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In this paper, we establish the regularized gap function for a class of mixed parametric vector quasiequilibrium problems. Some examples are given to illustrate our results. Our main results extend and differ from those corresponding ones in the current literatures.
Dong Thap University Journal of Science, Vol 11, No 5, 2022, 03-08 ERROR BOUNDS FOR A CLASS OF MIXED PARAMETRIC VECTOR QUASI-EQUILIBRIUM PROBLEMS Nguyen Huynh Vu Duy1, Nguyen Ngoc Hien2, and Vo Minh Tam3* Office of Academic Affairs, Ho Chi Minh City Open University Academic Affairs Office, Dong Thap University Faculty of Mathematics - Informatics Teacher Education, Dong Thap University *Corresponding author: vmatm@dthu.edu.vn Article history Received:11/06/2021; Received in revised form: 02/12/2021; Accepted: 09/12/2021 Abstract In this paper, we establish the regularized gap function for a class of mixed parametric vector quasiequilibrium problems (briefly, (MPVQEP) ) Then an error bound is also provided for (MPVQEP) via this gap function under suitable assumptions Some examples are given to illustrate our results Our main results extend and differ from those corresponding ones in the current literatures Keywords: Error bound, mixed parametric vector quasi-equilibrium problems, regularized gap function, strongly monotone CẬN SAI SỐ CHO MỘT LỚP BÀI TOÁN TỰA CÂN BẰNG VÉCTƠ THAM SỐ HỖN HỢP Nguyễn Huỳnh Vũ Duy1, Nguyễn Ngọc Hiền2 Võ Minh Tâm3* Phòng Quản lý Đào tạo, Trường Đại học Mở Thành phố Hồ Chí Minh Phòng Đào tạo, Trường Đại học Đồng Tháp Khoa Sư phạm Toán - Tin, Trường Đại học Đồng Tháp *Tác giả liên hệ: vmtam@dthu.edu.vn Lịch sử báo Ngày nhận: 11/06/2021; Ngày nhận chỉnh sửa: 02/12/2021; Ngày duyệt đăng: 09/12/2021 Tóm tắt Trong báo này, chúng tơi thiết lập hàm gap chỉnh hóa cho lớp tốn tựa cân véctơ tham số hỗn hợp (viết tắt (MPVQEP) ) Khi đó, cận sai số thu cho tốn (MPVQEP) thơng qua hàm gap chỉnh hóa xem xét số giả thiết phù hợp Một số ví dụ đưa để mô tả kết đạt Các kết chúng tơi báo mở rộng khác với kết tương ứng nghiên cứu cơng trình gần Từ khóa: Cận sai số, tốn tựa cân véctơ tham số hỗn hợp, hàm gap chỉnh hóa, đơn điệu mạnh DOI: https://doi.org/10.52714/dthu.11.5.2022.974 Cite: Nguyen Huynh Vu Duy, Nguyen Ngoc Hien, and Vo Minh Tam (2022) Error bounds for a class of mixed parametric vector quasi-equilibrium problems Dong Thap University Journal of Science, 11(5), 3-8 Natural Sciences issue Introduction and preliminaries In 1997, Yamashita and Fukushima introduced a class of merit functions for variational inequality problems: ( x, ) sup{ h( x), x y ‖ x y‖ }, yK H ( x, y) T ( , x), ( y, x) int m , y K ( x, ) (1) Given S ( ) the solution set of (MPVQEP) we always assume that S ( ) for all To illustrate motivations for this setting, we provide some special cases of the problem (MPVQEP) : where is a nonnegative parameter, (, ) : n {} , K n , h : n n This function was first introduced by Auslender (1976) for and by Fukushima (1992) for The function (,0) is called the gap function, while the function (, ) is called the regularized gap function, with One of the many useful applications of gap functions is to derive the so-called error bounds as an upper estimation of the distance between the solution set and an arbitrary