References

Transcription

1 References 1. Aghezzaf, B., Hachimi, M.: Second-order optimality conditions in multiobjective optimization problems. J. Optim. Theory Appl. 102, (1999) 2. Ahmad, I., Gupta, S.K., Kailey, K., Agarwal, R.P.: Duality in nondifferentiable minmax fractional programming with B ( p, r) invexity. J. Inequal. Appl. 2011(75) (2011) 3. Ahmad, I., Husain, Z.: Optimality conditions and duality in nondifferentiable minmax fractional programming with generalized convexity. J. Optim. Theory Appl. 129(2), (2006) 4. Ahmad, I.: Optimality conditions and duality in fractional minmax programming involving generalized ρ-invexity. Int. J. Manag. Syst. 19, (2003) 5. Al-Khayyal, F.A., Kyparisis, J.: Finite convergence of algorithms for nonlinear programs and variational inequalities. J. Optim. Theory Appl. 70(2), (1991) 6. Antczak, T.: Modified ratio objective approach in mathematical programming. J. Optim. Theory Appl. 126(1), (2005) 7. Antczak, T.: On ( p, r)-invexity-type nonlinear programming problems. J. Math. Anal. Appl. 264(2001), (2001) 8. Antczak, T.: (p, r)-invex sets and functions. J. Math. Anal. Appl. 263, (2001) 9. Antczak, T.: Lipschitz r-invex functions and nonsmooth programming. Numer. Funct. Anal. Optim. 23, (2002) 10. Antczak, T.: Generalized ( p, r)-invexity in mathematical programming. Numer. Funct. Anal. Optim. 24, (2003) 11. Antczak, T.: A class of B-( p, r)-invex functions and mathematical programming. J. Math. Anal. Appl. 268, (2003) 12. Antczak, T.: B-( p, r)-pre-invex functions. Folia Math. Acta Univ. Lodziensis 11, 3 15 (2004) 13. Antczak, T.: Minimax programming under ( p, r)-invexity. Eur. J. Oper. Res. 158, 1 19 (2004) 14. Antczak, T.: r-pre-invexity and r-invexity in mathematical programming. Comput. Math. Appl. 50, (2005) 15. Antczak, T.: The notion of V-r-invexity in differentiable multiobjective programming. J. Appl. Anal. 11, (2005) 16. Antczak, T.: Optimality and duality for nonsmooth multiobjective programming problems with V-r-invexity. J. Glob. Optim. 45, (2009) 17. Antczak, T.: Generalized B-( p, r)-invexity functions and nonlinear mathematical programming. Numer. Funct. Anal. Optim. 30, 1 22 (2009) 18. Antczak, T.: The l 1 penalty function method for nonconvex differentiable optimization problems with inequality constraints. Asia-Pac. J. Oper. Res. 27, (2010) Springer Nature Singapore Pte Ltd R.U. Verma, Semi-Infinite Fractional Programming, Infosys Science Foundation Series in Mathematical Sciences, DOI /

2 284 References 19. Barros, A.I.: Discrete and Fractional Programming Techniques for Location Models. Kluwer Academic Publishers, Dordrecht-Boston-London (1998) 20. Bigi, G., Castellani, M.: Second order optimality conditions for differentiable multiobjective problems. RAIRO Oper. Res. 34, (2000) 21. Beck, A., Teboulle, M.: Global optimality conditions for quadratic optimization problems with binary constraints. SIAM J. Optim. 11(1), (2000) 22. Bector, C.R., Chandra, S.: Second-order duality for generalized fractional programming. Methods Oper. Res. 56, (1986) 23. Ben-Israel, A., Mond, B.: What is the invexity? J. Aust. Math. Soc. Ser. B 28, 1 9 (1986) 24. Bolintinéanu, S., El Maghri, M.: Second-order efficiency conditions and sensitivity of efficient points. J. Optim. Theory Appl. 98, (1998) 25. Bonnans, J.F., Cominetti, R., Shapiro, A.: Second-order optimality conditions based on parabolic second-order tangent sets. SIAM J. Optim. 9, (1999) 26. Burke, J.V.: Second-order necessary and sufficient conditions for convex NDO. Math. Prog. 38, (1987) 27. Burke, J.V., Ferris, M.C.: Characterization of solution sets of convex programs. Oper. Res. Lett. 10(1), (1991) 28. Burke, J.V., Ferris, M.C., Qian, M.: On the Clarke subdifferential of the distance function to a closed set. J. Math. Anal. Appl. 166, (1992) 29. Burke, J.V., Han, S.P.: A Gauss-Newton approach to solving generalized inequalities. Math. Oper. Res. 11, (1986) 30. Burke, J.V., More, J.J.: On the identification of active constraints. SIAM J. Numer. Anal. 25, (1988) 31. Cambini, A., Martein, L.: First and second order optimality conditions in vector optimization. J. Stat. Manag. Syst. 5, (2002) 32. Cambini, A., Martein, L., Vlach, M.: Second order tangent sets and optimality conditions. Math. Japonica 49, (1999) 33. Cambini, R.: Second order optimality conditions in multiobjective programming. Optimization 44, (1998) 34. Cambini, A., Martein, L.: First and second order optimality conditions in vector optimization. J. Stat. Manag. Syst. 5, (2002) 35. Chaney, R.: Second-order sufficiency conditions for nondifferentiable programming problems. SIAM J. Control Optim. 20, (1982) 36. Chaney, R.: Second order necessary conditions in constrained semismooth optimization. SIAM J. Control Optim. 25, (1987) 37. Chaney, R.: Second-order necessary conditions in semismooth optimization. Math. Prog. 40, (1988) 38. Chaney, R.: Second-order sufficient conditions in nonsmooth optimization. Math. Oper. Res. 13, (1988) 39. Charnes, A., Cooper, W.W.: Goal programming and multiobjective optimization, Part 1. Eur. J. Oper. Res. 1, (1977) 40. Chen, H., Hu, C.F.: On the resolution of the Vasicek-type interest rate model. Optimization 58, (2009) 41. Chinchuluum, A., Pardalos, P.M.: A survey of recent developments in multiobjective optimization. Ann. Oper. Res. 154, (2007) 42. Clarke, F.H.: Generalized gradients and applications. Trans. Am. Math. Soc. 205, (1975) 43. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983) 44. Cominetti, R.: Metric regularity, tangent sets, and second-order optimality conditions. Appl. Math. Optim. 21, (1990) 45. Craven, B.D.: Invex functions and constrained local minima. Bull. Aust. Math. Soc. 24(3), (1981) 46. Crespi, G., La Torre, D., Rocca, M.: Mollified derivatives and second-order optimality conditions. J. Nonlin. Convex Anal. 4, (2003)

3 References Daum, S., Werner, R.: A novel feasible discretization method for linear semi-infinite programming applied to basket option pricing. Optimization 60, (2011) 48. Dinkelbach, W.: On nonlinear fractional programming. Manag. Sci. 13(1967), (1967) 49. Dien, P.H., Sach, P.H.: Second-order optimality conditions for the extremal problem under inclusion constraints. Appl. Math. Optim. 20, (1989) 50. Egudo, R.R., Hanson, M.A.: Second order duality in multiobjective programming. Opsearch 30, (1993) 51. Floudas, C.A., Visweswaran, V.: Quadratic optimization. In: Horsi, R., Pardalos, P.M. (eds.) Handbook of Global Optimization, pp Kluwer Academic Publishers, Dordrecht 52. Ferris, M.C., Mangasarian, O.L.: Minmum principle sufficiency. Math. Program. 57(1), 1 14 (1992) 53. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to Theory of NP- Completeness. W.H. Freeman, San Francisco (1979) 54. Georgiev, P.G., Zlateva, N.: Second-order subdifferentials of C 1,1 functions and optimality conditions. Set-Valued Anal. 4, (1996) 55. Gfrerer, H.: Second-order optimality conditions for scalar and vector optimization problems in Banach spaces. SIAM J. Optim. 45, (2006) 56. Ginchev, I., Guerraggio, A.: Second-order optimality conditions in nonsmooth unconstrained optimization. Pliska Stud. Math. Bulg. 12, (1998) 57. Ginchev, I., Guerraggio, A., Rocca, M.: Second-order conditions for C 1,1 constrained vector optimization. Math. Prog. 104, (2005) 58. Ginchev, I., Guerraggio, A., Rocca, M.: Second-order conditions in C 1,1 vector optimization with inequality and equality constraints. In: Lecture Notes in Economics and Mathematical Systems, vol. 563, pp Springer, Berlin (2006) 59. Goberna, M.A., López, M.A.: Linear Semi-Infinite Optimization. Wiley, New York (1998) 60. Goberna, M.A., López, M.A.: Semi-infinite Programming-Recent Advances. Kluwer, Dordrecht (2001) 61. Goemans, M.X., Williamson, D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 42, (1995) 62. Guerraggio, A., Luc, D.T., Minh, N.B.: Second-order optimality conditions for C 1 multiobjective programming problems. Acta Math. Vietnam. 