Tbilisi Mathematical Journal

Higher-order parameter-free sufficient optimality conditions in discrete minmax fractional programming

Ram U. Verma and G. J. Zalmai

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

The purpose of this paper is to establish a fairly large number of sets of second-order parameter-free sufficient optimality conditions for a discrete minmax fractional programming problem. Our effort to accomplish this goal is by utilizing various new classes of generalized second-order $(\phi,\eta,\rho,\theta,m)$-invex functions, which generalize most of the concepts available in the literature.

Article information

Source
Tbilisi Math. J., Volume 10, Issue 2 (2017), 211-233.

Dates
Received: 4 December 2016
Accepted: 2 May 2017
First available in Project Euclid: 26 May 2018

Permanent link to this document
https://projecteuclid.org/euclid.tbilisi/1527300056

Digital Object Identifier
doi:10.1515/tmj-2017-0038

Mathematical Reviews number (MathSciNet)
MR3665635

Zentralblatt MATH identifier
1371.90113

Subjects
Primary: 90C26: Nonconvex programming, global optimization
Secondary: 90C30: Nonlinear programming 90C32: Fractional programming 90C46: Optimality conditions, duality [See also 49N15] 90C47: Minimax problems [See also 49K35]

Keywords
Discrete minmax fractional programming generalized second-order $(\phi,\eta,\rho,\theta,m)$-invex functions parameter-free sufficient optimality conditions

Citation

Verma, Ram U.; Zalmai, G. J. Higher-order parameter-free sufficient optimality conditions in discrete minmax fractional programming. Tbilisi Math. J. 10 (2017), no. 2, 211--233. doi:10.1515/tmj-2017-0038. https://projecteuclid.org/euclid.tbilisi/1527300056


