Well posedness for set optimization problems involving set order relations
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Well posedness for set optimization problems involving set order relations
CHAPTER IINTRODUCTIONSet-valued optimization plays an important role in optimization and has attracted a great deal of attent ion of mathematicians in Well posedness for set optimization problems involving set order relations n recent years. It deals wit h optimization problems which the objective map and/or the const raint 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 Well posedness for set optimization problems involving set order relations scalar optimization as well as vector optimization.It is worth noting that there are two approaches to formulate optimality notions for set-valued optWell posedness for set optimization problems involving set order relations
imization problems, namely t he vector approach and the set approach. These criteria depend on the way that the notion of minimality is defined. In thCHAPTER IINTRODUCTIONSet-valued optimization plays an important role in optimization and has attracted a great deal of attent ion of mathematicians in Well posedness for set optimization problems involving set order relations . 4] and the references therein). Hence, the set-valued optimization problem is called setvalued 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 Well posedness for set optimization problems involving set order relations et al. [r>] in 1997, ami originally studied independent ly by Young [6] and Nishnianidzc [7]. By set approach one compares all images of the set-valueWell posedness for set optimization problems involving set order relations
d objective map and hence in (his approach the set-valued opt imizat ion problem is called set optimization problem (sece.g.,[8, 9, 10, 11. 12] and thCHAPTER IINTRODUCTIONSet-valued optimization plays an important role in optimization and has attracted a great deal of attent ion of mathematicians in Well posedness for set optimization problems involving set order relations objective map is a vector single-valued map, (hey reduce to the known concept, of efficiency. In this t hesis, set approach Is applied and set order relat ions arc used to define optimality notions for set-valued optimization problems.We would like to give a brief review of set order relat ions. The Well posedness for set optimization problems involving set order relations int roduct ion of set order relations was presented by Kuroiwact al. |5], by Young [6] and Nishnianidze2[7]. Knroiwa [11’ showed six relations amongWell posedness for set optimization problems involving set order relations
sets including lower set. less relation, upper set less relat ion and set less relat ion (combinat ion of t he lower and t he upper set less relation)CHAPTER IINTRODUCTIONSet-valued optimization plays an important role in optimization and has attracted a great deal of attent ion of mathematicians in Well posedness for set optimization problems involving set order relations ate 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 relat ion and lipper set less relat Well posedness for set optimization problems involving set order relations ion were studied in some publical ions [13, 14] and the references t herein. To t he best, of our knowledge, there has been no publication about well-Well posedness for set optimization problems involving set order relations
posedness, solution existence condit ion and stability for optimization problems involving the set less relation and new types of set order relations CHAPTER IINTRODUCTIONSet-valued optimization plays an important role in optimization and has attracted a great deal of attent ion of mathematicians in Well posedness for set optimization problems involving set order relations han vector approach whenever wo need IO consider preferences over sets. Set optimization problems play important roles and useful applications in the practical situations. Many important and significant applications of set order relations wore studied and discussed. The possibly less order relation Well posedness for set optimization problems involving set order relations has been applied to interval arithmetic and Fortran compiler Í95 of Sun Microsystems, order relat ions was presented by Neukel 15] in the project inveWell posedness for set optimization problems involving set order relations
stigating relationship between noise dist urbance and quality of life in the region surrounding the FYankfurt Airport in Germany. Another application CHAPTER IINTRODUCTIONSet-valued optimization plays an important role in optimization and has attracted a great deal of attent ion of mathematicians in Well posedness for set optimization problems involving set order relations et order relation can 1)0 found and applied in daily activities, for instance, comparing teams of football players, ranking group of students on the same academic level,... For furt her reading and references, wo refer to studies [13, 14, 17, 18, 19, 20. 21. 22] and the reference therein.With the fi Well posedness for set optimization problems involving set order relations rst introduction was given two decades ago, set optimization can be seen as a very young direction in the field of optimization. However, in the role3Well posedness for set optimization problems involving set order relations