feasible point Since then, many authors investigated the regularized gap functions and error bounds for various kinds of optimization problems, variational inequality problems and equilibrium problems (see, for example, Anh et al (2018), Bigi and Passacantando (2016), Gupta and Mehra (2012), Hung et al (2020a, 2020b, 2021), Khan and Chen (2015a, 2015b), Mastroeni (2003) and the references therein) (a) If m , K ( x, ) A, H1 , T1 ( , x) T1 ( x) ( y, x) y x, , x, y A, then (MPVQEP) reduces to the following variational n Throughout this paper, let be the ndimensional Euclidean space with the inner product , and norm‖ ‖ , respectively provide some examples to support the results presented in this paper Our main results extend and differ from those corresponding ones in the current literatures Let m {( y1 , , ym ) m : yi 0, i 1,2, , m} be the nonnegative orthant of m , A n be a nonempty, closed and convex set in n and be nonempty subsets of a finite dimensional space i {1,2, , m}, let Ti : A n , For each Hi : A A be continuous bifunctions such that Hi ( x, x) for all x A and : A A n be a continuous bifunction such that ( x, y) ( y, x) n for all x, y A Let K : A A , H : ( H1 , H , , H m ), T : (T1 , T2 , , Tm ) and for any x, n , T ( , x), ) : (T1 ( , x), , T2 ( , x), , , Tm ( , x), ) We now consider the following generalized parametric vector quasi-equilibrium problem (briefly, (MPVQEP) ) in finding x K ( x, ) for each parameter fixed such that inequality problem (briefly, (VIP)) studied in Yamashita and Fukushima (1997) of finding x A such that T1 ( x), y x 0, y A (b) If m 1, 0, K ( x, ) K ( x) , , x A then the problem (MPVQEP) reduces to the following abstract quasiequilibrium problem (briefly, (QEP)) studied in Bigi and Passacantando (2016) of finding x K ( x) such that H1 ( x, y) 0, y K ( x) In this paper, we study regularized gap functions and error bounds for the problem (MPVQEP) under suitable assumptions We also We recall some notations and definitions used in the sequel Definition (See Rockafellar and Wets (1998)) A real valued function f : A is said to be convex if f ( x (1 ) y) f ( x) (1 ) f ( y) for every x, y A and [0,1] Definition Let f : A A and : A A Then T : A n, n be functions (i) (See Mastroeni (2003)) f is said to be strongly monotone with modulus if, for each ( x, y) A A , f ( x, y) f ( y, x) ‖ x y‖ ; Dong Thap University Journal of Science, Vol 11, No 5, 2022, 03-08 (ii) T is said to be strongly monotone with modulus if, for each , ( x, y) A A, T ( , y) T ( , x), ( y, x) ‖ x y‖ Theorem Assume that (i) K has compact and convex values; (ii) H i is convex in the second component for all i 1,2, , m ; (iii) for each t n and x A , the function t , ( y, x) is convex Regularized gap functions and error bounds for (MPVQEP) y In the section, we propose the regularized gap function and error bound for MPVQEP Then, for each and , the function defined by (2) is a gap function for (MPVQEP) Motivated by Auslender (1976), Bigi and Passacantando (2016), we consider the following definition of gap functions Let Proof (a) For each fixed, it is clear that for any x ( ) , i.e., x K ( x, ) and so ( ) x A : x K ( x, ) , ( , x) max h( , x, y ) ‖ x y‖ yK ( x , ) h( , x, x) and we assume that ( ) , Definition A real valued function is said to be a gap