26, (2001) 63. Gutierrez, C., Jiménez, B., Novo, V.: New second-order derivatives and optimality conditions for scalar and vector optimization. J. Optim. Theory Appl. 142, (2009) 64. Hanson, M.A.: On sufficiency of the Kuhn-Tucker conditions. J. Math. Anal. Appl. 80, (1981) 65. Hanson, M.A.: Second order invexity and duality in mathematical programming. Opsearch 30, (1993) 66. Hanson, M.A., Mond, B.: Further generalization of convexity in mathematical programming. J. Inf. Optim. Sci. 3, (1982) 67. Hettich, R.: Semi-infinite Programming. Lecture Notes in Control and Information Sciences, vol. 7. Springer, Berlin (1976) 68. Hettich, R., Still, G.: Second order optimality conditions for generalized semi-infinite programming problems. Optimization 34, (1995) 69. Hettich, R.: An implementation of a discretization method for semi-infinite programming. Math. Program. 34, (1986) 70. Husain, Z., Ahmad, I., Sharma, S.: Second order duality for minmax fractional programming. Optim. Lett (2009) 71. Ito, S.A.: A dual parametrization method for convex semi-infinite programming. Ann. Oper. Res. 98, (2000) 72. Ivanov, V.I.: Second-order optimality conditions in nonsmooth optimization. C. R. Acad. Bulgare Sci. 60, (2007) 73. Jagannathan, R., Schaible, S.: Duality in generalized fractional programming via Farkas lemma. J. Optim. Theory Appl. 41(3), (1983)

4 286 References 74. Jeyakumar, V.: Strong and weak invexity in mathematical programming. Methods Oper. Res. 55, (1985) 75. Jeyakumar, V.: First and second order fractional programming duality. Opsearch 22, (1985) 76. Jiménez, B., Novo, V.: First and second order sufficient conditions for strict minimality in multiobjective programming. Numer. Funct. Anal. Optim. 23, (2002) 77. Jiménez, B., Novo, V.: First and second order sufficient conditions for strict minimality in nonsmooth vector optimization. J. Math. Anal. Appl. 284, (2003) 78. Jiménez, B., Novo, V.: Second-order necessary conditions in set constrained differentiable vector optimization. Math. Methods Oper. Res. 58, (2000) 79. Jiménez, B., Novo, V.: Optimality conditions in differentiable optimization via second-order tangent sets. Appl. Math. Optim. 49, (2004) 80. Jongen, HTh, Stein, O.: On generic one-parametric semi-infinite optimization. SIAM J. Optim. 7(4), (1997) 81. Jongen, HTh, Jonker, P., Twilt, F.: Nonlinear Optimization in Finite Dimension. Kluwer, Dordrecht (2000) 82. Jongen, H.Th., Wetterling, W., Zwier, G.: On sufficient conditions for local optimality in semi-infinite programming. Optimization 18, Kanniappan, P., Pandian, P.: On generalized convex functions in optimization theory-a survey. Opsearch 33, (1996) 84. Karney, D.F.: Clarke s theorem for semiinfinite convex programs. Adv. Appl. Math. 2, 7 12 (1981) 85. Karney, D.F.: A duality theorem for semi-infinite convex programs and their finite subprograms. Math. Program. 27, (1983) 86. Kawasaki, H.: Second-order necessary conditions of the Kuhn-Tucker type under new constraint qualifications. J. Optim. Theory Appl. 57, (1988) 87. Kawasaki, H.: An envelope-like effect of infinitely many inequality constraints on second order necessary conditions for minimization problems. Math. Prog. 41, (1988) 88. Kawasaki, H.: Second-order necessary and sufficient optimality conditions for minimizing a sup-type function. J. Appl. Math. Optim. 