| รหัสโครงการ : | R000000166 |
| 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 |
| คำสำคัญของโครงการ(Keyword) : | Variational inequality problem; uniformly prox-regular sets; optimization; fuzzy mapping; nonconvex set |
| หน่วยงานเจ้าของโครงการ : | คณะวิทยาศาสตร์และเทคโนโลยี > ภาควิชาวิทยาศาสตร์ สาขาวิชาคณิตศาสตร์และสถิติ |
| ลักษณะโครงการวิจัย : | โครงการวิจัยเดี่ยว |
| ลักษณะย่อยโครงการวิจัย : | ไม่อยู่ภายใต้แผนงานวิจัย/ชุดโครงการวิจัย |
| ประเภทโครงการ : | โครงการวิจัยใหม่ |
| สถานะของโครงการ : | propersal |
| งบประมาณที่เสนอขอ : | 325000 |
| งบประมาณทั้งโครงการ : | 325,000.00 บาท |
| วันเริ่มต้นโครงการ : | 01 ธันวาคม 2557 |
| วันสิ้นสุดโครงการ : | 30 พฤศจิกายน 2558 |
| ประเภทของโครงการ : | การวิจัยพื้นฐาน |
| กลุ่มสาขาวิชาการ : | วิศวกรรมศาสตร์และเทคโนโลยี |
| สาขาวิชาการ : | สาขาวิทยาศาสตร์กายภาพและคณิตสาศตร์ |
| กลุ่มวิชาการ : | คณิตศาสตร์ |
| ลักษณะโครงการวิจัย : | ระดับชาติ |
| สะท้อนถึงการใช้ความรู้เชิงอัตลักษณ์ : | สะท้อนถึงการใช้ความรู้เชิงอัตลักษณ์ |
| สร้างความร่วมมือประหว่างประเทศ GMS : | ไม่สร้างความร่วมมือทางการวิจัยระหว่างประเทศ |
| นำไปใช้ในการพัฒนาคุณภาพการศึกษา : | นำไปใช้ประโยชน์ในการพัฒนาณภาพการศึกษา |
| เกิดจากความร่วมมือกับภาคการผลิต : | ไม่เกิดจากความร่วมมือกับภาคการผลิต |
| 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.
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| จุดเด่นของโครงการ : | - |
| 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.
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| 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 : | - |
| การทบทวนวรรณกรรม/สารสนเทศ : | 9.1 The classical variational inequality problem and its generalizations
Let be a real Hilbert space whose inner product and norm are denoted by and , respectively. Let be a nonempty closed convex set in
9.1.1 The classical variational inequality problem
Let be a nonlinear mapping. The variational inequality is formulated as finding a point such that
----------------- (VI)
The (classical) variational inequality (VI) was initially studied by Stampacchia [32] and ever since have been widely studied. Such a problem is connected with the convex minimization problem, the complementarity problem, the problem of finding a point satisfying and so on. Also, its theory combines novel theoretical and algorithmic advances with new domains of applications. As a result of interaction between different branches of mathematical and engineering sciences, we now have a variety of iterative algorithms including the projection methods and its variant forms, the Wiener–Hopf equations techniques, auxiliary principle technique and decomposition methods for solving variational inequalities and related optimization problems, see, for example, [5, 7-9, 11-14, 31] and the references therein. Also, it is well known that variational inequalities are equivalent to the fixed point. This alternative technique has played an important and vital role in developing several iterative schemes for solving (general) variational inequalities.
9.1.2 Generalized variational inequality problem
9.1.2.1 Noor variational inequalities
In 1988, Noor [16] presented the useful generalization of the variational inequality which is known as Noor general (single-valued) variational inequality problem, as following: Let and be nonlinear mappings. The Noor variational inequality is formulated as finding a point such that
----------------- (NVI)
It turned out that odd-order, nonsymmetric obstacle, free, unilateral, nonlinear equilibrium, and moving boundary problems arising in various branches of pure and applied sciences can be studied via the generalized variational inequalities in this sense, see [29].