function of p: n A (MPVQEP) if it satisfies the following conditions: for each fixed, (a) p( , x) 0, for all x ( ) (b) for any x0 ( ) , p( , x0 ) if and only if x0 is a solution of (MPVQEP) Inspired by the approaches of Yamashita and Fukushima (1997), we construct a regularized gap function for (MPVQEP) Suppose that K ( x, ) is a compact set for each x A and , then for each and fixed, we consider a function : A defined by ( , x) max h( , x, y) ‖ x y‖ yK ( x , ) , (2) where h( , x, y) H i ( x, y) Ti ( , x), ( x, y) 1i m Remark The function in (2) is welldefined Indeed, as H i , Ti and are continuous for any i 1,2, , m , the function h is continuous Combine the continuity of h, ‖ ‖ and the compactness of K ( x, ) for each x A and , we have is well-defined We show that is a gap function for (MPVQEP) under suitable conditions (3) We have h( , x, x) H i ( x, x) Ti ( , x), ( x, x) 1i m Then, from (3), we conclude that ( , x) for any x ( ) (b) If there exists x0 ( ) , i.e., x0 K ( x0 , ) such that ( , x0 ) , then h( , x0 , y) ‖ x0 y‖ 0, y K ( x0 , ) or min{ H i ( x0 , y ) Ti ( , x0 ), ( x0 , y )} 1i m ‖ x0 y‖ , y K ( x0 , ) For arbitrary x K ( x0 , ) and (0,1) , let y x ( x0 x) Since K ( x0 , ) is a convex set, we get y K ( x0 , ) and so H i ( x0 , y ) Ti ( , x0 ), ( x0 , y ) 1i m ‖ x0 y‖ (4) Since H i is convex in the second component for all i 1,2, , m , we have H i ( x0 , y ) H i ( x0 , x0 ) (1 ) H i ( x0 , x) (1 ) H i ( x0 , x) (5) It follows from condition (iii) that Ti ( , x0 ), ( y , x0 ) (1 )Ti ( , x0 ), ( x, x0 ) Ti ( , x0 ), ( x0 , x0 ) Natural Sciences issue Since ( x, y) ( y, x) we have ( x0 , x0 ) n n for all x, y A, and so Ti ( , x0 ), ( x0 , y ) (1 )Ti ( , x0 ), ( x0 , x) (6) We have ‖ x0 y‖ x0 x ( x0 x) = (1 ) ‖ x0 x‖ 2 (7) (1 ) Hi ( x0 , x) (1 )Ti ( , x0 ), ( x0 , x) (1 )2 ‖ x0 x‖ (1 ) H i ( x0 , x) Ti ( , x0 ), ( x0 , x) 1i m (1 ) ‖ x0 x‖ for all y [0,1 x] It follows from some direct computations that ( ) {0} Indeed, ( , x) max h( , x, y ) ‖ x y‖ So, yK ( x , ) min{ H i ( x0 , x) Ti ( , x0 ), ( x0 , x)} max min{( x y)((3 ) x y), 1i m (1 )‖ x0 x‖ It is clear that all assumptions imposed in Theorem are satisfied Hence, the function defined by (2) is the gap function for (MPVQEP) Equivalently, y[0,1 x ] ( x y)((8 2 ) x y)} ( x y) (8) Taking the limit in (8) as 1 , we obtain H i ( x0 , x) Ti ( , x0 ), ( x0 , x) 1i m x K ( x0 , ) , there exits (2 ) x Next, we investigate the error bound for (MPVQEP) via the gap function For each i 1, , m , we now consider the Hi0 ( x0 , x) Ti0 ( , x0 ), ( x, x0 ) 0, following problem GPVQEP : find x K ( x, ) (i ) for each parameter fixed such that that is, m , x K ( x0 , ) Hence, x0 S ( ) Conversely, if x0 S ( ), then there exists i0 {1, , m} such that Hi0 ( x0 , y) Ti0 ( , x0 ), ( y, x0 ) 0, y K ( x0 , ) This means that H i ( x0 , y ) Ti ( , x0 ), ( x0 , y ) ‖ x0 y‖ 0, 1i m y K ( x0 , ) or max H i ( x0 , y ) Ti ( , x0 ), ( x0 , y ) ‖ x0 y‖ yK ( x0 , ) 1i m Hence, ( , x0 ) Since ( , x) for any x K ( x, ) , ( , x0 ) H ( x, y ) T ( , x), ( y, x) (( y x)((3 ) x y),( y x)((8 2 ) x y)) -int 1i m H ( x0 , x) T ( , x0 ), ( x, x0 ) int H ( x) y xy 8x and ( y, x) y x for all x, y A, Then, the problem (MPVQEP) is equivalent to finding x [0,1 x] [0,1] such that From (4)-(7), we get that Then, for any i0 m such that