in Theorem 4.10 is false. In fact, for any and , , since . But . This implies that the conditions of Theorem 4.10 are not necessary.

Example 4.15.

Let , , , , and . We consider . It is easy to see that . But , and the condition (ii) of Theorem 4.7 is false. In fact, if we take , then, , and . Therefore, .

Example 4.16.

Let , , , , , and . We consider . It is easy to see that and . That is, the condition (i) of Theorem 4.7 is valid, and , for all .

Theorem 4.17.

Let , and .

(i)If , for any unit vector , then .

(ii)If , for any unit vector , then .

Where, .

Proof.
  1. (i)
    Suppose, on the contrary, that , then, there exists , and such that . Let and , then, , and . Hence,
    (4.28)
     
Since , there exists such that , for all . Hence,
(4.29)
Therefore,
(4.30)
Which is a contradictions to the assumption. This completes the proof.
  1. (ii)
    Similar to the proof of (i), we have there exists and such that . Hence, there exists such that , for all . It is easy to see that, if we take an appropriate subsequences and of and , respectively, then there exist an index , and such that
    (4.31)
     

Therefore, , for all , and , which is a contradiction to the assumption. This completes the proof.

Remark 4.18.

The following necessary conditions for -local weakly (efficient) solutions may not be true.
(4.32)

See the following example.

Example 4.19.

Let ,
(4.33)
, , , . Consider the following problem:
(MP)
It is easy to see that is an -efficient solution of (MP), but, . In fact,
(4.34)
, for all . It is obvious that . On the other hand, . Hence, .

Declarations

Acknowledgments

This work was partially supported by the National Science Foundation of China (no. 10771228 and 10831009), the Research Committee of The Hong Kong Polytechnic University, the Doctoral Foundation of Chongqing Normal University (no.10XLB015) and the Natural Science Foundation project of CQ CSTC (no. CSTC. 2010BB2090).

Authors’ Affiliations

(1)
Department of Mathematics, Chongqing Normal University, Chongqing, China
(2)
Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Hong Kong

