Document Type: Research Paper

**Authors**

College of Applied Mathematics, Chengdu University of Information Technology, Chengdu 610225, P.R. China

**Abstract**

The aim of present paper is to study a constrained programming with generalized $\alpha-$univex fuzzy mappings. In this paper we introduce the concepts of $\alpha-$univex, $\alpha-$preunivex, pseudo $\alpha-$univex and $\alpha-$unicave fuzzy mappings, and we discover that $\alpha-$univex fuzzy mappings are more general than univex fuzzy mappings. Then, we discuss the relationships of generalized $\alpha-$univex fuzzy mappings and get some properties. In the last, we derive necessary and sufficient Karush-Kuhn-Tucker conditions and its dual problems with generalized differentiable $\alpha-$univex fuzzy mappings for fuzzy constrained programming problem.

**Keywords**

[1] B. Aghezzaf, M. Hachimi, Generalized invexity and duality in multiobjective programming problems, Journal of

Global Optimization, 18(1) (2000), 91-101.

[2] T. Antczak, Relationships between pre-invex concepts, Nonlinear Analysis, Theory, Methods and Applications, 60(2)

(2005), 349-367.

[3] M. S. Bazaraa, H. D. Sherali, C. M. Shetty, Nonlinear programming: Theory and algorithms, Wiley Interscience

Publishers, 2006.

[4] C. R. Bector, S. Chandra, S. Gupt, S. K. Suneja, Univex functions and univex nonlinear programming, In: Proceedings

of the Administrative Sciences Association of Canada, 1 (1992), 115-124.

[5] B. Bede, S. G. Gal, Generalizations of the differentiability of fuzzy-number-valued functions with applications to

fuzzy differential equations, Fuzzy Sets and Systems, 151(3) (2005), 581-599.

[6] A. Benisrael, B. Mond, What is invexity?, Journal of the Australian Mathematical Society Ser. B, 28(1) (1986),

1-9.

[7] C. R. Bector, C. Singh, B-vex functions, Journal of Optimization Theory and Applications, 71(2) (1991), 237-253.

[8] B. Bede, L. Stefanini, Generalized differentiability of fuzzy-valued functions, Fuzzy Sets and Systems, 230 (2013),

119-141.

[9] C. R. Bector, S. K. Suneja, C. S. Lalitha, Generalized B-vex functions and generalized B-vex programming, Journal

of Optimization Theory and Applications, 76(3) (1993), 561-576.

[10] Y. Chalcocano, W. A. Lodwick, R. Osuna-Gomez, A. Ruanlizana, The Karush-Kuhn-Tucker optimality conditions

for fuzzy optimization problems, Fuzzy Optimization and Decision Making, 15(1) (2016), 57-73.

[11] Y. Chalcocano, W. A. Lodwick, H. Romanflores, The Karush-Kuhn-Tucker optimality conditions for a class of

fuzzy optimization problems using strongly generalized derivative, Joint Ifsa World Congress and Naps Annual

Meeting, (2013), 203-208.

[12] Y. Chalcocano, W. A. Lodwick, A. Rufianlizana, Optimality conditions of type KKT for optimization problem with

interval-valued objective function via generalized derivative, Fuzzy Optimization and Decision Making, 12(3) (2013),

305-322.

[13] Y. Chalcocano, H. Roman-Flores, M. D. Jimenez-Gamero, Generalized derivative and π-derivative for set-valued

functions, Information Sciences, 181(11) (2011), 2177-2188.

Global Optimization, 18(1) (2000), 91-101.

[2] T. Antczak, Relationships between pre-invex concepts, Nonlinear Analysis, Theory, Methods and Applications, 60(2)

(2005), 349-367.

[3] M. S. Bazaraa, H. D. Sherali, C. M. Shetty, Nonlinear programming: Theory and algorithms, Wiley Interscience

Publishers, 2006.

[4] C. R. Bector, S. Chandra, S. Gupt, S. K. Suneja, Univex functions and univex nonlinear programming, In: Proceedings

of the Administrative Sciences Association of Canada, 1 (1992), 115-124.

[5] B. Bede, S. G. Gal, Generalizations of the differentiability of fuzzy-number-valued functions with applications to

fuzzy differential equations, Fuzzy Sets and Systems, 151(3) (2005), 581-599.

[6] A. Benisrael, B. Mond, What is invexity?, Journal of the Australian Mathematical Society Ser. B, 28(1) (1986),

1-9.

[7] C. R. Bector, C. Singh, B-vex functions, Journal of Optimization Theory and Applications, 71(2) (1991), 237-253.

