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March  2020, 16(2): 623-631. doi: 10.3934/jimo.2018170

Optimality conditions for multiobjective fractional programming, via convexificators

Mansoureh Alavi Hejazi 1, and  Soghra Nobakhtian 1,2,,

1. 

Department of Mathematics, University of Isfahan, P.O. Box: 81745-163, Isfahan, Iran
2. 

School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box: 19395-5746, Tehran, Iran

* Corresponding author: S. Nobakhtian

Received  February 2016 Revised  July 2017 Published  March 2020 Early access  October 2018

In this paper, the idea of convexificators is used to derive the Karush-Kuhn-Tucker necessary optimality conditions for local weak efficient solutions of multiobjective fractional problems involving inequality and equality constraints. In this regard, several well known constraint qualifications are generalized and relationships between them are investigated. Moreover, some examples are provided to clarify our results.

Keywords: Multiobjective fractional optimization problem, convexificator, necessary optimality conditions, nonsmooth analysis.
Mathematics Subject Classification: Primary: 90C32, 49J52, 90C46.
Citation: Mansoureh Alavi Hejazi, Soghra Nobakhtian. Optimality conditions for multiobjective fractional programming, via convexificators. Journal of Industrial and Management Optimization, 2020, 16 (2) : 623-631. doi: 10.3934/jimo.2018170
References:
[1]

V. Azhmyakov, An approach to controlled mechanical systems based on the multiobjective optimization technique, J. Ind. Manag. Optim., 4 (2008), 697-712.  doi: 10.3934/jimo.2008.4.697.

[2]

M. S. Bazaraa, H. D. Sherali and C. M. Shetty, Nonlinear Programming: Theory and Algorithms (3rd ed.), John Wiley and Sons, Inc., Hoboken, New Jersey, 2006. doi: 10.1002/0471787779.

[3]

G. Bigi, Componentwise versus global approaches to nonsmooth multiobjective optimization, J. Ind. Manag. Optim., 1 (2005), 21-32.  doi: 10.3934/jimo.2005.1.21.

[4]

F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley-Interscience, New York, 1983.

[5]

C. Chen, T. C. E. Cheng, S. Li and X. Yang, Nonlinear augmented Lagrangian for nonconvex multiobjective optimization, J. Ind. Manag. Optim., 7 (2011), 151-174.  doi: 10.3934/jimo.2011.7.157.

[6]

V. F. Demyanov, Convexification and Concavification of a Positivity Homogeneous Function by the Same Family of Linear Functions, Universia di Pisa, Report 3, 1994.

[7]

V. F. Demyanov and V. Jeyakumar, Hunting for a smaller convex subdifferential, J. Glob. Optim., 10 (1997), 305-326.  doi: 10.1023/A:1008246130864.

[8]

V. F. Demyanov and A. M. Rubinov, Constructive Nonsmooth Analysis, Verlag Peter Leng, Franfurt, 1995.

[9]

J. Dutta and S. Chandra, Convexifcators, generalized convexity and vector optimzation, Optimization, 53 (2004), 77-94.  doi: 10.1080/02331930410001661505.

[10]

N. Gadhi, Necessary and sufficient optimality conditions for fractional multi-objective problems, Optimization, 57 (2008), 527-537.  doi: 10.1080/02331930701455945.

[11]

M. Golestani and S. Nobakhtian, Convexificators and strong Kuhn-Tucker conditions, Comput. Math. Appl., 64 (2012), 550-557.  doi: 10.1016/j.camwa.2011.12.047.

[12]

V. Jeyakumar and D. T. Luc, Approximate Jacobian matrices for nonsmooth continuous maps and C1-optimization, SIAM J. Control Optim., 36 (1998), 1815-1832.  doi: 10.1137/S0363012996311745.

[13]

V. Jeyakumar and D. T. Luc, Nonsmooth calculus, minimality, and monotonicity of convexificators, J. Optim. Theory Appl., 101 (1999), 599-621.  doi: 10.1023/A:1021790120780.

[14]

X. F. Li and J. Z. Zhang, Stronger Kuhn-Tucker type conditions in nonsmooth multiobjective optimization: Locally Lipschitz case, J. Optim. Theory Appl., 127 (2005), 367-388.  doi: 10.1007/s10957-005-6550-9.

[15]

Z. A. Liang, H. X. Huang and P. M. Pardalos, Efficiency conditions and duality for a class of multiobjective fractional programming problems, J. Glob. Optim., 27 (2003), 447-471.  doi: 10.1023/A:1026041403408.

[16]

X. J. Long, N. J. Huang and Z. B. Liu, Optimality conditions, duality and saddle points for nondifferentiable multiobjective fractional programs, J. Ind. Manag. Optim., 4 (2008), 287-298.  doi: 10.3934/jimo.2008.4.287.

[17]

D. V. Luu, Convexificators and necessary conditions for efficiency, Optimization, 63 (2014), 321-335.  doi: 10.1080/02331934.2011.648636.

[18]

P. Michel and J. P. Penot, A generalized derivative for calm and stable functions, Differ. Integr. Equ., 5 (1992), 433-454. 

[19]

B. S. Mordukhovich and Y. H. Shao, On nonconvex subdifferential calculus in Banach space, J. Convex Anal., 2 (1995), 211-227. 

