Error Bounds and Merit Functions for Exponentially General Variational Inequalities
Abstract
In this paper, some new classes of classes of exponentially general variational inequalities are introduced. It is shown that the odd-order and nonsymmetric exponentially boundary value problems can be studied in the framework of exponentially general variational inequalities. We consider some classes of merit functions for exponentially general variational inequalities. Using these functions, we derive error bounds for the solution of exponentially general variational inequalities under some mild conditions. Since the exponentially general variational inequalities include general variational inequalities, quasi-variational inequalities and complementarity problems as special cases, results proved in this paper hold for these problems. Results obtained in this paper represent a refinement of previously known results for several classes of variational inequalities and related optimization problems.
References
G. Alirezaei and R. Mazhar, On exponentially concave functions and their impact in information theory, J. Inform. Theory Appl. 9(5) (2018), 265-274. https://doi.org/10.1109/ita.2018.8503202
T. Antczak, On $(p,r)$-invex sets and functions, J. Math. Anal. Appl. 263 (2001), 355-379.
M. Avriel, $r$-Convex functions, Math. Program. 2 (1972), 309-323.
M. U. Awan, M. A. Noor, K. I. Noor, Y-M. Chu and S. Ellahi, On $G^{(sigma,h)$-convexity of functions and applications to Hermite-Hadamard inequality, in: Approximation and Computation in Science and Engineering (Edits: N. J. Daras and Th. M. Rassias), Springer Optimization and its Applications 180 (2022), 927-944. https://doi.org/10.1007/978-3-030-84122-5_43
S. Batool, M. A. Noor and K. I. Noor, Merit functions for absolute value variational inequalities, AIMS Math. 6(11) (2021), 12133-12147. https://doi.org/10.3934/math.2021704
S. N. Bernstein, Sur les fonctions absolument monotones, Acta Math. 52 (1929), 1-66. https://doi.org/10.1007/bf02592679
G. L. Balnkenship and J. L. Menaldi, Optimal stochastic scheduling of power generation system with scheduling delays and large cost differentials, SIAM Optim. 22(1984), 121-132. https://doi.org/10.1137/0322009
F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley, NY, 1983.
R. W. Cottle, J. S. Pang and R. E. Stone, The Linear Complementarity Problem, Academic Press, New York, 1992.
G. Cristescu and L. Lupsa, Non Connected Convexities and Applications, Kluwer Academic Publisher, Dordrechet, 2002.
N. J. Daras and Th. M. Rassias (Editors), Approximation and Computation in Science and Engineering, Springer Optimization and Its Applications, 180, Springer, Cham, 2022.
V. M. Filippov, Variational Principles for Nonpotential Operators, Vol. 77, American Math. Soc., USA, 1989.
M. Fukushima, Equivalent differentiable Optimization problems and descent methods for asymmetric variational inequality problems, Math. Program. 53 (1992), 99-110. https://doi.org/10.1007/bf01585696
R. Glowinski, J. J. Lions and R. Tremolieres, Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam, 1981.
S. Karamardian, Generalized complementarity problem, J. Optim. Theory Appl. 8 (1971), 161-168. https://doi.org/10.1007/bf00932464
G. M. Korpelevich, The extragradient method for finding saddle points and other problems, Ekonomika i Matematicheskie Metody 12 (1976), 747-756.
J. L. Lions and G. Stampacchia, Variational inequalities, Comm. Pure Appl. Math. 20 (1967), 493-512. https://doi.org/10.1002/cpa.3160200302
B. Martinet, Regularization d'inequations variationnelles par approximations successive, Revue Fran. d'Informat. Rech. Opers. 4 (1970), 154-159.
C. F. Niculescu and L. E. Persson, Convex Functions and Their Applications, Springer-Verlag, New York, 2018.
M. A. Noor, On variational Inequalities, PhD Thesis, Brunel University, London, U.K., 1975.
M. A. Noor, General variational inequalities, Appl. Math. Letters 1 (1988), 119-121.
M. A. Noor, Merit functions for general variational inequalities, J. Math. Anal. Appl. 316(2) (2006), 736-752. https://doi.org/10.1016/j.jmaa.2005.05.011
M. A. Noor, On merit fuunctions for quasi variational inequalities, J. Math. Inequal. 1 (2007), 259-276.
M. A. Noor, Variational inequalities in physical oceanography: in Ocean Waves Engineering (Edited by M. Rahman), Computational Mechanics Publications, Southampton, England (1994), 201-226.
M. A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl. 251 (2000), 217-229. https://doi.org/10.1006/jmaa.2000.7042
M. A. Noor, Some developments in general variational inequalities, Appl. Math. Comput. 152 (2004), 199-277.
M. A. Noor and K. I. Noor, On exponentially Convex Functions, J. Orissa Math. Soc. 38(01-02) (2019), 33-35.
M. A. Noor and K. I. Noor, Strongly exponentially convex functions, U.P.B. Bull. Sci. Appl. Math. Series A 81(4) (2019), 75-84.
M. A. Noor and K. I. Noor, Strongly exponentially convex functions and their properties, J. Advanc. Math. Studies 12(2) (2019), 177-185.
M. A. Noor and K. I. Noor, New classes of exponentially general convex functions, U.P.B. Bull. Sci. Appl. Math. Series A 82(3) (2020), 117-128.
M. A. Noor and K. I. Noor, Exonentially general variational inequalities, J. Advan. Math. Stud. 16(1) (2023).
M. A. Noor and K. I. Noor, Some new trends in mixed variational inequalities, J. Advan. Math. Stud. 15(2) (2022), 105-140.
