American Institute of Mathematical Sciences

Advanced
  • Previous Article
    A new method for ranking decision making units using common set of weights: A developed criterion
  • JIMO Home
  • This Issue
  • Next Article
    Evaluation strategy and mass balance for making decision about the amount of aluminum fluoride addition based on superheat degree
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 & 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

[1]

Gaoxi Li, Zhongping Wan, Jia-wei Chen, Xiaoke Zhao. Necessary optimality condition for trilevel optimization problem. Journal of Industrial & Management Optimization, 2020, 16 (1) : 55-70. doi: 10.3934/jimo.2018140
[2]

Truong Q. Bao, Boris S. Mordukhovich. Refined necessary conditions in multiobjective optimization with applications to microeconomic modeling. Discrete & Continuous Dynamical Systems, 2011, 31 (4) : 1069-1096. doi: 10.3934/dcds.2011.31.1069
[3]

Xian-Jun Long, Nan-Jing Huang, Zhi-Bin Liu. Optimality conditions, duality and saddle points for nondifferentiable multiobjective fractional programs. Journal of Industrial & Management Optimization, 2008, 4 (2) : 287-298. doi: 10.3934/jimo.2008.4.287
[4]

Nuno R. O. Bastos, Rui A. C. Ferreira, Delfim F. M. Torres. Necessary optimality conditions for fractional difference problems of the calculus of variations. Discrete & Continuous Dynamical Systems, 2011, 29 (2) : 417-437. doi: 10.3934/dcds.2011.29.417
[5]

Xian-Jun Long, Jing Quan. Optimality conditions and duality for minimax fractional programming involving nonsmooth generalized univexity. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 361-370. doi: 10.3934/naco.2011.1.361
[6]

Henri Bonnel, Ngoc Sang Pham. Nonsmooth optimization over the (weakly or properly) Pareto set of a linear-quadratic multi-objective control problem: Explicit optimality conditions. Journal of Industrial & Management Optimization, 2011, 7 (4) : 789-809. doi: 10.3934/jimo.2011.7.789
[7]

Giancarlo Bigi. Componentwise versus global approaches to nonsmooth multiobjective optimization. Journal of Industrial & Management Optimization, 2005, 1 (1) : 21-32. doi: 10.3934/jimo.2005.1.21
[8]

Piernicola Bettiol, Nathalie Khalil. Necessary optimality conditions for average cost minimization problems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2093-2124. doi: 10.3934/dcdsb.2019086
[9]

Shahlar F. Maharramov. Necessary optimality conditions for switching control problems. Journal of Industrial & Management Optimization, 2010, 6 (1) : 47-55. doi: 10.3934/jimo.2010.6.47
[10]

Dariusz Idczak, Stanisław Walczak. Necessary optimality conditions for an integro-differential Bolza problem via Dubovitskii-Milyutin method. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2281-2292. doi: 10.3934/dcdsb.2019095
[11]

Sofia O. Lopes, Fernando A. C. C. Fontes, Maria do Rosário de Pinho. On constraint qualifications for nondegenerate necessary conditions of optimality applied to optimal control problems. Discrete & Continuous Dynamical Systems, 2011, 29 (2) : 559-575. doi: 10.3934/dcds.2011.29.559
[12]

Monika Dryl, Delfim F. M. Torres. Necessary optimality conditions for infinite horizon variational problems on time scales. Numerical Algebra, Control & Optimization, 2013, 3 (1) : 145-160. doi: 10.3934/naco.2013.3.145
[13]

Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2021, 11 (4) : 739-769. doi: 10.3934/mcrf.2020045
[14]

Yibing Lv, Zhongping Wan. Linear bilevel multiobjective optimization problem: Penalty approach. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1213-1223. doi: 10.3934/jimo.2018092
[15]

Ying Gao, Xinmin Yang, Kok Lay Teo. Optimality conditions for approximate solutions of vector optimization problems. Journal of Industrial & Management Optimization, 2011, 7 (2) : 483-496. doi: 10.3934/jimo.2011.7.483
[16]

Nazih Abderrazzak Gadhi, Khadija Hamdaoui. Optimality results for a specific fractional problem. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020157
[17]

Jianxiong Ye, An Li. Necessary optimality conditions for nonautonomous optimal control problems and its applications to bilevel optimal control. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1399-1419. doi: 10.3934/jimo.2018101
[18]

Stepan Sorokin, Maxim Staritsyn. Feedback necessary optimality conditions for a class of terminally constrained state-linear variational problems inspired by impulsive control. Numerical Algebra, Control & Optimization, 2017, 7 (2) : 201-210. doi: 10.3934/naco.2017014
[19]

M. Soledad Aronna. Second order necessary and sufficient optimality conditions for singular solutions of partially-affine control problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1233-1258. doi: 10.3934/dcdss.2018070
[20]

Yasmine Cherfaoui, Mustapha Moulaï. Biobjective optimization over the efficient set of multiobjective integer programming problem. Journal of Industrial & Management Optimization, 2021, 17 (1) : 117-131. doi: 10.3934/jimo.2019102

2020 Impact Factor: 1.801

Tools

Metrics

  • PDF downloads (448)
  • HTML views (1228)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences