รายละเอียดโครงการวิจัย
- ชื่อโครงการ (ภาษาไทย) :
- การมีอยู่จริงและรูปแบบการประมาณค่าที่เหมาะสมของคำตอบสำหรับบางระบบของปัญหาอสมการการแปรผันบนวงศ์ของเซตที่ไม่คอนเวกซ์
- ชื่อโครงการ (ภาษาอังกฤษ) :
- The existence and global optimization schemes for solutions of some systems of variational inequalities problems on a class of nonconvex sets
- หน่วยงานเจ้าของโครงการ :
- คณะวิทยาศาสตร์และเทคโนโลยี
- ลักษณะโครงการวิจัย :
- โครงการวิจัยเดี่ยว
- ลักษณะย่อยโครงการวิจัย :
- ไม่อยู่ภายใต้แผนงานวิจัย/ชุดโครงการวิจัย
- ประเภทโครงการ :
- โครงการวิจัยใหม่
- วันเริ่มต้นโครงการ :
- 1 ธันวาคม 2557
- วันสิ้นสุดโครงการ :
- 30 พฤศจิกายน 2558
- ประเภทของการวิจัย :
- การวิจัยพื้นฐาน
- ความสำคัญและที่มาของปัญหา :
- Nowadays variational inequality theory, which was introduced by Stampacchia [G. Stampacchia, Formes bilineaires coercivities sur les ensembles convexes, C.R. Acad. Sci. Paris 258 (1964) 4413-4416.] in 1964, has emerged as a fascinating and interesting branch of mathematical and engineering sciences with a wide range of applications in industry, finance, economics, social, ecology, regional, pure and applied sciences. Roughly speaking, the ideas and techniques of the variational inequalities are being applied in a variety of diverse areas of sciences and proved to be productive and innovative. It has been shown that this theory provides a simple, natural and unified framework for a general treatment of unrelated problems. These activities have motived to generalize and extend the variational inequalities and related optimization problems in several directions using new and novel techniques. Especially, in 1985, Pang [J.-S. Pang, Asymmetric variational inequalities over product of sets: applications and iterative methods, Math. Prog. 31 (1985) 206–219.] showed that a variety of equilibrium models, for example, the traffic equilibrium problem, the spatial equilibrium problem, the Nash equilibrium problem and the general equilibrium programming problem can be uniformly modelled as a variational inequality defined on the product sets. He decomposed the original variational inequality into a system of variational inequalities and discussed the convergence of method of decomposition for system of variational inequalities. Later, it was noticed that variational inequality over product sets and the system of variational inequalities both are equivalent. Since then many authors studied the existence theory of various classes of system of variational inequalities, by exploiting fixed-point theorems and minimax theorems. On the other hand, in 1965, Zadeh [L.A. Zadeh, Fuzzy sets, Inform and Control 8 (1965) 338-353.] introduced the concept of fuzzy sets as an extension of crisp sets, the usual two-valued sets in ordinary set theory, by enlarging the truth value set to the real unit interval . Ordinary fuzzy sets are characterized by, and mostly identified with, mapping called “membership function” into . The basic operations and properties of fuzzy sets or fuzzy relations are defined by equations or inequalities between the membership functions. The applications of fuzzy set theory can be found in many branches of mathematical and engineering sciences including artificial intelligence, computer science, control engineering, management science and operations research. In 1989, motivated and inspired by recent research work in these two fields, Chang [S. Chang, Y. Zhu, On variational inequalities for fuzzy mappings, Fuzzy Sets and Systems 32 (1989) 359-367.] first introduced the concepts of variational inequalities on fuzzy set. Since then, due to the significant of this fuzzy type that appear to play an important role in many areas, several classes of variational inequalities of fuzzy type were considered and constantly developed. However, we would like to point out that almost all the results regarding the existence and iterative schemes for solving those (fuzzy) (system) variational inequalities and related optimizations problems are being consider in the convexity setting. In fact, this is because, the most of results are based on the properties of the projection operator over convex sets, which may not hold when the sets are nonconvex. Notice that the convexity assumption, made by researchers, has been used for guaranteeing the well definedness of the proposed iterative scheme which depends on the projection mapping. Actually, the convexity assumption may not require for the well definedness of the projection mapping because it may be well defined, even in the nonconvex case(e.g., when the considered set is a closed subset of a finite dimensional space or a compact subset of a Hilbert space, etc.). In this project, motivated by above observations, we will mainly focus our study to the system of variational inequalities by using the concept of fuzzy set theory on nonconvex sets. We noticed that, since the variational inequality problem usually a reformulation of some minimization problem of some functional over convex sets, it does not make sense to generalize the variational inequality problem by just replacing the convex sets by nonconvex ones. Also, as we mentioned-above, a straightforward generalization to the nonconvex case of the techniques used when set is convex cannot be done. For these reasons, we have some plans to make use of some recent techniques and ideas from nonsmooth analysis to overcome the difficulties that arise from the nonconvexity of the set Moreover, for the purpose of maximum benefit from this project, we should consider at least a class of nonconvex sets that properly includes the class of convex sets, such as uniformly prox-regular sets. In this point of view, together with the significant of both the variational inequality theory and fuzzy theory, our results will be very useful for updating all recent results and provide more choices of tool implements for the better applications. Of course, the results obtianed in this project are new, generalize, improve, and unify a number of recent results.
- วัตถุประสงค์ของโครงการ :
- 1. We will introduce the form of the systems of (fuzzy) variational inequality problem for nonconvex sets in Hilbert spaces. 2. The existence theorems and global optimization schemes will be provided for this new problem in the setting of uniformly prox-regular subset of Hilbert spaces. 3. We will consider the existence theorems and global optimization schemes of some generalization forms of such new systems of (fuzzy) variational inequality problem. 4. We intend to check the potentials of our iterative schemes, that will be presented corresponding to each of the problem, by using Mathematic Programs.
- ขอบเขตของโครงการ :
- Introduce and consider the systems of (fuzzy) variational inequality problem and its various forms for nonconvex subset of Hilbert spaces. Provide existence theorems and consider the global optimization for the solutions of those introduced problems on the uniformly prox-regular subset of Hilbert spaces, and check the potentials of the proposed iterative schemes by using Mathematic Programs.
- ผลที่คาดว่าจะได้รับ :
- วิธีการดำเนินการวิจัย และสถานที่ทำการทดลอง/เก็บข้อมูล :
- คำอธิบายโครงการวิจัย (อย่างย่อ) :
- -
- รายชื่อนักวิจัยในโครงการ :
-
ลำดับที่ รายชื่อ ประเภทนักวิจัย บทบาทหน้าที่ สัดส่วน 1 นายสมบูรณ์ นิยม นักวิจัยภายในมหาวิทยาลัย หัวหน้าโครงการวิจัย 50% 2 นายนรินทร์ เพชร์โรจน์ นักวิจัยภายนอกมหาวิทยาลัย ผู้ร่วมวิจัย 50%
สถาบันวิจัยและพัฒนา
มหาวิทยาลัยราชภัฏนครสวรรค์ ศูนย์การศึกษาย่านมัทรี398/1 หมู่ 3 ตำบลย่านมัทรี อำเภอพยุหะคีรี
จังหวัดนครสวรรค์ 60130