[8] B. Bede, L. Stefanini, Generalized differentiability of fuzzy-valued functions, Fuzzy Sets and Systems, 230 (2013),

119-141.

[9] C. R. Bector, S. K. Suneja, C. S. Lalitha, Generalized B-vex functions and generalized B-vex programming, Journal

of Optimization Theory and Applications, 76(3) (1993), 561-576.

[10] Y. Chalcocano, W. A. Lodwick, R. Osuna-Gomez, A. Ruanlizana, The Karush-Kuhn-Tucker optimality conditions

for fuzzy optimization problems, Fuzzy Optimization and Decision Making, 15(1) (2016), 57-73.

[11] Y. Chalcocano, W. A. Lodwick, H. Romanflores, The Karush-Kuhn-Tucker optimality conditions for a class of

fuzzy optimization problems using strongly generalized derivative, Joint Ifsa World Congress and Naps Annual

Meeting, (2013), 203-208.

[12] Y. Chalcocano, W. A. Lodwick, A. Rufianlizana, Optimality conditions of type KKT for optimization problem with

interval-valued objective function via generalized derivative, Fuzzy Optimization and Decision Making, 12(3) (2013),

305-322.

[13] Y. Chalcocano, H. Roman-Flores, M. D. Jimenez-Gamero, Generalized derivative and π-derivative for set-valued

functions, Information Sciences, 181(11) (2011), 2177-2188.

[14] S. S. Chang, L. A. Zadeh, On fuzzy mapping and control, IEEE Transactions on Systems, Man and Cybernetics,

2(1) (1972), 30-34.

[15] P. Diamond, P. Kloeden, Metric spaces of fuzzy sets: Theory and applications, World Scientic Publishing Co.,

Inc., River Edge, N,J., 1994.

[16] R. Goetschel, W. Voxman, Elementary fuzzy calculus, Fuzzy Sets and Systems, 18(1) (1986), 31-43.

[17] M. A. Hanson, On sufficiency of the Kuhn-Tucker conditions, Journal of Mathematical Analysis and Applications,

80(2) (1981), 545-550.

[18] A. Jayswal, R. Kumar, Some nondifferentiable multiobjective programming under generalized d-V-type-I univexity,

Journal of Computational and Applied Mathematics, 229(1) (2009), 175-182.

[19] O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems, 24(3) (1987), 301-317.

[20] L. F. Li, S. Y. Liu, J. K. Zhang, On fuzzy generalized convex mappings and optimality conditions for fuzzy weakly

univex mappings, Fuzzy Sets and Systems, 280 (2015), 107-132.

[21] C. P. Liu, Some characterizations and applications on strongly α-preinvex and strongly α-invex functions, Journal

of Industrial & Management Optimization, 4(4) (2008), 727-738.

[22] S. K. Mishra, R. P. Pant, J. S. Rautela, Generalized α-invexity and nondifferentiable minimax fractional program-

ming, Journal of Computational and Applied Mathematics, 206(1) (2007), 122-135.

[23] S. K. Mishra, R. P. Pant, J. S. Rajendra, Generalized α-univexity and duality for nondifferentiable minimax

fractional programming, Nonlinear Analysis, Theory, Methods and Applications, 70(1) (2009), 144-158.

[24] S. K. Mishra, S. Y. Wang, K. K. Lai, Optimality and duality for multiple-objective optimization under generalized

type I univexity, Journal of Mathematical Analysis and Applications, 303(1) (2005), 315-326.

[25] S. R. Mohan, S. K. Neogy, On invex sets and preinvex functions, Journal of Mathematical Analysis and Applications,

189(3) (1995), 901-908.

[26] S. Nanda, K. Kar, Convex fuzzy mappings, Fuzzy Sets and Systems, 48(1) (1992), 129-132.

[27] M. A. Noor, Fuzzy preinvex functions, Fuzzy Sets and Systems, 64(1) (1994), 95-104.

[28] M. A. Noor, On generalized preinvex functions and monotonicities, Journal of Inequalities in Pure and Applied

Mathematics, 5(4) (2004), 1-9.

[29] R. Osuna-Gomez, Y. Chalcocano, B. Hernandezjimenez, I. Aguirrecipe, Optimality conditions for fuzzy constrained

programming problems, Fuzzy Sets and Systems, 362 (2019), 35-54.

[30] R. Osuna-Gomez, Y. Chalcocano, A. Ruanlizana, B. Hernandezjimenez, Necessary and sufficient conditions for

fuzzy optimality problems, Fuzzy Sets and Systems, 296 (2016), 112-123.