of an important and prospective issuso, it has attracted a groat, deal of attention of mathematicians. As pointed out by Jahn in [23], the set optimiCHAPTER IINTRODUCTIONSet-valued optimization plays an important role in optimization and has attracted a great deal of attent ion of mathematicians in Well posedness for set optimization problems involving set order relations ch as solution existence [24, 25. 26). optimality conditions [27, 28. 29], nonlinear scalaxization [9, 30. 31], Ekeland variational principles [8, 32], duality theory [33], well-posedness ‘13, 34, 35], and stability [36, 37, 38. 39].Well-posedncss plays an important role in both theory results and n Well posedness for set optimization problems involving set order relations umerical methods. Many mathematicians have paid much attention on this topic (see e.g., [40. 41. 42, 43] and the reference therein). In 1966, TikhonovWell posedness for set optimization problems involving set order relations
[44] introduced a definition of woH-posedness for unconst .rained optimization problems which is called Tikhonov well-posedness. This concept requireCHAPTER IINTRODUCTIONSet-valued optimization plays an important role in optimization and has attracted a great deal of attent ion of mathematicians in Well posedness for set optimization problems involving set order relations quence to the unique solution. On this topic, many results have been devoted to a lot of important problems such as variational inequalities [15]. equilibrium problems [46], inclusion problems [17] and the references therein. Later on. generalizations of Tikhonov well-posedness were introduced and s Well posedness for set optimization problems involving set order relations tudied widely. In 1995, Loridan 48] considered the vector optimization problem and introduced a definition of well-posed noss based on the convergenceWell posedness for set optimization problems involving set order relations
of a subsequence of a minimizing sequence. One of extensions of 1'ikhonov well-posedness is the so-called R-uudl-posedness proposed by Bcelnarczuck fCHAPTER IINTRODUCTIONSet-valued optimization plays an important role in optimization and has attracted a great deal of attent ion of mathematicians in Well posedness for set optimization problems involving set order relations 47, 50, 51. 52, 53]. An initial concept of the Uícil-pơsedne..Well posedness for set optimization problems involving set order relations
uced a now not ion of well-4posedness that strengthened Tikhonov's concept as it required the convergence to the optimal solution of each sequence belCHAPTER IINTRODUCTIONSet-valued optimization plays an important role in optimization and has attracted a great deal of attent ion of mathematicians in Well posedness for set optimization problems involving set order relations n be outside of t he 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. opt Well posedness for set optimization problems involving set order relations imization problems, the concept of well-posedncss was firstly given by Zhang el al. [61] in 2009. In this publication, the authors introduced three kiWell posedness for set optimization problems involving set order relations
nds of well-posedncss for set optimization problems involving lower set less relation. Using the scalarization method [62], they established relationsCHAPTER IINTRODUCTIONSet-valued optimization plays an important role in optimization and has attracted a great deal of attent ion of mathematicians in Well posedness for set optimization problems involving set order relations some criteria for set optimization problems to be well-posed. Lalor, Gutierrez et al. [34] extended some results of Zhang et al. [61] by using another form of the scalarization function due to [9]. Recently, Long ami Peng [13] introduced three kinds of well-posedness in t he sense of Bednarczuk, nam Well posedness for set optimization problems involving set order relations ed B-well-poscdness. for set optimization problems involving upper set less relation and established some relationships among them. They also gave chaWell posedness for set optimization problems involving set order relations
racterizations of well-ị rosed ness for these problems. After that, Crespi Ct al. [63] introduced a definition of wcll-posedness which slightly generaCHAPTER IINTRODUCTIONSet-valued optimization plays an important role in optimization and has attracted a great deal of attent ion of mathematicians in Well posedness for set optimization problems involving set order relations i 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-posedncss for set optimization problems is a prospective topic, and hence it shou Well posedness for set optimization problems involving set order relations ld lx* study more on different types of well-posedness for these problems involving various kinds of set order relations. This t hesis focuses mainlyWell posedness for set optimization problems involving set order relations
on three types of well-posedness. namely L-well-poscdness, B-well-poscdness. LP well-posedness. involvinghttps://khothuvien.cori!5three kinds of set oGọi ngay
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