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Sufficient efficiency conditions associated with a multidimensional multiobjective fractional variational problem

Year 2018, Volume: 1 Issue: 1, 1 - 13, 01.08.2018

Abstract

This paper presents a study on sufficient efficiency conditions for a class of multidimensional vector ratio optimization problems, identified by (MFP), of minimizing a vector of path-independent curvilinear integral functionals quotients subject to PDE and/or PDI constraints involving higher-order partial derivatives. Under generalized (\rho, b)-quasiinvexity assumptions, sufficient conditions of efficiency are formulated for a feasible solution in (MFP).

References

  • \bibitem{aaia} R.P. Agarwal, I. Ahmad, A. Iqbal, S. Ali, \textit{Generalized invex sets and preinvex functions on Riemannian manifolds}, Taiwanese Journal of Mathematics, \textbf{16}, 5, (2012), 1719-1732.
  • \bibitem{bp} A. Barani, M.R. Pouryayevali, \textit{Invex sets and preinvex functions on Riemannian manifolds}, J. Math. Anal. Appl., \textbf{328}, 2, (2007), 767-779.
  • \bibitem{ta} T. Antczak, \textit{$G$-pre-invex functions in mathematical programming}, J. Comput. Appl. Math., \textbf{217}, 1, (2008), 212-226.
  • \bibitem{AOR1} M. Arana-Jim\'{e}nez, R. Osuna-G\'{o}mez, G. Ruiz-Garz\'{o}n, M. Rojas-Medar, \textit{On variational problems: Characterization of solutions and duality}, J. Math. Anal. Appl., \textbf{311}, 1, (2005) 1-12.
  • \bibitem{CP} A. Chinchuluun, P.M. Pardalos, \textit{A survey of recent developments in multiobjective optimization}, Ann. Oper. Res., \textbf{154}, 1, (2007), 29-50.
  • \bibitem{MAH1} M.A. Hanson, \textit{On sufficiency of Kuhn-Tucker conditions}, J. Math. Anal. Appl., \textbf{80}, 2, (1981), 545-550.
  • \bibitem{RJ} R. Jagannathan, \textit{Duality for nonlinear fractional programming}, Z. Oper. Res., \textbf{17}, (1973), 1-3.
  • \bibitem{vj} V. Jeyakumar, \textit{Strong and weak invexity in mathematical programming}, Math. Oper. Res., \textbf{55}, (1985), 109-125.
  • \bibitem{LHP} Z.A. Liang, H.X. Huang, P.M. Pardalos, \textit{Efficiency conditions and duality for a class of multiobjective fractional programming problems}, J. Glob. Optim., \textbf{27}, 4, (2003), 447-471.
  • \bibitem{StS} \c St. Mititelu, S. Trean\c t\u{a}, \textit{Efficiency conditions in vector control problems governed by multiple integrals}, J. Appl. Math. Comput., https://doi.org/10.1007/s12190-017-1126-z, (2017), 1-19.
  • \bibitem{MO} B. Mond and I. Husain, \textit{Sufficient optimality criteria and duality for variational problems with generalised invexity}, J. Aust. Math. Soc. (B), \textbf{31}, 1, (1989), 108-121.
  • \bibitem{nn} M.A. Noor, K.I. Noor, \textit{Some characterizations of strongly preinvex functions}, J. Math. Anal. Appl., \textbf{316}, 2, (2006), 697-706.