[31] R. Osuna-Gomez, B. Hernandezjimenez, Y. Chalcocano, G. Ruizgarzon, New efficiency conditions for multiobjective

interval-valued programming problems, Information Sciences, 420 (2017) 235-248.

[32] M. Panigrahi, G. Panda, S. Nanda, Convex fuzzy mapping with differentiability and its application in fuzzy opti-

mization, European Journal of Operational Research, 185(1) (2008), 47-62.

[33] R. P. Pant, J. S. Rautela, α-invexity and duality in mathematical programming, Journal of Mathematical Analysis

and Approximation Theory, 2 (2006), 104-114.

[34] M. L. Puri, D. A. Ralescu, Differentials of fuzzy functions, Journal of Mathematical Analysis and Applications,

91(2) (1983), 552-558.

[35] J. S. Rautela, R. P. Pant, Duality in mathematical programming under α-univexity, Journal of Mathematical

Analysis and Approximation Theory, 1 (2007), 72-83.

[36] L. Stefanini, M. Arana-Jimenez, Karush-Kuhn-Tucker conditions for interval and fuzzy optimization in several

variables under total and directional generalized differentiability, Fuzzy Sets and Systems, 362 (2019), 1-34.

2(1) (1972), 30-34.

[15] P. Diamond, P. Kloeden, Metric spaces of fuzzy sets: Theory and applications, World Scientic Publishing Co.,

Inc., River Edge, N,J., 1994.

[16] R. Goetschel, W. Voxman, Elementary fuzzy calculus, Fuzzy Sets and Systems, 18(1) (1986), 31-43.

[17] M. A. Hanson, On sufficiency of the Kuhn-Tucker conditions, Journal of Mathematical Analysis and Applications,

80(2) (1981), 545-550.

[18] A. Jayswal, R. Kumar, Some nondifferentiable multiobjective programming under generalized d-V-type-I univexity,

Journal of Computational and Applied Mathematics, 229(1) (2009), 175-182.

[19] O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems, 24(3) (1987), 301-317.

[20] L. F. Li, S. Y. Liu, J. K. Zhang, On fuzzy generalized convex mappings and optimality conditions for fuzzy weakly

univex mappings, Fuzzy Sets and Systems, 280 (2015), 107-132.

[21] C. P. Liu, Some characterizations and applications on strongly α-preinvex and strongly α-invex functions, Journal

of Industrial & Management Optimization, 4(4) (2008), 727-738.

[22] S. K. Mishra, R. P. Pant, J. S. Rautela, Generalized α-invexity and nondifferentiable minimax fractional program-

ming, Journal of Computational and Applied Mathematics, 206(1) (2007), 122-135.

[23] S. K. Mishra, R. P. Pant, J. S. Rajendra, Generalized α-univexity and duality for nondifferentiable minimax

fractional programming, Nonlinear Analysis, Theory, Methods and Applications, 70(1) (2009), 144-158.

[24] S. K. Mishra, S. Y. Wang, K. K. Lai, Optimality and duality for multiple-objective optimization under generalized

type I univexity, Journal of Mathematical Analysis and Applications, 303(1) (2005), 315-326.

[25] S. R. Mohan, S. K. Neogy, On invex sets and preinvex functions, Journal of Mathematical Analysis and Applications,

189(3) (1995), 901-908.

[26] S. Nanda, K. Kar, Convex fuzzy mappings, Fuzzy Sets and Systems, 48(1) (1992), 129-132.

[27] M. A. Noor, Fuzzy preinvex functions, Fuzzy Sets and Systems, 64(1) (1994), 95-104.

[28] M. A. Noor, On generalized preinvex functions and monotonicities, Journal of Inequalities in Pure and Applied

Mathematics, 5(4) (2004), 1-9.

[29] R. Osuna-Gomez, Y. Chalcocano, B. Hernandezjimenez, I. Aguirrecipe, Optimality conditions for fuzzy constrained

programming problems, Fuzzy Sets and Systems, 362 (2019), 35-54.

[30] R. Osuna-Gomez, Y. Chalcocano, A. Ruanlizana, B. Hernandezjimenez, Necessary and sufficient conditions for

fuzzy optimality problems, Fuzzy Sets and Systems, 296 (2016), 112-123.

[31] R. Osuna-Gomez, B. Hernandezjimenez, Y. Chalcocano, G. Ruizgarzon, New efficiency conditions for multiobjective

interval-valued programming problems, Information Sciences, 420 (2017) 235-248.

[32] M. Panigrahi, G. Panda, S. Nanda, Convex fuzzy mapping with differentiability and its application in fuzzy opti-

mization, European Journal of Operational Research, 185(1) (2008), 47-62.