26, (1992) 89. Kelly, T.K., Kupperschmid, M.: Numerical verification of second-order sufficiency conditions for nonlinear programming. SIAM Rev. 40, (1998) 90. Khanh, P.Q., Tuan, N.D.: First and second-order optimality conditions using approximations for nonsmooth vector optimization in Banach spaces. J. Optim. Theory Appl. 130, (2006) 91. Khanh, P.Q., Tuan, N.D.: Optimality conditions for nonsmooth multiobjective optimization using Hadamard directional derivatives. J. Optim. Theory Appl. 133, (2007) 92. Kornbluth, J.S.H.: A survey of goal programming. Omega 1, (1973) 93. Li, Z.: Duality theorems for a class of generalized convex multuiobjective programming. Acta Sci. Natur. Univ. Neimenggu 24(2), (1993) 94. Kuk, H., Kim, H.-J.: Optimality conditions and duality for nonsmooth generalized fractional programming problems. J. Stat. Manag. 6, (2003) 95. Lee, G.M.: Nonsmooth invexity in multiobjective programming. J. Inf. Optim. Sci. 15, (1994) 96. Li, X.F., Dong, J.L., Liu, Q.H.: Lipschitz B vex functions and nonsmooth programming. JOTA 93, (1997) 97. Liang, Z.A., Shi, Z.W.: Optimality conditions and duality for minmax fractional programming with generalized convexity. JMAA 277, (2003) 98. Liu, J.C.: Second order duality for minimax programming. Util. Math. 56, (1999) 99. Liu, J.C.: Optimality and duality for generalized fractional programming involving nonsmooth (F,ρ)-convex functions. Comput. Math. Appl. 32, (1996) 100. Liu, J.C.: Optimality and duality for generalized fractional programming involving nonsmooth pseudoinvexity. JMAA 202, (1998)

5 References Liu, J.C., Wu, C.S.: On minmax fractional optimality conditions with ( f,ρ)-convex functions. JMAA 219, (1998) 102. Lopez, M., Still, G.: Semi-infinite programming. Eur. J. Oper. Res. 180, (2007) 103. Malivert, C.: First and second order optimality conditions in vector optimization. Ann. Sci. Math. Québec 14, (1990) 104. Maruyama, Y.: Second-order necessary conditions for nonlinear optimization problems in Banach spaces by the use of Neustadt derivative. Math. Japonica 40, (1994) 105. Maurer, H.: First- and second-order sufficient optimality conditions in mathematical programming and optimal control. Math. Prog. Study 14, (1981) 106. Maurer, H., Zowe, J.: First and second order necessary and sufficient optimality conditions for infinite-dimensional programming problems. Math. Prog. 16, (1979) 107. Mangasarian, O.L.: Nonlinear Programming. McGraw-Hill, New York (1969) 108. Mangasarian, O.L.: Second- and higher-order duality theorems in nonlinear programming. J. Math. Anal. Appl. 51, (1975) 109. Mangasarian, O.L.: Error bounds of nondegenerate monotone linear complementarity problems. Math. Program. 48, (1990) 110. Mangasarian, O.L.: A simple characterization of solution sets of convex programs. Oper. Res. Lett. 7(1), (1988) 111. Mangasarian, O.L., Meyer, R.R.: Nonlinear perturbation of linear programs. SIAM J. Control Optim. 17(6), (1979) 112. Martin, D.H.: The essence of invexity. J. Optim. Theory Appl. 47, (1985) 113. Miettinen, K.M.: Nonlinear Multiobjective Optimization. Kluwer Academic Publishers, Boston (1999) 114. Mishra, S.K.: Generalized proper efficiency and duality for a class of nondifferentiable multiobjective variational problems with V-invexity. J. Math. Anal. Appl. 2, (1996) 115. Mishra, S.K., Mukherjee, R.N.: On efficiency and duality for multiobjective variational problems. J. Math. Anal. Appl. 187, (1994) 116. Mishra, S.K., Mukherjee, R.N.: Duality for multiobjective fractional variational problems. J. Math. Anal. Appl. 186, (1994) 117. Mishra, S.K., Rueda, N.G.: Higher-order generalized invexity and duality in mathematical programming. J. Math. Anal. Appl. 247, (2000) 118. Mishra, S.K., Rueda, N.G.: Second-order duality for nondifferential minimax programming involving generalized type I functions. J. Optim. Theory Appl. 130, (2006) 119. Mishra, S.K., Jaiswal, M.: Pankaj: Optimality conditions for multiple objective fractional subset programming with invex and related non-convex