# Mathematical Programming Based on Sufficient Optimality Conditions and Higher Order Exponential Type Generalized Invexities

### Abstract

First, a class of comprehensive higher order exponential type generalized $B$-($b,$ $\rho,$ $\eta,$ $\omega,$ $\theta,$ $\tilde{p},$ $\tilde{r},$ $\tilde{s}$)-invexities is introduced, which encompasses most of the existing generalized invexity concepts in the literature, including the Antczak type first order $B$-($b,$ $\eta,$ $\tilde{p},$ $\tilde{r}$)-invexities as well as the Zalmai type $(\alpha,$ $\beta,$ $\gamma,$ $\eta,$ $\rho,$ $\theta$)-invexities, and then a wide range of parametrically sufficient optimality conditions leading to the solvability for discrete minimax fractional programming problems are established with some other related results. To the best of our knowledge, the obtained results are new and general in nature relating the investigations on generalized higher order exponential type invexities.### References

T. Antczak, A class of B-(p,r)-invex functions and mathematical programming, J. Math. Anal. Appl. 268 (2003), 187-206.

T. Antczak, Generalized fractional minimax programming with B (p,r)-invexity, Computers and Mathematics with Applications, 56 (2008), 1505-1525.

T. Antczak, Generalized B-(p,r)-invexity functions and nonlinear mathematical programming, Numer. Funct. Anal. Optim. 30 (2009), 1-22.

A. Ben-Israel and B. Mond What is the invexity? Journal of Australian Mathematical Society Ser. B 28 (1986), 1-9.

C. Canning, R Canning, Absence of pure Nash equilibria in a class of co-ordination games, Statistics, Optimization & Information Computing 1 (1) (2013), 1-7.

M.A Hanson, On sufficiency of the Kuhn-Tucker conditions, Journal of Mathematical Analysis and Applications 80 (1985), 545–550.

V. Jeyakumar, Strong and weak invexity in mathematical programming, Methods Oper. Res.55 (1985), 109-125.

M.H. Kim, G.S. Kim and G.M. Lee, On ϵ−optimality conditions for multiobjective fractional optimization problems, Fixed Point Theory & Applications 2011:6 doi:10.1186/1687-1812-2011-6.

G.S. Kim and G.M. Lee, On ϵ−optimality theorems for convex vector optimization problems, Journal of Nonlinear and Convex Analysis 12 (3) (2013), 473-482.

Q Li, A modified Fletcher-Reeves-type method for nonsmooth convex optimization, Statistics, Optimization & Information Computing 2(3)(2014), 200-210.

O.L. Mangasarian, Second- and higher-order duality theorems in nonlinear programming, J. Math. Anal. Appl. 51 (1975), 607-620.

S.K. Mishra, Second order generalized invexity and duality in mathematical programming, Optimization 42 (1997), 51-69.

S.K. Mishra, Second order symmetric duality in mathematical programming with F-convexity, European J. Oper. Res. 127 (2000), 507-518.

S.K. Mishra and N. G. Rueda, Higher-order generalized invexity and duality in mathematical programming, J. Math. Anal. Appl., 247 (2000), 173-182.

S.K. Mishra and N. G. Rueda, Second-order duality for nondifferential minimax programming involving generalized type I functions, J. Optim. Theory Appl., 130 (2006), 477-486.

S.K. Mishra, M. Jaiswal and Pankaj, Optimality conditions for multiple objective fractional subset programming with invex and related non-convex functions, Communications on Applied Nonlinear Analysis 17 (3) (2010), 89–101.

B. Mond and T. Weir, Generalized convexity and higher-order duality, J. Math. Sci. 16-18 (1981-1983), 74-94.

B. Mond and J. Zhang, Duality for multiobjective programming involving second-order V-invex functions, in Proceedings of the Optimization Miniconference II (B. M. Glover and V. Jeyakumar, eds.), University of New South Wales, Sydney, Australia ,1997, pp. 89-100.

B. Mond and J. Zhang, Higher order invexity and duality in mathematical programming, in Generalized Convexity, Generalized Monotonicity: Recent Results (J. P. Crouzeix, et al., eds.), Kluwer Academic Publishers, printed in the Netherlands, 1998, pp. 357-372.

R.B. Patel, Second order duality in multiobjective fractional programming, Indian J. Math. 38 (1997), 39-46.

M.K. Srivastava and M. Bhatia, Symmetric duality for multiobjective programming using second order (F,ρ)-convexity, Opsearch 43 (2006), 274-295.