[33] R. P. Pant, J. S. Rautela, α-invexity and duality in mathematical programming, Journal of Mathematical Analysis

and Approximation Theory, 2 (2006), 104-114.

[34] M. L. Puri, D. A. Ralescu, Differentials of fuzzy functions, Journal of Mathematical Analysis and Applications,

91(2) (1983), 552-558.

[35] J. S. Rautela, R. P. Pant, Duality in mathematical programming under α-univexity, Journal of Mathematical

Analysis and Approximation Theory, 1 (2007), 72-83.

[36] L. Stefanini, M. Arana-Jimenez, Karush-Kuhn-Tucker conditions for interval and fuzzy optimization in several

variables under total and directional generalized differentiability, Fuzzy Sets and Systems, 362 (2019), 1-34.

[37] L. Stefanini, A generalization of Hukuhara difference and division for interval and fuzzy arithmetic, Fuzzy Sets and

Systems, 161(11) (2010), 1564-1584.

[38] Y. Syau, Generalization of preinvex and B-vex fuzzy mappings, Fuzzy Sets and Systems, 120(3) (2001), 533-542.

[39] Y. Syau, Invex and generalized convex fuzzy mappings, Fuzzy Sets and Systems, 115(3) (2000), 455-461.

[40] A. K. Tripathy, G. Devi, Mixed type duality for nondifferentiable multiobjective fractional programming under

generalized (d, ρ, φ, θ)-type I univex function, Applied Mathematics and Computation, 219 (2013), 9196-9201.

[41] G. X. Wang, C. X. Wu, Directional derivatives and subdifferential of convex fuzzy mappings and application in

convex fuzzy programming, Fuzzy Sets and Systems, 138(3) (2003), 559-591.

[42] H. Wu, The Karush-Kuhn-Tucker optimality conditions for multi-objective programming problems with fuzzy-valued

objective functions, Fuzzy Optimization and Decision Making, 8(1) (2009), 1-28.

[43] H. Wu, The Karush-Kuhn-Tucker optimality conditions for the optimization problem with fuzzy-valued objective

function, Mathematical Methods of Operations Research, 66(2) (2007), 203-224.

[44] Z. Z.Wu, J. P. Xu, Generalized convex fuzzy mappings and fuzzy variational-like inequality, Fuzzy Sets and Systems,

160(11) (2009), 1590-1619.

[45] Z. Z. Wu, J. P. Xu, Nonconvex fuzzy mappings and the fuzzy pre-variational inequality, Fuzzy Sets and Systems,

159(16) (2008), 2090-2103.

[46] X. M. Yang, D. Li, On properties of preinvex functions, Journal of Mathematical Analysis and Applications, 256(1)

(2001), 229-241.

Systems, 161(11) (2010), 1564-1584.

[38] Y. Syau, Generalization of preinvex and B-vex fuzzy mappings, Fuzzy Sets and Systems, 120(3) (2001), 533-542.

[39] Y. Syau, Invex and generalized convex fuzzy mappings, Fuzzy Sets and Systems, 115(3) (2000), 455-461.

[40] A. K. Tripathy, G. Devi, Mixed type duality for nondifferentiable multiobjective fractional programming under

generalized (d, ρ, φ, θ)-type I univex function, Applied Mathematics and Computation, 219 (2013), 9196-9201.

[41] G. X. Wang, C. X. Wu, Directional derivatives and subdifferential of convex fuzzy mappings and application in

convex fuzzy programming, Fuzzy Sets and Systems, 138(3) (2003), 559-591.

[42] H. Wu, The Karush-Kuhn-Tucker optimality conditions for multi-objective programming problems with fuzzy-valued

objective functions, Fuzzy Optimization and Decision Making, 8(1) (2009), 1-28.

[43] H. Wu, The Karush-Kuhn-Tucker optimality conditions for the optimization problem with fuzzy-valued objective

function, Mathematical Methods of Operations Research, 66(2) (2007), 203-224.

[44] Z. Z.Wu, J. P. Xu, Generalized convex fuzzy mappings and fuzzy variational-like inequality, Fuzzy Sets and Systems,

160(11) (2009), 1590-1619.

[45] Z. Z. Wu, J. P. Xu, Nonconvex fuzzy mappings and the fuzzy pre-variational inequality, Fuzzy Sets and Systems,

159(16) (2008), 2090-2103.

[46] X. M. Yang, D. Li, On properties of preinvex functions, Journal of Mathematical Analysis and Applications, 256(1)

(2001), 229-241.

Volume 16, Issue 5

September and October 2019

Pages 77-95