functions. Commun. Appl. Nonlinear Anal. 17(3), (2010) 120. Mishra, S.K.: Second order generalized invexity and duality in mathematical programming. Optimization 42, (1997) 121. Mishra, S.K.: Second order symmetric duality in mathematical programming with F- convexity. Eur. J. Oper. Res. 127, (2000) 122. Mititelu, Ş.: Invex functions. Rev. Roum. Math. Pures Appl. 49, (2004) 123. Mititelu, Ş.: Invex sets and nonsmooth invex functions. Rev. Roum. Math. Pures Appl. 52, (2007) 124. Mititelu, Ş., Postolachi, M.: Nonsmooth invex functions via upper directional derivative of Dini. J. Adv. Math. Stud. 4, (2011) 125. Mititelu, Ş., Stancu-Minasian, I.M.: Invexity at a point: generalizations and classification. Bull. Austral. Math. Soc. 48, (1993) 126. Mond, B.: Second order duality for nonlinear programs. Opsearch 11, (1974) 127. Mond, B., Smart, I.: Duality and sufficiency in control problems with invexity. JMAA 136, (1988) 128. Mond, B., Weir, T.: Generalized convexity and higher-order duality. J. Math. Sci , ( ) 129. Mordukhovich, B.S.: Approximation Methods in Problems of Optimization and Control. Nauka, Moscow, Russian (1988)

6 288 References 130. Mukherjee, R.N., Mishra, S.K.: Duality for multiobjective fractional variational problems. J. Math. Anal. Appl. 186, (1994) 131. Nahak, C., Nanda, S.: Duality for multiobjective variational problems with pseudoinvexity. Optimization 41, (1997) 132. Nahak, C., Nanda, S.: On efficiency and duality for multiobjective variational control problems with convexity. JMAA 209, (1997) 133. Nahak, C., Nanda, S.: Duality for multiobjective variational control problems with invexity. Optimization 36, (1996) 134. Pandian, P.: On sufficiency and duality for (b, F, ρ)-convex multiobjective programs. Indian J. Pure Appl. Math. 33, (2002) 135. Patel, R.B.: Second order duality in multiobjective fractional programming. Indian J. Math. 38, (1997) 136. Pitea, A., Postolache, M.: Duality theorems for a new class of multitime multiobjective variational problems. J. Glob. Optim. 54, (2012) 137. Preda, V.: On efficiency and duality for multiobjective programs. J. Math. Anal. Appl. 166, (1992) 138. Reemtsen, R., Rückmann, J.-J.: Semi-Infinite Programming. Kluwer, Boston (1998) 139. Robinson, S.M.: Local structure of feasible sets in nonlinear programming. Part II: Nondegeneracy. Math. Program. 22(1), (1984) 140. Rockafellar, R.T.: Convex Analysis. Princeton University Press (1970) 141. Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, New York (2004) 142. Rückman, J.J., Shapiro, A.: Second-order optimality conditions in generalized semi-infinite programming. Set-Valued Anal. 9, (2001) 143. Sawaragi, Y., Nakayama, H., Tanino, T.: Theory of Multiobjective Optimization. Academic Press, New York (1986) 144. Schaible, S.: Parameter-free convex equivalent and dual programs of fractional programming problems. Z. Oper. Res. 18, (1974) 145. Schaible, S.: Minimization of ratios. J. Optim. Theory Appl. 19(2), (1976) 146. Schaible, S.: Fractional programming I, duality. Manag. Sci. 22(8), (1976) 147. Schaible, S.: Fractional programming: applications and algorithms. Eur. J. Oper. Res. 7(2), (1981) 148. Schaible, S., Ibaraki, T.: Fractional programming. Eur. J. Oper. Res. 12(3), (1983) 149. Schaible, S., Shi, J.: Fractional programming: the sum of ratios case. Optim. Methods Soft. 18(2), (2003) 150. Schaible, S.: Fractional programming: a recent survey. J. Stat. Manag. Syst. 5, (2002) 151. Schaible, S., Shi, J.: Recent developments in fractional programming: single ratio and maxmin case. In: Nonlinear Analysis and Convex Analysis, pp Yokohama Publishers, Yokohama (2004) 152. Sagan, H.: Introduction to the Calculus of Variations. McGraw-Hill, New York (1969) 153. Shapiro, A.: On duality theory of convex semi-infinite programming. Optimization 54, (2005) 154. Shapiro, A.: Semi-infinite programming, duality, discretization and optimality conditions. Optimization 58, (2009) 155. Shapiro, A.: On concepts of directional differentiability. J. Math. Anal. Appl. 66, (1990) 156. Shi, A.: A combined algorithm for fractional programming. Ann. Oper. Res. 103, (2001) 157. Shor, N.Z.: On a bounding method for quadratic extremal problems with 0 1 variables. Kibernetika 2, (1985) 158. Srivastava, M.K., Bhatia, M.: Symmetric duality for multiobjective programming using second order (F,ρ)-convexity. Opsearch 43, (2006) 159. Srivastava, K.K., Govil, M.G.: Second order duality for multiobjective programming involving (F,ρ,σ)-type I functions. Opsearch 37, (2000)

7 References Suneja, S.K., Lalitha, C.S., Khurana, S.: Second order symmetric duality in multiobjective programming. Eur. J. Oper. Res. 144, (2003) 161. Stancu-Minasian, I.M.: Fractional Programming: Theory Models and Applications. Kluwer, Dordrecht (1997) 162. Stein, O.: Bilevel Strategies in Semi-infinite Programming. Kluwer, Boston (2003) 163. Stein, O., Still, G.: On optimality conditions for generalized semi-infinite programming problems. J. Optim. Theory Appl. 104, (2000) 164. Stein, O., Still, G.: On generalized semi-infinite optimization and bilevel optimization. Eur. J. Oper. Res. 142(3), (2002) 165. Stein, O., Still, G.: Solving semi-infinite optimization problems with interior point techniques. SIAM J. Control Optim. 42(3), (2003) 166. Still, G.: Generalized semi-infinite programming: theory and methods. Eur. J. Oper. Res. 119, (1999) 167. Still, G.: Discretization in semi-infinite programming: The rate of convergence. Math. Program. Ser. A 91(1), (2001) 168. Vartak, M.N., Gupta, I.: Duality theory for fractional programming problems under η- convexity. Opsearch 24, (1987) 169. Verma, R.U.: Generalized higher order invexities in parametric optimality conditions for discrete minmax fractional programming. J. Basic Appl. Sci. 12, (2016) 170. Verma, R.U.: Parametric duality models for discrete minmax fractional programming problems on higher order invex functions. Ann. TPS 14, 3 19 (2016) 171. Verma, R.U.: Generalized hybrid invexities and efficiency conditions for multiobjective fractional programming. Tbilisi Math. J. 8(2), (2015) 172. Verma, R.U.: New ɛ optimality conditions for multiobjective fractional subset programming problems. Trans. Math. Program. Appl. 1(1), (2013) 173. Verma, R.U.: Second order (,,ρ,η,θ)-invexity frameworks and ɛ efficiency conditions for multiobjective fractional programming. Theory Appl. Math. Comput. Sci. 2(1), (2012) 174. Verma, R.U.: Role of second order B (ρ, η, θ, p, r)-invexities and parametric sufficient conditions in semiinfinite minimax fractional programming. Trans. Math. Program. Appl. 1(2), (2013) 175. Verma, R.U.: A generalization to Zalmai type second order univexities and applications to parametric duality models to discrete minimax fractional programming. Adv. Nonlinear Var. Inequal. 15(2), (2012) 176. Verma, R.U.: Generalized (G, b, β, φ, h, ρ, θ)-univexities with applications to parametric duality models for discrete minimax fractional programming. Trans. Math. Program. Appl. 1(1), 1 14 (2013) 177. Verma, R.U.: Advanced higher order univexities with applications to parametric duality models in discrete minimax fractional programming. Trans. Math. Program. Appl. 1(3), (2013) 178. Verma, R.U.: Hybrid (G, β, φ, ρ, θ, p, r)-univexities of higher-orders and applications to minimax fractional programming. Trans. Math. Program. Appl. 1(5), (2013) 179. Verma, R.U., Liu, J.: Higher-order parametric optimality conditions for discrete minmax fractional programming based on univexity. PanAm. Math. J. 27(1), (2017) 180. Verma, R.U., Mohapatra, R.N.: Generalized higher order hybrid invexities with second order parametric optimality criteria for discrete minmax