K.K. Srivastava and M. G. Govil, Second order duality for multiobjective programming involving (F,ρ,σ)-type I functions, Opsearch 37 (2000), 316-326.

S.K. Suneja, C. S. Lalitha, and S. Khurana, Second order symmetric duality in multiobjective programming, European J. Oper. Res. 144 (2003), 492-500.

R.U. Verma and G.J. Zalmai, Generalized parametric duality models in discrete minmax fractional programming based on second order optimality conditions, Submitted for publication.

R.U. Verma, Weak ϵ− efficiency conditions for multiobjective fractional programming, Applied Mathematics and Computation, 219 (2013), 6819-6827.

R.U. Verma, A generalization to Zalmai type second order univexities and applications to parametric duality models to discrete minimax fractional programming, Advances in Nonlinear Variational Inequalities 15 (2) (2012), 113-123.

R.U. Verma, Parametric duality models for multiobjective fractional programming basedd on new generation hybrid invexities, Journal of Applied Functional Analysis 10 (3-4)(2015), 234-253.

R.U. Verma, Generalized (G,b,β,ϕ,h,ρ,θ)-Univexities with Applications to Parametric Duality Models for Discrete Minimax Fractional Programming, Transactions on Mathematical Programming and Applications 1 (1) (2013), 1-14.

R.U. Verma, New ϵ−optimality conditions for multiobjective fractional subset programming problems, Transactions on Mathematical Programming and Applications 1 (1) (2013), 69-89.

R.U. Verma, Second order (Φ,Ψ,ρ,η,θ)−invexity frameworks and ϵ−efficiency conditions for multiobjective fractional programming, Theory and Applications of Mathematics & Computer Science 2 (2)(2012), 31-47.

X.M. Yang, Second order symmetric duality for nonlinear programs, Opsearch 32 (1995), 205-209.

X.M. Yang, On second order symmetric duality in nondifferentiable multiobjective programming, J. Ind. Manag. Optim. 5 (2009), 697-703.

X.M. Yang and S. H. Hou, Second-order symmetric duality in multiobjective programming, Appl. Math. Lett. 14 (2001), 587-592.

X.M.Yang,K.L.Teo and X.Q.Yang, Higher order generalized convexity and duality in nondifferentiable multiobjective mathematical programming, J. Math. Anal. Appl. 297 (2004), 48-55.

X.M. Yang, X.Q. Yang and K.L. Teo, Nondifferentiable second order symmetric duality in mathematical programming with F-convexity, European J. Oper. Res. 144 (2003), 554-559.

X.M. Yang, X.Q. Yang and K.L. Teo, Huard type second-order converse duality for nonlinear programming, Appl. Math. Lett. 18 (2005), 205-208.

X.M. Yang, X.Q. Yang and K.L. Teo, Higher-order symmetric duality in multiobjective programming with invexity, J. Ind. Manag. Optim. 4 (2008), 385-391.

X.M. Yang, X.Q. Yang K.L. Teo and S.H. Hou, Second order duality for nonlinear programming, Indian J. Pure Appl. Math. 35 (2004), 699-708.

K. Yokoyama, Epsilon approximate solutions for multiobjective programming problems, Journal of Mathematical Analysis and Applications 203 (1) (1996), 142–149.

G.J. Zalmai, General parametric sufficient optimality conditions for discrete minmax fractional programming problems containing generalized (θ,η,ρ)-V-invex functions and arbitrary norms Journal of Applied Mathematics & Computing 23 (1-2) (2007), 1-23.

G.J. Zalmai, Hanson-Antczak-type generalized invex functions in semiinfinte minmax fractional programming, Part I: Sufficient optimality conditions, Communications on Applied Nonlinear Analysis, 19 (4) (2012), 1-36.

G.J. Zalmai and Q. Zhang, Global nonparametric sufficient optimality conditions for semiinfinite discrete minmax fractional programming problems involving generalized (η,ρ)−invex functions, Numerical Functional Analysis and Optimization 28 (2007), 173-209.

J. Zhang and B. Mond, Second order b-invexity and duality in mathematical programming, Utilitas Math. 50 (1996), 19-31.

J. Zhang and B. Mond, Second order duality for multiobjective nonlinear programming involving generalized convexity, in Proceedings of the Optimization Miniconference III (B. M. Glover, B. D. Craven, and D. Ralph, eds.), University of Ballarat, (1997), pp. 79-95.

*Statistics, Optimization & Information Computing*,

*3*(3), 276-293. https://doi.org/10.19139/soic.v3i3.139

- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).