It is easy to see that For where is the identity operator, problem (NVI) is equivalent to (VI). Moreover, if is a polar (dual) cone of a convex cone in then
problem (NVI) is equivalent to finding such that
and ------------- (GCP)
which is known as the general complementarity problem. For where is a point-to point mapping, problem (GCP) is called the implicit (quasi-)complementarity problem. If then problem (GCP) is known as the generalized complementarity problem. Such problems have been studied extensively in the literature. In fact, for suitable and appropriate choice of the operators and spaces, one can obtain several classes of variational inequalities and related optimization problems.
In 2007, by using some fixed point formulation, Noor [27] suggested the following three-step iterative method: for a given compute the approximate solution by the iterative schemes
------------------- (1)
where for all and is an operator. The algorithm (1) is a three-step predictor–corrector method. He used algorithm (1) for finding the common elements of the set of the solutions of the Noor variational inequalities problems and the set of the fixed points of nonexpansive mappings. He also considered the convergence analysis of this suggested iterative schemes under some mild conditions. Further, he showed that the suggested algorithm recovers many algorithms suggested by many authors. Moreover, since the Noor variational inequalities include variational inequalities and complementarity problems as special cases, results obtained in this paper continue to hold for these problems. For more information, one may see [20-21, 23-26, 28].
9.1.2.2 Multivalued (general) variational inequalities problems
Let be the family of all nonempty subsets of Let be a multivalued operator, and let be a single-valued operator. Let be a nonempty, closed, and convex set in H. We consider the problem of finding such that
------------------ (MGVIP)
An inequality of type (MVIP) is called a general multivalued general variational inequality. Notice that, if is a single-valued operator, then problem (MVIP) is equivalent to the problem (NVI). If the identity operator, then problem (MVIP) is equivalent to finding such that
------------------(MVIP)
which is the multivalued variational inequality introduced and studied by Fang and Peterson [6]. For the applications, numerical methods, and formulations, see [6, 18], and references therein.
In [22], the author used the auxiliary principle technique of Glowinski, Lions, and Tremolieres [10] to suggest a new class of predictor-corrector algorithms for solving the problem (MGVIP). In fact, he considered the following problem: for a given consider the problem of finding a unique satisfying the auxiliary variational inequality
where is a constant. By using an observation, that is, if then clearly is a solution of the problem (MGVIP), he suggested the following predictor–corrector methods for solving the general multivalued variational inequality (MGVIP): for a given compute the approximate solution by the iterative scheme
--------------- (2)
where are constants and is the Hausdorff metric on He showed that convergence of the proposed methods (2), under some mild conditions. Moreover, in such the presented paper, it also pointed that the algorithm (2) recovers a useful iterative method, so-called a three-step forward-backward splitting method. Consequently, his presented results represent an improvement and refinement of the previously known results and can be considered as an extension of solving general variational inequalities and complementarity problems.
9.1.2.3 Fuzzy variational inequalities problems
In what follows, we denote the collection of all fuzzy sets on by that is, For and the set
Is called a -cut set of A mapping from to is called a fuzzy mapping. If is a fuzzy mapping, then the set for is a fuzzy set in (in the sequel we denote by ) and is the degree of membership of in
In [4], Chang et. al., introduced the concept of fuzzy generalized set-valued mixed variational-like inequality problem as the following: Let be a real Hilbert space, be a nonempty closed convex subset of Let be two fuzzy mappings and are functions. For given nonlinear mappings we consider the problem of finding and such that
------------- (FVLVIP)
where is a function. The problem (FVLVIP) is called fuzzy generalized set-valued mixed variational-like inequality problem. In such a paper, by using the general auxiliary principle technique, Ky Fan’s KKM theorem, Nadler’s fixed point theorem and some new analytic techniques, they proved some existence theorems and some iterative approximation schemes for this kind of fuzzy variational-like inequalities in Hilbert spaces setting.
It is important to see that, the problem of type (FVLVIP) is very useful and general setting one. To see this, let us consider some special cases of the problem (FVLVIP).
