Stability Analysis for Higher-Order Adjacent Derivative in Parametrized Vector Optimization
© X. K. Sun and S. J. Li. 2010
Received: 29 March 2010
Accepted: 3 August 2010
Published: 16 August 2010
By virtue of higher-order adjacent derivative of set-valued maps, relationships between higher-order adjacent derivative of a set-valued map and its profile map are discussed. Some results concerning stability analysis are obtained in parametrized vector optimization.
Research on stability and sensitivity analysis is not only theoretically interesting but also practically important in optimization theory. A number of useful results have been obtained in scalar optimization (see [1, 2]). Usually, by stability, we mean the qualitative analysis, which is the study of various continuity properties of the perturbation or marginal function or map of a family of parametrized optimization problems. On the other hand, by sensitivity, we mean the quantitative analysis, which is the study of derivatives of the perturbation function.
Some authors have investigated the sensitivity of vector optimization problems. In , Tanino studied some results concerning the behavior of the perturbation map by using the concept of contingent derivative of set-valued maps for general multiobjective optimization problems. In , Shi introduced a weaker notion of set-valued derivative TP-derivative and investigated the behavior of contingent derivative for the set-valued perturbation maps in a nonconvex vector optimization problem. Later on, Shi also established sensitivity analysis for a convex vector optimization problem see  . In , Kuk et al. investigated the relationships between the contingent derivatives of the perturbation maps i.e., perturbation map, proper perturbation map, and weak perturbation map and those of feasible set map in the objective space by virtue of contingent derivative, TP-derivative and Dini derivative. Considering convex vector optimization problems, they also investigated the behavior of the above three kinds of perturbation maps under some convexity assumptions see ).
On the other hand, some interesting results have been proved for stability analysis in vector optimization problems. In , Tanino studied some qualitative results concerning the behavior of the perturbation map in convex vector optimization. In , Li investigated the continuity and the closedness of contingent derivative of the marginal map in multiobjective optimization. In , Xiang and Yin investigated some continuity properties of the mapping which associates the set of efficient solutions to the objective function by virtue of the additive weight method of vector optimization problems and the method of essential solutions.
To the best of our knowledge, there is no paper to deal with the stability of higher-order adjacent derivative for weak perturbation maps in vector optimization problems. Motivated by the work reported in [3–9], in this paper, by higher-order adjacent derivative of set-valued maps, we first discuss some relationships between higher-order adjacent derivative of a set-valued map and its profile map. Then, by virtue of the relationships, we investigate the stability of higher-order adjacent derivative of the perturbation maps.
The rest of this paper is organized as follows. In Section 2, we recall some basic definitions. In Section 3, after recalling the concept of higher-order adjacent derivative of set-valued maps, we provide some relationships between the higher-order adjacent derivative of a set-valued map and its profile map. In Section 4, we discuss some stability results of higher-order adjacent derivative for perturbation maps in parametrized vector optimization.
Throughout this paper, let and be two finite dimensional spaces, and let be a pointed closed convex cone with a nonempty interior , where is said to be pointed if . Let be a set-valued map. The domain and the graph of are defined by and , respectively. The so-called profile map is defined by for all
At first, let us recall some important definitions.
Definition 2.1 (see ).
Let be a nonempty subset of . An elements is said to be a minimal point (resp. weakly minimal point) of if resp., . The set of all minimal points resp., weakly minimal point of is denoted by (resp., .
Definition 2.2 (see ).
Definition 2.3 (see ).
Definition 2.4 (see ).
Definition 2.5 (see ).
Definition 2.6 (see ).
3. Higher-Order Adjacent Derivatives of Set-Valued Maps
In this section, we recall the concept of higher-order adjacent derivative of set-valued maps and provide some basic properties which are necessary in the following section. Throughout this paper, let be an integer number and .
Definition 3.1 (see ).
Definition 3.2 (see ).
The proof follows on the lines of Proposition in  by replacing contingent derivative by th-order adjacent derivative.
