Luận án: Khảo sát tính đặt vấn đề cho bài toán tối ưu tập hợp với các quan hệ thứ tự tập hợp

CHAPTER I INTRODUCTION Set-valued optimization plays an important role in optimization and has attracted a great deal of attention of mathematicians in recent years. It deals with optimization problems which the objective map and/or the constraint map(s) are set-valued maps. It is clear that single-valued maps work as a particular case of set- valued maps, and hence set-valued optimization provides an important generalization and unification of scalar optimization as well as vector optimization. It is worth noting that there are two approaches to formulate optimality notions for set-valued optimization problems, namely the vector approach and the set approach.

These criteria depend on the way that the notion of minimality is defined. In the vector approach, optimal solutions are defined as the efficient points of the union of all images of the set-valued objective map (see e. Hence, the set-valued optimization problem is called set- valued vector optimization problem in this approach. On the other hand, the set approach is based on set order relations defined on the power set of the objective space, which was firstly introduced by Kuroiwa et al.

By set approach one compares all images of the set-valued objective map and hence in this approach the set-valued optimization problem is called set optimization problem (see e. Both of them may be considered as generalizations of the usual definition of minimality in vector optimization. In fact, when objective map is a vector single-valued map, they reduce to the known concept of efficiency. In this thesis, set approach is applied and set order relations are used to define optimality notions for set-valued optimization problems.

We would like to give a brief review of set order relations. The introduction of set order relations was presented by Kuroiwa et al. Later on, Jahn and Ha [10] introduced some new set order relations. As pointed out in [10], the set less relation is generalized and more appropriate in practical problems than both the lower and upper set less relations.

Furthermore, the set less relation plays a center role in relationships with other new order relations for sets proposed in [10] which are more useful in set optimization. Lower set less relation and upper set less relation were studied in some publications [13, 14] and the references therein. To the best of our knowledge, there has been no publication about well-posedness, solution existence condition and stability for optimization problems involving the set less relation and new types of set order relations proposed in [10]. Due to a prospective issue, it should study further about these topics.

The set approach seems to be more natural and interesting than vector ap- proach whenever we need to consider preferences over sets. Set optimization problems play important roles and useful applications in the practical situations. Many impor- tant and significant applications of set order relations were studied and discussed. The possibly less order relation has been applied to interval arithmetic and Fortran compiler f95 of Sun Microsystems.

order relations was presented by Neukel [15] in the project investigating relationship between noise disturbance and quality of life in the region surrounding the Frankfurt Airport in Germany. Another application of set order relations in the field of finance about measures of risk was found by Hamel and Heyde [16]. Some simple but very useful applications of set order relation can be found and applied in daily activities, for instance, comparing teams of football players, ranking group of students on the same academic level,. For further reading and references, we refer to studies [13, 14, 17, 18, 19, 20, 21, 22] and the reference therein.

With the first introduction was given two decades ago, set optimization can be seen as a very young direction in the field of optimization. However, in the role 3 of an important and prospective issuse, it has attracted a great deal of attention of mathematicians. As pointed out by Jahn in [23], the set optimization criterion opens a new and wide field of research. Many important and interesting results have been obtained in different topics in this area such as solution existence [24, 25, 26], optimality conditions [27, 28, 29], nonlinear scalarization [9, 30, 31], Ekeland variational principles [8, 32], duality theory [33], well-posedness [13, 34, 35], and stability [36, 37, 38, 39].

Well-posedness plays an important role in both theory results and numer- ical methods. Many mathematicians have paid much attention on this topic (see e. In 1966, Tikhonov [44] introduced a definition of well-posedness for unconstrained optimization problems which is called Tikhonov well-posedness. This concept requires two conditions.

The first condition is the existence and uniqueness of the solution and the second one is the convergence of each minimizing sequence to the unique solution. On this topic, many results have been devoted to a lot of important problems such as variational inequalities [45], equilibrium problems [46], inclusion problems [47] and the references therein. Later on, generalizations of Tikhonov well-posedness were introduced and studied widely. In 1995, Loridan [48] considered the vector optimization problem and intro- duced a definition of well-posedness based on the convergence of a subsequence of a minimizing sequence.

One of extensions of Tikhonov well-posedness is the so-called B-well-posedness proposed by Bednarczuck for vector optimization problems in [49]. After that, B-well-posedness has been intensively considered for various problems re- lated to optimization [47, 50, 51, 52, 53]. An initial concept of the well-posedness by perturbations (also named extended well-posedness) for a optimization problem is proposed by Zolezzi [54, 55]. This concept established a form of continuous de- pendence of the solutions upon a parameter.

It should be noted that the fundamen- tal requirement in the Tikhonov well-posedness is that every minimizing sequence must lie in the feasible set. Levitin-Polyak [40] introduced a new notion of well- 4 posedness that strengthened Tikhonov’s concept as it required the convergence to the optimal solution of each sequence belonging to a larger set of minimizing se- quences. More precisely, the well-posedness in the sense of Levitin-Polyak allows the minimizing sequences can be outside of the feasible set and requires the distance of the minimizing sequence from the feasible region to approach zero. There is a large amount of works related to well-posedness for different important optimization problems [41, 42, 46, 48, 50, 56, 57, 58, 59, 60].