(I) Let be two ordinary multivalued mappings to a closed bounded subset of and , are given mappings. Now we define two fuzzy mappings as follows:
where and are the characteristic functions of the sets and respectively. If are constant functions on which are defined by and for all respectively. Then
Thus, the problem (FVLVIP) is equivalent to finding and such that
-------------(SNMVIP)
This kind of problem is called the set-valued strongly nonlinear mixed variational-like inequality and was introduced and studied by Noor [17] under the additional condition that is compact-valued. It is also considered by Zeng [34].
(II) If and and are single valued, then problem (SNMVIP) is equivalent to finding a such that
---------------(MVIP)
This is called the mixed variational-like inequality problem and was studied by Ansari and Yao [1].
(III) If and then problem (FVLVIP) is equivalent to finding such that
--------------------(VLIP)
This is also a class of special fuzzy variational-like inequalities. The case of ordinary set valued mappings (i.e., in the nonfuzzy case) was considered by Noor [19].
9.1.2.4 System of variational inequilities problems
Let be a real Hilbert space, and be nonlinear operator. Let denote the subdifferential of function where is a proper convex lower semicontinuous function on We consider the following type of variational inequality problem: for fix positive real numbers , find such that
---------------------(SMVIP)
This problem is called system of mixed variational inequality problem. The approximate solvability of problem (SMVIP), by using the technique so-called resolvent operator technique, has been considered. It is should be pointed out that, the results presented in this paper are more general and include many previously known results as special cases. Some of them are as followings:
(I) If is closed convex set in and for all where is the indicator function of defined by if and otherwise. then the problem (SMVIP) is equivalent to find such that
----------------(SVIP)
The problem (SVIP) have been studied by Chang et al. [3].
(II) If is an univariate mappings, then (SMVIP) problem reduces to finding such that
This problem was studied by Noor [30].
(III) If is an univariate mappings, then (SVIP) problem reduces to finding such that
This problem was studied by Verma [33].
Moreover, it is easy to see that, the classical variational inequality, introduced and studied by Stampacchia [32] in 1964, is a particular case of the problem (SMVIP). This shows that the problem (SMVIP) is more general and includes several classes of variational inequalities and related optimization problems as special cases.
9.2 The system of variational inequilities problems on nonconvex sets
Let be a real Hilbert space whose inner product and norm are denoted by and , respectively. Let be a nonempty closed set in . In 2009, Moudafi [15] considered the following important problem: find such that
-------------(SNVI)
Where is a mapping on a closed set of a Hilbert space and is the proximal normal cone to at . Moudafi considered the problem (SNVI) on a class of closed set, as uniformly prox-regular sets. It is worth to point out that, this class is sufficiently large to include the class of convex sets, -convex sets, submanifolds (possibly with boundary) of the images under a diffeomorphism of convex sets and many other nonconvex sets; see [2]. Observe that, if is a closed convex set then the problem (SNVI) is reduced to
which is nothing but the system of variational inequality problem introduced by Verma [33]. Notice that, the methods proposed by Verma contain several known as well as new projection schemes as special cases, while some have been applied to problems arising, especially from complementarity problems, convex quadratic programming and other variational problems
It is well known that, the two-step models for nonlinear variational inequalities are relatively more challenging than the usual variational inequalities. However, until now, there are only few researches considered the system of variational inequalities problems on nonconvex setting. Meanwhile, both on existences of solutions of problems and the construction of solutions, a tremendous amount of research on fuzzy variational inequalities problems has been done in the case of convex sets. Thus, in this project, we intend to consider fuzzy variational inequality and system of variational inequality on nonconvex setting. Of course, as far as we know, knowledge that will be obtained by this project are new, and consequently, due to the significant of both the system variational inequality theory and fuzzy theory, our obtained results will be very useful and provide the better applications.
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| ทฤษฎี สมมุติฐาน กรอบแนวความคิด : | Under some suitable conditions, we can control the existence of solution of variational inequality problems and by using some scalars control conditions we can construct an algorithm for finding such the existence solutions. |
| วิธีการดำเนินการวิจัย และสถานที่ทำการทดลอง/เก็บข้อมูล : | - |
| คำอธิบายโครงการวิจัย (อย่างย่อ) : | - |
| จำนวนเข้าชมโครงการ : | 411 ครั้ง |
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