Export citation

References

  • R. Andreani, J.M. Martinez and M.L. Schuverdt, On second-order optimality conditions for nonlinear programming, Optimization 56 (2007), 529-542.
  • W. Dinkelbach, On nonlinear fractional programming, Manag. Sci. 13 (1967), 492-498.
  • M.A. Hanson, Second order invexity and duality in mathematical programming, Opsearch 30 (1993), 313-320.
  • R. Hettich and G. Still, Second order optimality conditions for generalized semi-infinite programming problems, Optimization 34 (1995), 195-211.
  • H. Kawasaki, Second-order necessary and sufficient optimality conditions for minimizing a sup-type function, J. Appl. Math. Optim. 26 (1992), 195-220.
  • T.K. Kelly and M. Kupperschmid, Numerical verification of second-order sufficiency conditions for nonlinear programming, SIAM Review 40 (1998), 310-314.
  • A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies 204, Elsevier Science B.V. Amsterdam, 2006.
  • J.-L. Liu and H.M. Srivastava, Certain properties of Dziok-Srivastava operator, Applied Mathematics and Computation 159 (2004), 485-493.
  • M. Lopez and G. Still, Semi-infinite programming, European Journal of Operational Research 180 (2007), 491-518.
  • D. Luu and N. Hung, On alternative theorems and necessary conditions for efficiency, Optimization 58 (2009), 49-62.
  • C. Malivert, First and second order optimality conditions in vector optimization, Ann. Sci. Math. Québec 14 (1990), 65-79.
  • H. Maurer, First-and second-order sufficient optimality conditions in mathematical programming and optimal control, Math. Prog. Study 14 (1981), 163-177.
  • H. Maurer and J. Zowe, First and second order necessary and sufficient optimality conditions for infinite-dimensional programming problems, Math. Prog. 16 (1979), 98-110.
  • E.J. Messerli and E. Polak, On Second-order necessary conditions of optimality, SIAM J. Control 7 (1969), 272-291.
  • B. Mond, Second order duality for nonlinear programs, Opsearch 11 (1974), 90-99.
  • J.P. Penot, Second order conditions for optimization problems with constraints, SIAM J. Control Optim. 37 (1998), 303-318.
  • A. Pitea and M. Postolache, Duality theorems for a new class of multitime multiobjective variational problems, Journal of Global Optimization 54(1) (2012), 47-58.
  • J.J. Rückman and A. Shapiro, Second-order optimality conditions in generalized semi-infinite programming, Set-Valued Anal. 9 (2001), 169-186.
  • S. Schaible, Fractional programming: A recent survey, J. Stat. Manag. Syst. 5 (2002), 63-86.
  • I.M. Stancu-Minasian, Fractional Programming: Theory, Models and Applications, Kluwer, Dordrecht, 1997.
  • R.U. Verma, Generalized hybrid $B-(b,\rho,\theta,\tilde{p},\tilde{r})$-invexities and efficiency conditions for multiobjective fractional programming, Tbilisi Mathematical Journal 8(2) (2015), 159-180.
  • R.U. Verma, Second-order ($\Phi,$ $\eta,$ $\rho,$ $\theta$)-invexities and parameter-free $\epsilon-$efficiency conditions for multiobjective discrete minmax fractional programming problems, Advances in Nonlinear Variational Inequalities, 17 (1) (2014), 27-46.
  • R U. Verma, New $\epsilon-$optimality conditions for multiobjective fractional subset programming problems, Transactions on Mathematical Programming and Applications 1 (1) (2013), 69-89.
  • R.U. Verma, Parametric duality models for multiobjective fractional programming based on new generation hybrid invexities, Journal of Applied Functional Analysis 10 (3-4) (2015), 234-253.
  • R.U. Verma, Multiobjective fractional programming problems and second order generalized hybrid invexity frameworks, Statistics, Optimization & Information Computing 2 (4) (2014), 280-304.
  • R.U. Verma and G.J. Zalmai, Second-order parametric optimality conditions in discrete minmax fractional programming, Communications on Applied Nonlinear Analysis 23(3) (2016), 1-32.
  • G.-W. Weber, Generalized semi-infinite optimization: theory and applications in optimal control and discrete optimization, J. Stat. Manag. Syst. 5 (2002), 359-388.
  • G.-W. Weber, P. Taylan, Z. Alparslan-Gök, S. Özögür-Akyüz, and B. Akteke-Öztürk, Optimization of gene-environment networks in the presence of errors and uncertainty with Chebyshev approximation, TOP 16 (2008), 284-318.
  • G.-W. Weber and A. Tezel, On generalized semi-infinite optimization of genetic networks, TOP 15 (2007), 65-77.
  • G.-W. Weber, A. Tezel, P. Taylan, A.Söyler, and M. Çetin, Mathematical contributions to dynamics and optimization of gene-environment networks, Optimization 57 (2008), 353-377.
  • D.J. White, Optimality and Efficiency, Wiley, New York, 1982.
  • A. Winterfeld, Application of general semi-infinite programming to lapidary cutting problems, European J. Oper. Res. 191 (2008), 838-854.
  • P.L. Yu, Multiple-Criteria Decision Making: Concepts, Techniques, and Extensions, Plenum Press, New York, 1985.
  • G.J. Zalmai, Proper efficiency principles and duality models for a class of continuous-time multiobjective fractional programming problems with operator constraints, J. Stat. Manag. Syst. 1 (1998), 11-59.
  • G.J. Zalmai, Hanson-Antczak-type generalized $(\alpha,\beta,\gamma,\xi,\eta,\rho,\theta)$-V-invex functions in semiinfinite multiobjective fractional programming, Part III: Second order parametric duality models, Advances in Nonlinear Variational Inequalities 16(2) (2013), 91-126.
  • G.J. Zalmai, Hanson-Antczak-type generalized $(\alpha,\beta,\gamma,\xi,\eta,\rho,\theta)$-V-invex functions in semiinfinite multiobjective fractional programming, Part I: Sufficient efficiency conditions, Advances in Nonlinear Variational Inequalities 16(1) (2013), 91-114.
  • G.J. Zalmai, Generalized ($\mathcal{F},$ $b,$ $\phi,$ $\rho,$ $\theta)$-univex n-set functions and semiparametric duality models in multiobjective fractional subset programming, International Journal of Mathematics and Mathematical Sciences 2005(7)(2005), 1109-1133.
  • G.J. Zalmai, Optimality conditions and duality for constrained measurable subset selection problems with minmax objective functions, Optimization 20 (1989), 377-395.
  • G.J. Zalmai and Q. Zhang, Necessary sufficiency conditions for semiinfinite multiobjective optimization problems involving Hadamard directionally differentiable functions, Transactions on Mathematical Programming and Applications 1(1) (2013), 129-147.
  • L. Zhian and Y. Qingkai, Duality for a class of multiobjective control problems with generalized invexity, Journal of Mathematical Analysis and Applications 256 (2001), 446-461.