Example Let n 1, m , A [0,1], , 1, , K ( x, ) [0,1 x], T1 ( , x) x, T2 ( , x) 2 x, H1 ( x, y) y xy 3x , Hi ( x, y) Ti ( , x), ( y, x) 0, y K ( x, ) S (i ) ( ) Given GPVQEP (i ) the solution set of Remark If x0 im1 S (i ) ( ), then x0 is the same solution of GPVQEP for all i 1, , m (i ) Thus, it is clear that x0 is a solution of the problem (MPVQEP) Theorem For each , let x0 be a solution of the problem (MPVQEP) Suppose that all the conditions of Theorem hold and for each with i 1,2, , m , H i is strongly monotone modulus i and Ti is -strongly monotone with modulus i Let min1i m i and Dong Thap University Journal of Science, Vol 11, No 5, 2022, 03-08 min1i m i Assume further that m (i ) i 1 S ( ) , x0 K ( x, ) for any x K ( x0 , ) and satisfying ( , x) ( ) || x x0 ||2 Therefore, ‖ x x0‖ Then, for each x K ( x0 , ) , we have ‖ x x0‖ ( , x) (9) Proof Since S ( ) , all the problems m i 1 GPVQEP (i ) (i ) have the same solution Without ( , x) and hence the proof is completed n, m, , A, , K , T1 , Example Let T2 , H1 , H , as be in Example From Example 1, we have the solution of (GPVQEP) γ , S ( ) {0} loss of generality, we assume that x0 is the same solution Since x0 K ( x, ) for any x K ( x0 , ) , and the regularized gap function of (MPVQEP) is ( , x) defined by max min{ H i ( x, y ) Ti ( , x), ( x, y )} ‖ x y‖ yK ( x , ) 1 i m min{ H i ( x, x0 ) Ti ( , x), ( x, x0 )} ‖ x x0‖ 10 1 i m Then, we can assume that there exists i0 {1, , m} such that min{ H i ( x, x0 ) Ti ( , x), ( x, x0 )} 1i m H i0 ( x, x0 ) Ti0 ( , x), ( x, x0 ) and so, (10) follows that ( , x) H i0 ( x, x0 ) Ti0 ( , x), ( x, x0 ) ‖ x x0‖ (11) ( , x) (2 ) x2 It is easy to check that H1 and H are strongly monotone with moduli 1 and 7, respectively Also T1 and T2 are -strongly monotone with the moduli 1 and 2 2 , respectively Then, and Therefore, the assumptions of Theorem are satisfied, and so Theorem holds Some numerical results of Theorem are shown in Table Table Illustrate the error bounds given by (9) with 0.15 , 0.3 and 0.5 As x0 S (i0 ) ( ) , we obtain Hi0 ( x0 , x) Ti0 ( , x0 ), ( x, x0 ) (12) Since H i0 is strongly monotone with modulus i , we conclude that Hi0 ( x0 , x) Hi0 ( x, x0 ) i0 ‖ x x0‖ (13) It follows from the -strong monotonicity of Ti0 with modulus i0 that Ti0 ( , x), ( x, x0 ) Ti0 ( , x0 ), ( x, x0 ) i0 ‖ x x0‖ (14) Employing (12) – (14), we obtain H i0 ( x, x0 ) Ti0 ( , x), ( x, x0 ) ( i0 i0 )‖ x x0‖ ( )‖ x x0‖ From (11) and (15), we get (15) x x x0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Error bounds 0.15 0.3 0.5 0.000 0.137 0.273 0.410 0.547 0.684 0.820 0.957 1.094 1.231 1.367 0.000 0.133 0.266 0.399 0.532 0.665 0.798 0.931 1.064 1.197 1.330 0.000 0.129 0.258 0.387 0.516 0.645 0.775 0.904 1.033 1.162 1.291 Remark In special cases of (a), (b) mentioned in Sect 1, the regularized gap function for (GPVQEP) γ reduces to the regularized gap Natural Sciences issue function for (VIP) and (QEP) considered in Bigi and Passacantando (2016) and Yamashita and Fukushima (1997), respectively Therefore, for these cases, Theorem and Theorem extend to the existing ones in Bigi and Passacantando (2016) and Yamashita and Fukushima (1997), and are different from the corresponding results in Anh et al (2018), Hung et al (2020a, 2020b, 2021) and Khan and Chen (2015b) in the form of the problem (GPVQEP) γ perturbed by parameters Conclusions The class of mixed parametric vector quasiequilibrium problems (GPVQEP) γ is introduced in this paper Regularized gap functions and error bounds are stated for this kind of problems under suitable assumptions Examples are given to support the results presented here It would be interesting to consider the study of Levitin-Polyak well-posedness by perturbations and Hölder continuity of solution mapping for the class of mixed parametric vector quasi-equilibrium problems (GPVQEP) γ based on regularized gap functions Acknowledgements This work is supported by the Ministry of Education and Training of Vietnam under Grant No B2021.SPD.03 The authors are grateful to the anonymous referees for their valuable remarks which improved the results and presentation of the paper References Anh, L.Q., Hung, N.V., and Tam, V.M (2018) Regularized gap functions and error bounds for generalized mixed strong vector quasiequilibrium problems Comput Appl Math, 37, 5935-5950 Auslender, A (1976) Optimisation: Méthodes Numériques Paris: Masson Bigi, G and Passacantando, M (2016) Gap functions for quasiquilibria J Global Optim., 66, 791-810 Fukushima, M (1992) Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems Math Program., 53(1992), 99-110 Gupta, R and Mehra, A (2012) Gap functions and error bounds for quasi variational inequalities J Glob Optim., 53, 737-748 Hung, N.V., Tam, V.M., and Zhou, Y (2021) A new class of strong mixed vector GQVIPgeneralized quasi-variational inequality problems in fuzzy environment with regularized gap functions based error bounds J.Compu Appl Math., 381,113055 Hung, N.V., Tam, V.M., Tuan, N., and O'Regan, D (2020a) Regularized gap functions and error bounds for generalized mixed weak vector quasi variational inequality problems in fuzzy environments Fuzzy Sets Syst., 400, 162-176 Hung, N.V., Tam, V.M., and Pitea, A (2020b) Global error bounds for mixed quasihemivariational inequality problems on Hadamard manifolds Optimization, 69, 2033-2052 Khan, S.A., and Chen, J.W (2015a) Gap function and global error bounds for generalized mixed quasivariational inequalities Appl Math Comput., 260, 71-81 Khan, S.A., and Chen, J.W (2015b) Gap functions and error bounds for generalized mixed vector equilibrium problems J Optim Theory Appl., 166, 767-776 Mastroeni G (2003) Gap functions for equilibrium problems J Glob Optim., 27, 411-426 Rockafellar, R.T., and Wets, R.J.B (1998) Variational analysis Berlin: Springer Yamashita, Y., and Fukushima, M (1997) Equivalent unconstraint minimization and global error bounds for variational inequality problems SIAM J Control Optim., 35, 273-284 ... Global error bounds for mixed quasihemivariational inequality problems on Hadamard manifolds Optimization, 69, 2033-2052 Khan, S .A. , and Chen, J.W (201 5a) Gap function and global error bounds for. .. and Passacantando (2016) and Yamashita and Fukushima (1997), respectively Therefore, for these cases, Theorem and Theorem extend to the existing ones in Bigi and Passacantando (2016) and Yamashita... perturbed by parameters Conclusions The class of mixed parametric vector quasiequilibrium problems (GPVQEP) γ is introduced in this paper Regularized gap functions and error bounds are stated for this