In the field of set optimization problems, the concept of well-posedness was firstly given by Zhang et al. In this publication, the authors intro- duced three kinds of well-posedness for set optimization problems involving lower set less relation. Using the scalarization method [62], they established relationships between the well-posedness of a set optimization problem and the well-posedness of scalar optimization problems. Moreover, the authors provided some criteria for set optimization problems to be well-posed.

Later, Gutiérrez et al. [34] extended some results of Zhang et al. Recently, Long and Peng [13] introduced three kinds of well-posedness in the sense of Bednarczuk, named B-well-posedness, for set optimization problems involving up- per set less relation and established some relationships among them. They also gave characterizations of well-posedness for these problems.

After that, Crespi et al. [63] introduced a definition of well-posedness which slightly generalizes the one in Zhang et al. [61], and investigated well-posedness properties of set optimization problems. Very recently, Khoshkhabar-Amiranloo and Khorram [64] studied pointwise Levitin- Polyak well-posedness for set optimization problems involving lower set less relation and obtained some criteria and characterizations of this issue.

In a word, studying on well-posedness for set optimization problems is a prospective topic, and hence it should be study more on different types of well-posedness for these problems involving various kinds of set order relations. This thesis focuses mainly on three types of well- posedness, namely L-well-posedness, B-well-posedness, LP well-posedness, involving 5 three kinds of set order relations introduced by Kuroiwa under the names lower set less relation, upper set less relation and set less relation. In a close relationship with well-posedness, the stability analysis of the solu- tion maps for optimization problems is an important direction in studying optimiza- tion problems. As we know, the study of stability concerns on how the solution set maps vary with the changes in the parameters.

Generally speaking, stability requires that for a small perturbations or a small measurement error of the parameters, the induced perturbations of solution set map is also very small. Investigating on sta- bility of optimization problems can be understood as studying on some properties of the solution maps such as the upper semicontinuity, lower semicontinuity, continuity, Hölder/ Lipschitz continuity or (generalized) differentiability [36, 37, 38, 65, 66]. In this research, we aim to give connections between well-posedness and stability for concerned problems in the sense of upper semicontinuity of approximating solution map. Solution existence is a crucial topic in studying any mathematics model.

In set optimization, existence results for problems involving lower and/or upper set less relation were presented in [67] under assumptions on compactness of the domain of objective map and reported in [24, 25] with conditions about the cone-regularity and cone-semicompactness. On the other hand, using nonlinear scalarization method, existence conditions of optimal solutions were established in [9, 68]. However, there is no publications mentioned conditions to guarantee existence conditions of solutions of set optimization problems involving set less relation. Therefore, our goals include giving solution existence conditions for these problems instead of just focusing on well- posedness properties by a assumption that solution set is empty as many publications in the literature.

Due to the important role of set optimization problems and the high vol- ume of active research in optimization, there is a wide prospect to do research in optimization theory. These works mentioned above motivate strongly us to study 6 well-posedness as well as stability and solution existence for such problems involving different kinds of set order relations. We aim to investigate set optimization prob- lems involving different kinds of set order relations. We focus on the set less relation because it has more appropriate applications in real life and furthermore it is a gener- alization form of both lower set less relation and upper set less relation.

To the best of our knowledge, there is no publication about well-posedness of set optimization problem with the set less relation. In the following we give a description of the contents of this thesis. This chapter includes some notations, definitions, and some useful results that will be deplicated in later chapters. Studying on L-well-posedness for set optimization problems with three types of set order relations is the main goal of this chapter.

Various kinds of well-posedness for these problems (pointwise L-well-posedness, global L-well- posedness, metrically α-well-posedness, weak metrically α-well-posedness) and their relationships are concerned. We give necessary and sufficient conditions for set opti- mization problems to be pointwise/global L-well-posed. Moreover, Kuratowski mea- sure of noncompactness is applied to survey characterizations of L-well-posedness for set optimization problems. Furthermore, approximating solution maps and their sta- bility are researched to propose links between stability of the approximating problem and well-posedness of the set optimization problem.

In addition to studying well- posedness, existence conditions of such problems are given by employing properties of a KKM-map. B-well-posedness for set optimization problems involving three kinds of set order relations is presented in this chapter. We give characterizations and sufficient and/or necessary conditions of B-well-posedness. Moreover, relationships between pointwise L-well-posedness and pointwise B-well-posedness are studied.

This chapter investigates LP well-posedness for set optimiza- tion problems involving three types of set order relations. Necessary and sufficient 7 conditions for the reference problems to be LP well-posed are given. Furthermore, using the Kuratowski measure of noncompactness, we study characterizations of LP well-posedness for set optimization problems. Moreover, links between stability and LP well-posedness of such problems are established via the study on approximating solution maps.

We also study metrically LP well-posedness and bridges between it and LP well-posedness. We give the concluding remarks. Some obtained results in this study are summarized. Moreover, we also give some comments about prospect remarks related to what we done to develop this research.