Optimality Conditions for Approximate Solutions in Multiobjective Optimization Problems
© Ying Gao et al. 2010
Received: 18 July 2010
Accepted: 25 October 2010
Published: 27 October 2010
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.
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  and Huang and Liu  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  derived the cone characterizations for approximate solutions of vector optimization problems by using translated cones. In , 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.  and Penot  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 [7–9].
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  and has been used to establish vector variational principle, approximate Kuhn-Tucker-type conditions and approximate duality theorems, and so forth, (see [11–20]). Later, several authors have proposed other -efficiency concepts (see, e.g., White ; Helbig  and Tanaka ).
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.
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.
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 ) of (2.1) with respect to if . is a Henig' properly efficient solution (see ) of (2.1) with respect to if , for some convex cone with .
Lemma 2.4 (see ).
3. Cone Characterizations of Approximate Solutions: Convex Case
If , then the conditions of Theorems 3.1 and 3.2 are also necessary(see ). But for , these are not necessary conditions, see the following example.
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.
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, .
If and , then Theorems 3.1 and 3.5 reduce to the corresponding results in .
In , 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 ). Therefore, if and , Theorems 3.2 and 3.6 reduce to the corresponding results in .
4. Cone Characterizations of Approximate Solutions: Nonconvex Case
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 4.4 (see ).
Lemma 4.5 (see ).
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.
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.
The proof is similar to Theorem 4.7.
- (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
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.
Let , then, . But for all , the condition (ii) of Corollary 4.8 satisfies (see Example 3.7 in ), 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.
See the following example.
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).
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