M. A. Noor and K. I. Noor, Iterative methods and sensitivity analysis for exponential general variational inclusions, Earthline J. Math. Sci. 12(1) (2023), 53-103. https://doi.org/10.34198/ejms.12123.53107
M. A. Noor and K. I. Noor, Some novel aspects of quasi variational inequalities, Earthline J. Math. Sci. 10(1) (2022), 1-66. https://doi.org/10.34198/ejms.10122.166
M. A. Noor and K. I. Noor, Dynamical system for solving quasi variational inequalities, U.P.B. Sci. Bull. Series A 84(4) (2022), 55-66.
M. A. and K. I. Noor, Higher order generalized variational inequalities and nonconvex optimization, U.P.B. Sci. Bull. Series A 85(2)(2022), 77-88.
M. A. Noor and K. I. Noor, Iterative schemes for solving higher order hemivariational inequalities, Appl. Math. E-Notes 24(2024), in Press.
M. A. Noor, K. I. Noor and A. G. Khan, Merit functions for quasi variational inequalities, Appl. Comput. Math. 16(1) (2017), 19-32. https://doi.org/10.18576/amis/100621
M. A. Noor, R. Kamal and K. I. Noor, Error bounds for general variational inclusion involving difference of operators, Appl. Math. Inf. Sci. 10(6) (2016), 2189-2196. https://doi.org/10.18576/amis/100621
M. A. Noor, K. I. Noor and M. U. Awan, Some approximations schemes for solving exponentially variational inequalities, in: Trends in Applied Mathematical Analysis (Edit. P. M. Pardalos and Th. M. Rassias), Springer, 2023.
M. A. Noor, K. I. Noor and M. T. Rassias, New trends in general variational inequalities, Acta. Appl. Mathematicae 170(1) (2020), 981-164. https://doi.org/10.1007/s10440-020-00366-2
M. A. Noor, K. I. Noor and Th. M. Rassias, Some aspects of variational inequalities, J. Comput. Appl. Math. 47 (1993), 285-312. https://doi.org/10.1016/0377-0427(93)90058-j
M. A. Noor, K. I. Noor and M. Th. Rassias, General variational inequalities and optimization, Geometry and Noconvex Optimization (Edit. Themistocles M. Rassias), Springer, 2023. https://doi.org/10.1007/978-3-030-27407-8_23
M. A. Noor, K. I. Noor and Th. M. Rassias, Relative strongly exponentially convex functions, in: Nonlinear Analysis and Global Optimization (Edit. Th. M. Rassias and P. M. Pardalos), Springer (2020), 357-371. https://doi.org/10.1007/978-3-030-61732-5_16
S. Pal and T. K. Wong, On exponentially concave functions and a new information geometry, Annals. Prob. 46(2) (2018), 1070-1113. https://doi.org/10.1214/17-aop1201
M. Patriksson, Nonlinear Programming and Variational Inequality Problems: A Unified Approach, Kluwer Academic Publishers, Dordrecht, 1998.
J. Pecaric, F. Proschan and Y. L. Tong, Convex Functions, Partial Orderings and Statistical Applications, Academic Press, New York, 1992.
J. Pecaric and J. Jaksetic, On exponential convexity, Euler-Radau expansions and stolarsky means, Rad Hrvat. Matematicke Znanosti 17(515) (2013), 81-94.
A. Pitea, M. Postolache: Duality theorems for a new class of multitime multiobjective variational problems, J. Glob. Optim. 53(1) (2012), 47-58. https://doi.org/10.1007/s10898-011-9740-z
A. Pitea, M. Postolache: Minimization of vectors of curvilinear functionals on the second order jet bundle. Necessary conditions, Optim. Lett. 6(3) (2012), 459-470. https://doi.org/10.1007/s11590-010-0272-0
A. Pitea, M. Postolache: Minimization of vectors of curvilinear functionals on the second order jet bundle. Sufficient efficiency conditions, Optim. Lett. 6(8) (2012), 1657-1669. https://doi.org/10.1007/s11590-011-0357-4
M. V. Solodov and P. Tseng, Some methods based on the $D$-gap functions for solving monotone variational inequalities, Comput. Optim. Appl. 17 (2000), 255-277.
M. V. Solodov, Merit functions and error bounds for generalized variational inequalities, J. Math. Anal. Appl. 287 (2003), 405-414. https://doi.org/10.1016/s0022-247x(02)00554-1
G. Stampacchia, Formes bilineaires coercivites sur les ensembles convexes, C. R. Acad. Paris 258 (1964), 4413-4416.
E. Tonti, Variational formulation for every nonlinear problem, Intern. J. Engng. Sciences 22 (1984), 1343-1371. https://doi.org/10.1016/0020-7225(84)90026-0
N. Yamashita and M. Fukushima, Equivalent unconstrained minimization and global error bounds for variational inequality problems, SIAM J. Control Optim. 35 (1997), 273-284. https://doi.org/10.1137/s0363012994277645
N. H. Xiu and J. Z. Zhang, Global projection-type error bounds for general variational inequalities, J. Optim. Theory Appl. 112 (2002), 213-238. https://doi.org/10.1023/a:1013056931761
Y. X. Zhao, S. Y. Wang and L. Coladas Uria, Characterizations of $r$-convex functions, J. Optim. Theory Appl. 145 (2010), 186-195. https://doi.org/10.1007/s10957-009-9617-1
D. L. Zhu and P. Marcotte, Cocoercivity and its role in the convergence of iterative schemes for solving variational inequalities, SIAM J. Optim. 6 (1996), 714-726. https://doi.org/10.1137/s1052623494250415
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