• Previous Article
    Predicting 72-hour reattendance in emergency departments using discriminant analysis via mixed integer programming with electronic medical records
  • JIMO Home
  • This Issue
  • Next Article
    Fairness preference based decision-making model for concession period in PPP projects
doi: 10.3934/jimo.2018170

Optimality conditions for multiobjective fractional programming, via convexificators

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.

Citation: Mansoureh Alavi Hejazi, Soghra Nobakhtian. Optimality conditions for multiobjective fractional programming, via convexificators. Journal of Industrial & Management Optimization, 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. ChenT. C. E. ChengS. 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. LiangH. 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. LongN. 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. ChenT. C. E. ChengS. 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. LiangH. 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. LongN. 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.

[1]

Gaoxi Li, Zhongping Wan, Jia-wei Chen, Xiaoke Zhao. Necessary optimality condition for trilevel optimization problem. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-16. 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 - A, 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 - A, 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]

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

[9]

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 - A, 2011, 29 (2) : 559-575. doi: 10.3934/dcds.2011.29.559

[10]

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

[11]

Yibing Lv, Zhongping Wan. Linear bilevel multiobjective optimization problem: Penalty approach. Journal of Industrial & Management Optimization, 2018, 13 (5) : 1-11. doi: 10.3934/jimo.2018092

[12]

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

[13]

Jianxiong Ye, An Li. Necessary optimality conditions for nonautonomous optimal control problems and its applications to bilevel optimal control. Journal of Industrial & Management Optimization, 2018, 13 (5) : 1-21. doi: 10.3934/jimo.2018101

[14]

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

[15]

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

[16]

Ricardo Almeida. Optimality conditions for fractional variational problems with free terminal time. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 1-19. doi: 10.3934/dcdss.2018001

[17]

Xiao-Bing Li, Qi-Lin Wang, Zhi Lin. Optimality conditions and duality for minimax fractional programming problems with data uncertainty. Journal of Industrial & Management Optimization, 2018, 13 (5) : 1-19. doi: 10.3934/jimo.2018089

[18]

Ana Cristina Barroso, José Matias. Necessary and sufficient conditions for existence of solutions of a variational problem involving the curl. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 97-114. doi: 10.3934/dcds.2005.12.97

[19]

Monika Laskawy. Optimality conditions of the first eigenvalue of a fourth order Steklov problem. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1843-1859. doi: 10.3934/cpaa.2017089

[20]

Yong Xia. New sufficient global optimality conditions for linearly constrained bivalent quadratic optimization problems. Journal of Industrial & Management Optimization, 2009, 5 (4) : 881-892. doi: 10.3934/jimo.2009.5.881

2017 Impact Factor: 0.994

Metrics

  • PDF downloads (18)
  • HTML views (74)
  • Cited by (0)

Other articles
by authors

[Back to Top]