References

  1. Miettinen K, Mäkelä MM: On cone characterizations of weak, proper and Pareto optimality in multiobjective optimization. Mathematical Methods of Operations Research 2001, 53(2):233–245. 10.1007/s001860000109MathSciNetView ArticleMATHGoogle Scholar
  2. Huang LG, Liu SY: Cone characterizations of Pareto, weak and proper efficient points. Journal of Systems Science and Mathematical Sciences 2003, 23(4):452–460.MathSciNetMATHGoogle Scholar
  3. Engau A, Wiecek MM: Cone characterizations of approximate solutions in real vector optimization. Journal of Optimization Theory and Applications 2007, 134(3):499–513. 10.1007/s10957-007-9235-8MathSciNetView ArticleMATHGoogle Scholar
  4. Aghezzaf B, Hachimi M: Second-order optimality conditions in multiobjective optimization problems. Journal of Optimization Theory and Applications 1999, 102(1):37–50. 10.1023/A:1021834210437MathSciNetView ArticleMATHGoogle Scholar
  5. Cambini A, Martein L, Vlach M: Second-order tangent sets and optimaity conditions. Japan Advanced Studies of Science and Technology, Hokuriku, Japan; 1997.Google Scholar
  6. Penot J-P: Second-order conditions for optimization problems with constraints. SIAM Journal on Control and Optimization 1999, 37(1):303–318.MathSciNetView ArticleMATHGoogle Scholar
  7. Jiménez B, Novo V: Optimality conditions in differentiable vector optimization via second-order tangent sets. Applied Mathematics and Optimization 2004, 49(2):123–144.MathSciNetView ArticleMATHGoogle Scholar
  8. Gutiérrez C, Jiménez B, Novo V: New second-order directional derivative and optimality conditions in scalar and vector optimization. Journal of Optimization Theory and Applications 2009, 142(1):85–106. 10.1007/s10957-009-9525-4MathSciNetView ArticleMATHGoogle Scholar
  9. Bigi G: On sufficient second order optimality conditions in multiobjective optimization. Mathematical Methods of Operations Research 2006, 63(1):77–85. 10.1007/s00186-005-0013-9MathSciNetView ArticleMATHGoogle Scholar
  10. Kutateladze SS: Convex -programming. Soviet Mathematics. Doklady 1979, 20: 390–1393.MathSciNetGoogle Scholar
  11. Vályi I: Approximate saddle-point theorems in vector optimization. Journal of Optimization Theory and Applications 1987, 55(3):435–448. 10.1007/BF00941179MathSciNetView ArticleMATHGoogle Scholar
  12. Liu J-C: -properly efficient solutions to nondifferentiable multiobjective programming problems. Applied Mathematics 1999, 12(6):109–113.MathSciNetMATHGoogle Scholar
  13. Bolintinéanu S: Vector variational principles; -efficiency and scalar stationarity. Journal of Convex Analysis 2001, 8(1):71–85.MathSciNetMATHGoogle Scholar
  14. Dutta J, Vetrivel V: On approximate minima in vector optimization. Numerical Functional Analysis and Optimization 2001, 22(7–8):845–859. 10.1081/NFA-100108312MathSciNetView ArticleMATHGoogle Scholar
  15. Göpfert A, Riahi H, Tammer C, Zălinescu C: Variational Methods in Partially Ordered Spaces. Springer, New York, NY, USA; 2003:xiv+350.MATHGoogle Scholar
  16. Bednarczuk EM, Przybyła MJ: The vector-valued variational principle in Banach spaces ordered by cones with nonempty interiors. SIAM Journal on Optimization 2007, 18(3):907–913. 10.1137/060658989MathSciNetView ArticleMATHGoogle Scholar
  17. Chen G, Huang X, Yang X: Vector Optimization. Set-Valued and Variational Analysis, Lecture Notes in Economics and Mathematical Systems. Volume 541. Springer, Berlin, Germany; 2005:x+306.MATHGoogle Scholar
  18. Gupta D, Mehra A: Two types of approximate saddle points. Numerical Functional Analysis and Optimization 2008, 29(5–6):532–550. 10.1080/01630560802099274MathSciNetView ArticleMATHGoogle Scholar
  19. Gutiérrez C, Jiménez B, Novo V: A Set-valued ekeland's variational principle in vector optimization. SIAM Journal on Control and Optimization 2008, 47(2):883–903. 10.1137/060672868MathSciNetView ArticleMATHGoogle Scholar
  20. Gutiérrez C, López R, Novo V: Generalized -quasi-solutions in multiobjective optimization problems: existence results and optimality conditions. Nonlinear Analysis: Theory, Methods & Applications 2010, 72(11):4331–4346. 10.1016/j.na.2010.02.012MathSciNetView ArticleMATHGoogle Scholar
  21. White DJ: Epsilon efficiency. Journal of Optimization Theory and Applications 1986, 49(2):319–337. 10.1007/BF00940762MathSciNetView ArticleMATHGoogle Scholar
  22. Helbig S: One new concept for -efficency. talk at Optimization Days, Montreal, Canada, 1992Google Scholar
  23. Tanaka T: A new approach to approximation of solutions in vector optimization problems. In Proceedings of APORS. Volume 1995. Edited by: Fushimi M, Tone K. World Scientific, Singapore; 1994:497–504.Google Scholar
  24. Sawaragi Y, Nakayama H, Tanino T: Theory of Multiobjective Optimization, Mathematics in Science and Engineering. Volume 176. Academic Press, Orlando, Fla, USA; 1985:xiii+296.Google Scholar
  25. Rong WD, Ma Y: -properly efficient solution of vector optimization problems with set-valued maps. OR Transaction 2000, 4: 21–32.Google Scholar
  26. Borwein J: Proper efficient points for maximizations with respect to cones. SIAM Journal on Control and Optimization 1977, 15(1):57–63. 10.1137/0315004MathSciNetView ArticleMATHGoogle Scholar
  27. Huang LR: Separate necessary and sufficient conditions for the local minimum of a function. Journal of Optimization Theory and Applications 2005, 125(1):241–246. 10.1007/s10957-004-1726-2MathSciNetView ArticleMATHGoogle Scholar
  28. Rudin W: Functional Analysis, McGraw-Hill Series in Higher Mathematics. McGraw-Hill, New York, NY, USA; 1973:xiii+397.Google Scholar

Copyright

© Ying Gao et al. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Advertisement