  • \bibitem{rp} R. Pini, \textit{Convexity along curves and indunvexity}, Optimization, \textbf{29}, 4, (1994), 301-309.
  • \bibitem{PG1} V. Preda, \textit{On efficiency and duality for multiobjective programs}, J. Math. Anal. Appl., \textbf{166}, 2, (1992), 365-377.
  • \bibitem{PG3} V. Preda, \textit{Optimality and duality in fractional multiple objective programming involving semilocally preinvex and related functions}, J. Math. Anal. Appl., \textbf{288}, 2, (2003), 365-382.
  • \bibitem{tr} T. Rapcs\'{a}k, \textit{Smooth Nonlinear Optimization in $ R^{n} $}, Nonconvex Optimization and Its Applications, Kluwer Academic, 1997.
  • \bibitem{DS} D.J. Saunders, \textit{The Geometry of Jet Bundles}, Cambridge Univ. Press, 1989.
  • \bibitem{ty} W. Tang, X. Yang, \textit{The sufficiency and necessity conditions of strongly preinvex functions}, OR Transactions, \textbf{10}, 3, (2006), 50-58.
  • \bibitem{TA} T. Tanino and Y. Sawaragi, \textit{Duality theory in multiobjective programming}, J. Optim. Theory Appl., \textbf{27}, 4, (1979), 509-529.
  • \bibitem{ST4} S. Trean\c t\u{a} and C. Udri\c ste, \textit{On Efficiency Conditions for Multiobjective Variational Problems Involving Higher Order Derivatives}, Proceedings of the 15th International Conference on Automatic Control, Modelling $\&$ Simulation (ACMOS-13), June 1-3, Bra\c sov, Romania, (2013), 157-164.
  • \bibitem{ST3} S. Trean\c t\u{a}, \textit{PDEs of Hamilton-Pfaff type via multi-time optimization problems}, U.P.B. Sci. Bull., Series A: Appl. Math. Phys., \textbf{76}, 1, (2014), 163-168.
  • \bibitem{ST2} S. Trean\c t\u{a}, \textit{On a vector optimization problem involving higher order derivatives}, U.P.B. Sci. Bull., Series A: Appl. Math. Phys., \textbf{77}, 1, (2015), 115-128.
  • \bibitem{ST7} S. Trean\c t\u{a}, \textit{Multiobjective fractional variational problem on higher order jet bundles}, Commun. Math. Stat., \textbf{4}, 3, (2016), 323-340.
  • \bibitem{st} S. Trean\c t\u{a}, \textit{Higher-order Hamilton dynamics and Hamilton-Jacobi divergence PDE}, Computers and Mathematics with Applications, https://doi.org/10.1016/j.camwa.2017.09.033, (2017), 1-14.
  • \bibitem{stma} S. Trean\c t\u{a}, M. Arana-Jim\'{e}nez, \textit{KT-pseudoinvex multidimensional control problem}, Optim. Control Appl. Meth, https://doi.org/10.1002/oca.2410, (2018), 1-10.
  • \bibitem{cu} C. Udri\c ste, \textit{Convex Functions and Optimization Methods on Riemannian Manifolds}, Mathematics and Its Applications, Kluwer Academic, \textbf{297}, 1994.
  • \bibitem{WB} T. Weir, B. Mond, \textit{Pre-invex functions in multiple objective optimization}, J. Math. Anal. Appl., \textbf{136}, 1, (1988), 29-38.
  • \bibitem{WE} T. Weir and B. Mond, \textit{Generalized convexity and duality in multiple objective programming}, Bulletin of the Australian Mathematical Society, \textbf{39}, 2, (1989), 287-299.
Year 2018, Volume: 1 Issue: 1, 1 - 13, 01.08.2018

