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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  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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[8]

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

[9]

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

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[18]

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

[19]

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

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[23]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[8]

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

[9]

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

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[18]

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

[19]

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

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[23]

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

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