[20]

S. Nobakhtian, Optimality and duality for nonsmooth multiobjective fractional programming with mixed constraints, J. Glob. Optim., 41 (2008), 103-115.  doi: 10.1007/s10898-007-9168-7.

[21]

R. T. Rochafellar and R. J. B. Wets, Variational Analysis, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-642-02431-3.

[22]

S. Schaible, A Survey of Fractional Programming, In: S. Schaible, W.T. Ziemba, (eds.) Generalized Concavity in Optimization and Economics, Academic Press, New York, 1981.

[23]

J. S. Treiman, The linear nonconvex generalized gradient and Lagrange multipliers, SIAM J. Optim., 5 (1995), 670-680.  doi: 10.1137/0805033.

show all references

References:
[1]

V. Azhmyakov, An approach to controlled mechanical systems based on the multiobjective optimization technique, J. Ind. Manag. Optim., 4 (2008), 697-712.  doi: 10.3934/jimo.2008.4.697.

[2]

M. S. Bazaraa, H. D. Sherali and C. M. Shetty, Nonlinear Programming: Theory and Algorithms (3rd ed.), John Wiley and Sons, Inc., Hoboken, New Jersey, 2006. doi: 10.1002/0471787779.

[3]

G. Bigi, Componentwise versus global approaches to nonsmooth multiobjective optimization, J. Ind. Manag. Optim., 1 (2005), 21-32.  doi: 10.3934/jimo.2005.1.21.

[4]

F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley-Interscience, New York, 1983.

[5]

C. Chen, T. C. E. Cheng, S. Li and X. Yang, Nonlinear augmented Lagrangian for nonconvex multiobjective optimization, J. Ind. Manag. Optim., 7 (2011), 151-174.  doi: 10.3934/jimo.2011.7.157.

[6]

V. F. Demyanov, Convexification and Concavification of a Positivity Homogeneous Function by the Same Family of Linear Functions, Universia di Pisa, Report 3, 1994.

[7]

V. F. Demyanov and V. Jeyakumar, Hunting for a smaller convex subdifferential, J. Glob. Optim., 10 (1997), 305-326.  doi: 10.1023/A:1008246130864.

[8]

V. F. Demyanov and A. M. Rubinov, Constructive Nonsmooth Analysis, Verlag Peter Leng, Franfurt, 1995.

[9]

J. Dutta and S. Chandra, Convexifcators, generalized convexity and vector optimzation, Optimization, 53 (2004), 77-94.  doi: 10.1080/02331930410001661505.

[10]

N. Gadhi, Necessary and sufficient optimality conditions for fractional multi-objective problems, Optimization, 57 (2008), 527-537.  doi: 10.1080/02331930701455945.

[11]

M. Golestani and S. Nobakhtian, Convexificators and strong Kuhn-Tucker conditions, Comput. Math. Appl., 64 (2012), 550-557.  doi: 10.1016/j.camwa.2011.12.047.

[12]

V. Jeyakumar and D. T. Luc, Approximate Jacobian matrices for nonsmooth continuous maps and C1-optimization, SIAM J. Control Optim., 36 (1998), 1815-1832.  doi: 10.1137/S0363012996311745.

[13]

V. Jeyakumar and D. T. Luc, Nonsmooth calculus, minimality, and monotonicity of convexificators, J. Optim. Theory Appl., 101 (1999), 599-621.  doi: 10.1023/A:1021790120780.

[14]

X. F. Li and J. Z. Zhang, Stronger Kuhn-Tucker type conditions in nonsmooth multiobjective optimization: Locally Lipschitz case, J. Optim. Theory Appl., 127 (2005), 367-388.  doi: 10.1007/s10957-005-6550-9.

[15]

Z. A. Liang, H. X. Huang and P. M. Pardalos, Efficiency conditions and duality for a class of multiobjective fractional programming problems, J. Glob. Optim., 27 (2003), 447-471.  doi: 10.1023/A:1026041403408.

[16]

X. J. Long, N. J. Huang and Z. B. Liu, Optimality conditions, duality and saddle points for nondifferentiable multiobjective fractional programs, J. Ind. Manag. Optim., 4 (2008), 287-298.  doi: 10.3934/jimo.2008.4.287.

[17]

D. V. Luu, Convexificators and necessary conditions for efficiency, Optimization, 63 (2014), 321-335.  doi: 10.1080/02331934.2011.648636.

[18]

P. Michel and J. P. Penot, A generalized derivative for calm and stable functions, Differ. Integr. Equ., 5 (1992), 433-454. 

[19]

B. S. Mordukhovich and Y. H. Shao, On nonconvex subdifferential calculus in Banach space, J. Convex Anal., 2 (1995), 211-227. 

[20]

S. Nobakhtian, Optimality and duality for nonsmooth multiobjective fractional programming with mixed constraints, J. Glob. Optim., 41 (2008), 103-115.  doi: 10.1007/s10898-007-9168-7.

[21]

R. T. Rochafellar and R. J. B. Wets, Variational Analysis, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-642-02431-3.

[22]

S. Schaible, A Survey of Fractional Programming, In: S. Schaible, W.T. Ziemba, (eds.) Generalized Concavity in Optimization and Economics, Academic Press, New York, 1981.

[23]

J. S. Treiman, The linear nonconvex generalized gradient and Lagrange multipliers, SIAM J. Optim., 5 (1995), 670-680.  doi: 10.1137/0805033.

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