Abstract

References

  • \bibitem{aaia} R.P. Agarwal, I. Ahmad, A. Iqbal, S. Ali, \textit{Generalized invex sets and preinvex functions on Riemannian manifolds}, Taiwanese Journal of Mathematics, \textbf{16}, 5, (2012), 1719-1732.
  • \bibitem{bp} A. Barani, M.R. Pouryayevali, \textit{Invex sets and preinvex functions on Riemannian manifolds}, J. Math. Anal. Appl., \textbf{328}, 2, (2007), 767-779.
  • \bibitem{ta} T. Antczak, \textit{$G$-pre-invex functions in mathematical programming}, J. Comput. Appl. Math., \textbf{217}, 1, (2008), 212-226.
  • \bibitem{AOR1} M. Arana-Jim\'{e}nez, R. Osuna-G\'{o}mez, G. Ruiz-Garz\'{o}n, M. Rojas-Medar, \textit{On variational problems: Characterization of solutions and duality}, J. Math. Anal. Appl., \textbf{311}, 1, (2005) 1-12.
  • \bibitem{CP} A. Chinchuluun, P.M. Pardalos, \textit{A survey of recent developments in multiobjective optimization}, Ann. Oper. Res., \textbf{154}, 1, (2007), 29-50.
  • \bibitem{MAH1} M.A. Hanson, \textit{On sufficiency of Kuhn-Tucker conditions}, J. Math. Anal. Appl., \textbf{80}, 2, (1981), 545-550.
  • \bibitem{RJ} R. Jagannathan, \textit{Duality for nonlinear fractional programming}, Z. Oper. Res., \textbf{17}, (1973), 1-3.
  • \bibitem{vj} V. Jeyakumar, \textit{Strong and weak invexity in mathematical programming}, Math. Oper. Res., \textbf{55}, (1985), 109-125.
  • \bibitem{LHP} Z.A. Liang, H.X. Huang, P.M. Pardalos, \textit{Efficiency conditions and duality for a class of multiobjective fractional programming problems}, J. Glob. Optim., \textbf{27}, 4, (2003), 447-471.
  • \bibitem{StS} \c St. Mititelu, S. Trean\c t\u{a}, \textit{Efficiency conditions in vector control problems governed by multiple integrals}, J. Appl. Math. Comput., https://doi.org/10.1007/s12190-017-1126-z, (2017), 1-19.
  • \bibitem{MO} B. Mond and I. Husain, \textit{Sufficient optimality criteria and duality for variational problems with generalised invexity}, J. Aust. Math. Soc. (B), \textbf{31}, 1, (1989), 108-121.
  • \bibitem{nn} M.A. Noor, K.I. Noor, \textit{Some characterizations of strongly preinvex functions}, J. Math. Anal. Appl., \textbf{316}, 2, (2006), 697-706.
  • \bibitem{rp} R. Pini, \textit{Convexity along curves and indunvexity}, Optimization, \textbf{29}, 4, (1994), 301-309.
  • \bibitem{PG1} V. Preda, \textit{On efficiency and duality for multiobjective programs}, J. Math. Anal. Appl., \textbf{166}, 2, (1992), 365-377.
  • \bibitem{PG3} V. Preda, \textit{Optimality and duality in fractional multiple objective programming involving semilocally preinvex and related functions}, J. Math. Anal. Appl., \textbf{288}, 2, (2003), 365-382.
  • \bibitem{tr} T. Rapcs\'{a}k, \textit{Smooth Nonlinear Optimization in $ R^{n} $}, Nonconvex Optimization and Its Applications, Kluwer Academic, 1997.
  • \bibitem{DS} D.J. Saunders, \textit{The Geometry of Jet Bundles}, Cambridge Univ. Press, 1989.
  • \bibitem{ty} W. Tang, X. Yang, \textit{The sufficiency and necessity conditions of strongly preinvex functions}, OR Transactions, \textbf{10}, 3, (2006), 50-58.
  • \bibitem{TA} T. Tanino and Y. Sawaragi, \textit{Duality theory in multiobjective programming}, J. Optim. Theory Appl., \textbf{27}, 4, (1979), 509-529.
  • \bibitem{ST4} S. Trean\c t\u{a} and C. Udri\c ste, \textit{On Efficiency Conditions for Multiobjective Variational Problems Involving Higher Order Derivatives}, Proceedings of the 15th International Conference on Automatic Control, Modelling $\&$ Simulation (ACMOS-13), June 1-3, Bra\c sov, Romania, (2013), 157-164.
  • \bibitem{ST3} S. Trean\c t\u{a}, \textit{PDEs of Hamilton-Pfaff type via multi-time optimization problems}, U.P.B. Sci. Bull., Series A: Appl. Math. Phys., \textbf{76}, 1, (2014), 163-168.
  • \bibitem{ST2} S. Trean\c t\u{a}, \textit{On a vector optimization problem involving higher order derivatives}, U.P.B. Sci. Bull., Series A: Appl. Math. Phys., \textbf{77}, 1, (2015), 115-128.
  • \bibitem{ST7} S. Trean\c t\u{a}, \textit{Multiobjective fractional variational problem on higher order jet bundles}, Commun. Math. Stat., \textbf{4}, 3, (2016), 323-340.
  • \bibitem{st} S. Trean\c t\u{a}, \textit{Higher-order Hamilton dynamics and Hamilton-Jacobi divergence PDE}, Computers and Mathematics with Applications, https://doi.org/10.1016/j.camwa.2017.09.033, (2017), 1-14.
  • \bibitem{stma} S. Trean\c t\u{a}, M. Arana-Jim\'{e}nez, \textit{KT-pseudoinvex multidimensional control problem}, Optim. Control Appl. Meth, https://doi.org/10.1002/oca.2410, (2018), 1-10.
  • \bibitem{cu} C. Udri\c ste, \textit{Convex Functions and Optimization Methods on Riemannian Manifolds}, Mathematics and Its Applications, Kluwer Academic, \textbf{297}, 1994.
  • \bibitem{WB} T. Weir, B. Mond, \textit{Pre-invex functions in multiple objective optimization}, J. Math. Anal. Appl., \textbf{136}, 1, (1988), 29-38.
  • \bibitem{WE} T. Weir and B. Mond, \textit{Generalized convexity and duality in multiple objective programming}, Bulletin of the Australian Mathematical Society, \textbf{39}, 2, (1989), 287-299.
There are 28 citations in total.

