April  2019, 15(2): 791-815. doi: 10.3934/jimo.2018071

Henig proper efficiency in vector optimization with variable ordering structure

1. 

Faculty of Mathematics, "Alexandru Ioan Cuza" University, Bd. Carol Ⅰ, nr. 11, Iaşi, 700506, Romania

2. 

"Octav Mayer" Institute of Mathematics of the Romanian Academy, Bd. Carol Ⅰ, nr. 8, Iaşi, 700505, Romania

3. 

Department of Mathematics, "Gh. Asachi" Technical University, Bd. Carol Ⅰ, nr. 11, Iaşi, 700506, Romania

* Corresponding author: Marius Durea

Received  October 2016 Revised  March 2018 Published  June 2018

Fund Project: The work of the authors was supported by a grant of Romanian Ministry of Research and Innovation, CNCS-UEFISCDI, project number PN-Ⅲ-P4-ID-PCE-2016-0188, within PNCDI Ⅲ.

In this paper we introduce a notion of Henig proper efficiency for constrained vector optimization problems in the setting of variable ordering structure. In order to get an appropriate concept, we have to explore firstly the case of fixed ordering structure and to observe that, in certain situations, the well-known Henig proper efficiency can be expressed in a simpler way. Then, we observe that the newly introduced notion can be reduced, by a Clarke-type penalization result, to the notion of unconstrained robust efficiency. We show that this penalization technique, coupled with sufficient conditions for weak openness, serves as a basis for developing necessary optimality conditions for our Henig proper efficiency in terms of generalized differentiation objects lying in both primal and dual spaces.

Citation: Marius Durea, Elena-Andreea Florea, Radu Strugariu. Henig proper efficiency in vector optimization with variable ordering structure. Journal of Industrial & Management Optimization, 2019, 15 (2) : 791-815. doi: 10.3934/jimo.2018071
References:
[1]

J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkäuser, Basel, 1990. doi: 10.1007/978-0-8176-4848-0.  Google Scholar

[2]

F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983. doi: 10.1137/1.9781611971309.  Google Scholar

[3]

M. Durea, Some cone separation results and applications, Rev. Anal. Numer. Theor. Approx., 37 (2008), 37-46.   Google Scholar

[4]

M. DureaM. Panţiruc and R. Strugariu, Minimal time function with respect to a set of directions. Basic properties and applications, Optim. Methods Softw., 31 (2016), 535-561.  doi: 10.1080/10556788.2015.1121488.  Google Scholar

[5]

M. Durea and R. Strugariu, Generalized penalization and maximization of vectorial nonsmooth functions, Optimization, 66 (2017), 903-915.  doi: 10.1080/02331934.2016.1237515.  Google Scholar

[6]

M. DureaR. Strugariu and C. Tammer, On set-valued optimization problems with variable ordering structure, J. Global Optim., 61 (2015), 745-767.  doi: 10.1007/s10898-014-0207-x.  Google Scholar

[7]

G. Eichfelder, Variable Ordering Structures in Vector Optimization, Springer, Heidelberg, 2014. doi: 10.1007/978-3-642-54283-1.  Google Scholar

[8]

G. Eichfelder and T. Gerlach, Characterization of properly optimal elements with variable ordering structures, Optimization, 65 (2016), 571-588.  doi: 10.1080/02331934.2015.1040793.  Google Scholar

[9]

G. Eichfelder and R. Kasimbeyli, Properly optimal elements in vector optimization with variable ordering structures, J. Global Optim., 60 (2014), 689-712.  doi: 10.1007/s10898-013-0132-4.  Google Scholar

[10]

E.-A. Florea, Coderivative necessary optimality conditions for sharp and robust efficiencies in vector optimization with variable ordering structure, Optimization, 65 (2016), 1417-1435.  doi: 10.1080/02331934.2016.1152473.  Google Scholar

[11]

A. Göpfert, H. Riahi, C. Tammer and C. Zălinescu, Variational Methods in Partially Ordered Spaces, Springer, Berlin, 2003. doi: 10.1007/b97568.  Google Scholar

[12]

S. LiJ.-P. Penot and X. Xue, Codifferential calculus, Set-Valued Var. Anal., 19 (2011), 505-536.  doi: 10.1007/s11228-010-0171-7.  Google Scholar

[13]

B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, Vol. Ⅰ: Basic Theory, Springer, Berlin, 2006. doi: 10.1007/3-540-31247-1.  Google Scholar

[14]

H. V. NgaiH. T. Nguyen and M. Théra, Metric regularity of the sum of multifunctions and applications, J. Optim. Theory Appl., 160 (2014), 355-390.  doi: 10.1007/s10957-013-0385-6.  Google Scholar

[15]

J.-P. Penot, Cooperative behavior of functions, relations and sets, Math.Methods Oper. Res., 48 (1998), 229-246.  doi: 10.1007/s001860050025.  Google Scholar

[16]

R. T. Rockafellar, Proto-differentiability of set-valued mappings and its applications in optimization, Ann. Inst. H. Poincaré, 6 (1989), 449-482.  doi: 10.1016/S0294-1449(17)30034-3.  Google Scholar

[17]

Y. Xu and S. Li, Continuity of the solution mappings to parametric generalized non-weak vector Ky Fan inequalities, J. Ind. Manag. Optim., 13 (2017), 967-975.  doi: 10.3934/jimo.2016056.  Google Scholar

[18]

J. J. Ye, The exact penalty principle, Nonlinear Anal. TMA, 75 (2012), 1642-1654.  doi: 10.1016/j.na.2011.03.025.  Google Scholar

[19]

C. Zălinescu, Convex Analysis in General Vector Spaces, World Scientific, Singapore, 2002. doi: 10.1142/9789812777096.  Google Scholar

[20]

C. Zălinescu, Stability for a class of nonlinear optimization problems and applications, in Nonsmooth Optimization and Related Topics (Erice, 1989), Plenum, New York, 43 (1989), 437–458. doi: 10.1007/978-1-4757-6019-4_26.  Google Scholar

show all references

References:
[1]

J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkäuser, Basel, 1990. doi: 10.1007/978-0-8176-4848-0.  Google Scholar

[2]

F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983. doi: 10.1137/1.9781611971309.  Google Scholar

[3]

M. Durea, Some cone separation results and applications, Rev. Anal. Numer. Theor. Approx., 37 (2008), 37-46.   Google Scholar

[4]

M. DureaM. Panţiruc and R. Strugariu, Minimal time function with respect to a set of directions. Basic properties and applications, Optim. Methods Softw., 31 (2016), 535-561.  doi: 10.1080/10556788.2015.1121488.  Google Scholar

[5]

M. Durea and R. Strugariu, Generalized penalization and maximization of vectorial nonsmooth functions, Optimization, 66 (2017), 903-915.  doi: 10.1080/02331934.2016.1237515.  Google Scholar

[6]

M. DureaR. Strugariu and C. Tammer, On set-valued optimization problems with variable ordering structure, J. Global Optim., 61 (2015), 745-767.  doi: 10.1007/s10898-014-0207-x.  Google Scholar

[7]

G. Eichfelder, Variable Ordering Structures in Vector Optimization, Springer, Heidelberg, 2014. doi: 10.1007/978-3-642-54283-1.  Google Scholar

[8]

G. Eichfelder and T. Gerlach, Characterization of properly optimal elements with variable ordering structures, Optimization, 65 (2016), 571-588.  doi: 10.1080/02331934.2015.1040793.  Google Scholar

[9]

G. Eichfelder and R. Kasimbeyli, Properly optimal elements in vector optimization with variable ordering structures, J. Global Optim., 60 (2014), 689-712.  doi: 10.1007/s10898-013-0132-4.  Google Scholar

[10]

E.-A. Florea, Coderivative necessary optimality conditions for sharp and robust efficiencies in vector optimization with variable ordering structure, Optimization, 65 (2016), 1417-1435.  doi: 10.1080/02331934.2016.1152473.  Google Scholar

[11]

A. Göpfert, H. Riahi, C. Tammer and C. Zălinescu, Variational Methods in Partially Ordered Spaces, Springer, Berlin, 2003. doi: 10.1007/b97568.  Google Scholar

[12]

S. LiJ.-P. Penot and X. Xue, Codifferential calculus, Set-Valued Var. Anal., 19 (2011), 505-536.  doi: 10.1007/s11228-010-0171-7.  Google Scholar

[13]

B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, Vol. Ⅰ: Basic Theory, Springer, Berlin, 2006. doi: 10.1007/3-540-31247-1.  Google Scholar

[14]

H. V. NgaiH. T. Nguyen and M. Théra, Metric regularity of the sum of multifunctions and applications, J. Optim. Theory Appl., 160 (2014), 355-390.  doi: 10.1007/s10957-013-0385-6.  Google Scholar

[15]

J.-P. Penot, Cooperative behavior of functions, relations and sets, Math.Methods Oper. Res., 48 (1998), 229-246.  doi: 10.1007/s001860050025.  Google Scholar

[16]

R. T. Rockafellar, Proto-differentiability of set-valued mappings and its applications in optimization, Ann. Inst. H. Poincaré, 6 (1989), 449-482.  doi: 10.1016/S0294-1449(17)30034-3.  Google Scholar

[17]

Y. Xu and S. Li, Continuity of the solution mappings to parametric generalized non-weak vector Ky Fan inequalities, J. Ind. Manag. Optim., 13 (2017), 967-975.  doi: 10.3934/jimo.2016056.  Google Scholar

[18]

J. J. Ye, The exact penalty principle, Nonlinear Anal. TMA, 75 (2012), 1642-1654.  doi: 10.1016/j.na.2011.03.025.  Google Scholar

[19]

C. Zălinescu, Convex Analysis in General Vector Spaces, World Scientific, Singapore, 2002. doi: 10.1142/9789812777096.  Google Scholar

[20]

C. Zălinescu, Stability for a class of nonlinear optimization problems and applications, in Nonsmooth Optimization and Related Topics (Erice, 1989), Plenum, New York, 43 (1989), 437–458. doi: 10.1007/978-1-4757-6019-4_26.  Google Scholar

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