fractional programming. Commun. Appl. Nonlinear Anal. 22(4), (2015) 181. Verma, R.U., Seol, Y.: Some sufficient efficient conditions in semiinfinite multiobjective fractional programming based on exponential type invexities. J. Inequal. Appl. 2015, 252 (2015). doi: /s Verma, R.U., Zalmai, G.J.: Second-order parametric optimality conditions in discrete minmax fractional programming. Commun. Appl. Nonlinear Anal. 23(3), 1 32 (2016) 183. Verma, R.U., Zalmai, G.J.: Generalized second-order parametric optimality conditions in discrete minmax fractional programming and second order univexity. Stat. Optim. Inf. Comput. 4, (2016)

8 290 References 184. Verma, R.U., Zalmai, G.J.: Generalized parametric duality models in discrete minmax fractional programming based on second-order optimality conditions. Commun. Appl. Nonlinear Anal. 22(2), (2015) 185. Verma, R.U., Zalmai, G.J.: Generalized second-order parameter-free optimality conditions in discrete minmax fractional programming. Commun. Appl. Nonlinear Anal. 22(2), (2015) 186. Verma, R.U., Zalmai, G.J.: Parameter-free duality models in discrete minmax fractional programming based on second-order optimality conditions. Trans. Math. Program. Appl. 2(11), 1 37 (2014) 187. Verma, R.U., Zalmai, G.J.: Generalized parametric duality models in discrete minmax fractional programming based on second-order optimality conditions. PanAm. Math. J. 26(3), (2016) 188. Verma, R.U., Zalmai, G.J.: New generation parametric optimality conditions for semiinfinite discrete minmax fractional programming problems. Trans. Math. Program. Appl. 5(1), (2017) 189. von Neumann, J.: A model of general economic equilibrium. Rev. Econ. Stud. 13, 1 9 (1945) 190. Wang, S.Y.: Second order necessary and sufficient conditions in multiobjective programming. Numer. Funct. Anal. Optim. 12, (1991) 191. Weber, G.-W., Alparslan-Gök, S.Z., Söyler, A.: A new mathematical approach in environmental and life sciences: gene-environment networks and their dynamics. Environ. Model. Assess. 14, (2009) 192. Weber, G.-W., Tezel, A.: On generalized semi-infinite optimization of genetic networks. TOP 15, (2007) 193. Werner, J.: Duality in generalized fractional programming. Int. Ser. Numer. Anal. 84, (1988) 194. Xu, Z.: Mixed type duality in multiobjective programming problems. JMAA 198, (1996) 195. Yang, X.M., Yang, X.Q., Teo, K.L., Hou, S.H.: Second order duality for nonlinear programming. Indian J. Pure Appl. Math. 35, (2004) 196. Yang, X.: Generalized convex duality for multiobjective fractional programs. Opsearch 31, (1994) 197. Yang, X.Q.: Second-order conditions in C 1,1 optimization with applications. Numer. Funct. Anal. Optim. 14, (1993) 198. Yang, X.Q.: Second-order global optimality conditions of convex composite optimization. Math. Prog. 81, (1998) 199. Yang, X.Q.: Second-order global optimality conditions for optimization problems. J. Glob. Optim. 30, (2004) 200. Yokoyama, K.: Epsilon approximate solutions for multiobjective programming problems. J. Math. Anal. Appl. 203(1), (1996) 201. Yang, X.M., Teo, K.L., Yang, X.Q.: Higher-order generalized convexity and duality in nondifferentiable multiobjective mathematical programming. J. Math. Anal. Appl. 29, (2004) 202. Yang, X.M., Yang, X.Q., Teo, K.L.: Nondifferentiable second order symmetric duality in mathematical programming with F-convexity. Eur. J. Oper. Res. 144, (2003) 203. Yang, X.M., Yang, X.Q., Teo, K.L.: Huard type second-order converse duality for nonlinear programming. Appl. Math. Lett. 18, (2005) 204. Yang, X., Yang, X.Q., Teo, K.L.: Higher-order symmetric duality in multiobjective programming with invexity. J. Ind. Manag. Optim. 4, (2008) 205. Zalmai, G.J.: Optimality conditions and duality for constrained measurable subset selection problems with minmax objective functions. Optimization 20, (1989) 206. Zalmai, G.J.: Generalized second-order (F,β,φ,ρ,θ)-univex functions and parametric duality models in semiinfinite discrete minmax fractional programming. Adv. Nonlinear Var. Inequal. 15, (2012)