Details

Subjects Mathematical Sciences
Journal Section Articles
Authors

Savin Treanta

Publication Date August 1, 2018
Published in Issue Year 2018 Volume: 1 Issue: 1

Cite

APA Treanta, S. (2018). Sufficient efficiency conditions associated with a multidimensional multiobjective fractional variational problem. Journal of Multidisciplinary Modeling and Optimization, 1(1), 1-13.
AMA Treanta S. Sufficient efficiency conditions associated with a multidimensional multiobjective fractional variational problem. jmmo. August 2018;1(1):1-13.
Chicago Treanta, Savin. “Sufficient Efficiency Conditions Associated With a Multidimensional Multiobjective Fractional Variational Problem”. Journal of Multidisciplinary Modeling and Optimization 1, no. 1 (August 2018): 1-13.
EndNote Treanta S (August 1, 2018) Sufficient efficiency conditions associated with a multidimensional multiobjective fractional variational problem. Journal of Multidisciplinary Modeling and Optimization 1 1 1–13.
IEEE S. Treanta, “Sufficient efficiency conditions associated with a multidimensional multiobjective fractional variational problem”, jmmo, vol. 1, no. 1, pp. 1–13, 2018.
ISNAD Treanta, Savin. “Sufficient Efficiency Conditions Associated With a Multidimensional Multiobjective Fractional Variational Problem”. Journal of Multidisciplinary Modeling and Optimization 1/1 (August 2018), 1-13.
JAMA Treanta S. Sufficient efficiency conditions associated with a multidimensional multiobjective fractional variational problem. jmmo. 2018;1:1–13.
MLA Treanta, Savin. “Sufficient Efficiency Conditions Associated With a Multidimensional Multiobjective Fractional Variational Problem”. Journal of Multidisciplinary Modeling and Optimization, vol. 1, no. 1, 2018, pp. 1-13.
Vancouver Treanta S. Sufficient efficiency conditions associated with a multidimensional multiobjective fractional variational problem. jmmo. 2018;1(1):1-13.