Since is a closed convex cone contained in , has a compact base. It is clear that is a compact base for , where is a compact base for . In this proposition, we assume that is a compact base of . Since , there exist and such that . Since is compact, we may assume without loss of generality that .
may still not hold under the assumptions of Proposition 3.8.
4. Continuity of Higher-Order Adjacent Derivative for Weak Perturbation Map
Definition 4.1 (see ).
The following lemma palys a crucial role in this paper.
Thus, result (4.5) holds.
Since is a compact space, it follows from Corollary of Chapter in  and Theorem 4.5 that is u.s.c. on Thus, the proof is complete.
It is easy to see that Example 4.8 can also illustrate Theorem 4.9.
The authors would like to thank the anonymous referees for their valuable comments and suggestions which helped to improve the paper. This research was partially supported by the National Natural Science Foundation of China (Grant no. 10871216) and Chongqing University Postgraduates Science and Innovation Fund (Grant no. 201005B1A0010338).
- Bonnans JF, Shapiro A: Perturbation Analysis of Optimization Problems, Springer Series in Operations Research. Springer, New York, NY, USA; 2000:xviii+601.View ArticleMATHGoogle Scholar
- Fiacco AV: Introduction to Sensitivity and Stability Analysis in Nonlinear Programming, Mathematics in Science and Engineering. Volume 165. Academic Press, Orlando, Fla, USA; 1983:xii+367.Google Scholar
- Tanino T: Sensitivity analysis in multiobjective optimization. Journal of Optimization Theory and Applications 1988, 56(3):479–499. 10.1007/BF00939554MathSciNetView ArticleMATHGoogle Scholar
- Shi DS: Contingent derivative of the perturbation map in multiobjective optimization. Journal of Optimization Theory and Applications 1991, 70(2):385–396. 10.1007/BF00940634MathSciNetView ArticleMATHGoogle Scholar
- Shi DS: Sensitivity analysis in convex vector optimization. Journal of Optimization Theory and Applications 1993, 77(1):145–159. 10.1007/BF00940783MathSciNetView ArticleMATHGoogle Scholar
- Kuk H, Tanino T, Tanaka M: Sensitivity analysis in vector optimization. Journal of Optimization Theory and Applications 1996, 89(3):713–730. 10.1007/BF02275356MathSciNetView ArticleMATHGoogle Scholar
- Kuk H, Tanino T, Tanaka M: Sensitivity analysis in parametrized convex vector optimization. Journal of Mathematical Analysis and Applications 1996, 202(2):511–522. 10.1006/jmaa.1996.0331MathSciNetView ArticleMATHGoogle Scholar
- Tanino T: Stability and sensitivity analysis in convex vector optimization. SIAM Journal on Control and Optimization 1988, 26(3):521–536. 10.1137/0326031MathSciNetView ArticleMATHGoogle Scholar
- Li Shengjie: Sensitivity and stability for contingent derivative in multiobjective optimization. Mathematica Applicata 1998, 11(2):49–53.MathSciNetMATHGoogle Scholar
- Xiang SW, Yin WS: Stability results for efficient solutions of vector optimization problems. Journal of Optimization Theory and Applications 2007, 134(3):385–398. 10.1007/s10957-007-9214-0MathSciNetView ArticleMATHGoogle Scholar
- 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
- Holmes RB: Geometric Functional Analysis and Its Applications. Springer, New York, NY, USA; 1975:x+246.View ArticleGoogle Scholar
- Luc DT: Theory of Vector Optimization, Lecture Notes in Economics and Mathematical Systems. Volume 319. Springer, Berlin, Germany; 1989:viii+173.Google Scholar
- Aubin J-P, Ekeland I: Applied Nonlinear Analysis, Pure and Applied Mathematics. John Wiley & Sons, New York, NY, USA; 1984:xi+518.MATHGoogle Scholar
- Aubin J-P, Frankowska H: Set-Valued Analysis, Systems & Control: Foundations & Applications. Volume 2. Birkhäuser, Boston, Mass, USA; 1990:xx+461.MATHGoogle Scholar
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