9 References Zalmai, G.J.: General parametric sufficient optimality conditions for discrete minmax fractional programming problems containing generalized (θ,η,ρ)-v-invex functions and arbitrary norms. J. Appl. Math. Comput. 23(1 2), 1 23 (2007) 208. Zalmai, G.J.: Semiinfinite multiobjective fractional programming problems involving Hadamard directionally differentiable functions, Part I: Sufficient efficiency conditions. Trans. Math. Prog. Appl. 1(9), (2013) 209. Zalmai, G.J.: Semiinfinite multiobjective fractional programming problems involving Hadamard directionally differentiable functions, Part II: First-order parametric duality models. Trans. Math. Prog. Appl. 1(10), (2013) 210. Zalmai, G.J.: Zhang, Qinghong: Necessary efficiency conditions for semiinfinite multiobjective optimization problems involving Hadamard directionally differentiable functions. Trans. Math. Prog. Appl. 1(1), (2013) 211. Zalmai, G.J.: Hanson-Antczak-type generalized invex functions in semiinfinite minmax fractional programming, Part I: Sufficient optimality conditions. Commun. Appl. Nonlinear Anal. 19(4), 1 36 (2012) 212. Zalmai, G.J.: Hanson-Antczak-type generalized (α,β,γ,ξ,η,ρ,θ)-v- invex functions in semiinfinite multiobjective fractional programming Part I, Sufficient efficiency conditions. Adv. Nonlinear Var. Inequal. 16(1), (2013) 213. Zalmai, G.J.: Parametric duality models for discrete minmax fractional programming problems containing generalized (θ,η,ρ)-v-invex functions and arbitrary norms. J. Appl. Math. Comput. 24(1 2), (2007) 214. Zalmai, G.J., Zhang, Q.: Generalized (F,β,φ,ρ,θ)-univex functions and parametric duality models in semiinfinite discrete minmax fractional programming. Adv. Nonlinear Var. Inequal. 10(2), (2007) 215. Zalmai, G.J., Zhang, Q.: Global semiparametric sufficient effficient conditions for desiinfinite multiobjective fractional programming problems involving (α,η,ρ)-v-invex functions. Southeast Asian Bull. Math. 32, (2008) 216. Zalmai, G.J., Zhang, Q.: Semiinfinite multiobjective fractional programming, Part II: Duality models. J. Appl. Anal. 17, 1 35 (2011) 217. Zalmai, G.J., Zhang, Q.: Semiinfinite multiobjective fractional programming, Part I: Sufficient efficiency conditions. J. Appl. Anal. 16, (2010) 218. Zalmai, G.J.: Zhang, Qinghong: Generalized (F,β,φ,ρ,θ)-univex functions and global parametric sufficient optimality conditions in semiinfinite discrete minmax fractional programming. Panam. Math. J. 17, 1 26 (2007) 219. Zhang, J., Mond, B.: Second order b-invexity and duality in mathematical programming. Util. Math. 50, (1996) 220. Zhang, J., Mond, B.: Second order duality for multiobjective nonlinear programming involving generalized convexity. In: Glover, B.M., Craven, B.D., Ralph, D. (eds.) Proceedings of the Optimization Miniconference III, University of Ballarat, pp (1997) 221. Zalmai, G.J.: Zhang, Qinghong: Necessary efficiency conditions for semiinfinite multiobjective optimization problems involving Hadamard directionally differentiable functions. Trans. Math. Program. Appl. 1(1), (2013) 222. Zalmai, G.J., Verma, R.U.: Generalized parameter-free duality models in discrete minmax fractional programming based on second-order optimality conditions. Math. Sci. 10(4), (2016) 223. Zeidler, E.: Nonlinear Functional Analysis and its Applications III: Variational Methods and Optimization. Springer, New York (1985) 224. Zhang, J., Mond, B.: Duality for a nondifferentiable programming problem. Bull. Aust. Math. Soc. 55, (1997) 225. Zhian, L., Qingkai, Y.: Duality for a class of multiobjective control problems with generalized invexity. J. Math. Anal. Appl. 256, (2001)