doi: 10.3934/jimo.2021199
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Image space analysis for uncertain multiobjective optimization problems: Robust optimality conditions

1. 

College of Management, Chongqing College of Humanities, Science & Technology, Chongqing 401524, China

2. 

Department of Mathematics and Statistics, King Fahd University of Petroleum & Minerals, Dhahran, Saudi Arabia

3. 

Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, India

4. 

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

Corresponding author: Jiawei Chen

Received  April 2021 Revised  August 2021 Early access November 2021

We introduce the $ \mathcal{C} $-robust efficient solution and optimistic $ \mathcal{C} $-robust efficient solution of uncertain multiobjective optimization problems (UMOP). By using image space analysis, robust optimality conditions as well as saddle point sufficient optimality conditions for uncertain multiobjective optimization problems are established based on real-valued linear (regular) weak separation function and real-valued (vector-valued) nonlinear (regular) weak separation functions. We also introduce two inclusion problems by using the image sets of robust counterpart of (UMOP) and establish the relations between the solution of the inclusion problems and the $ \mathcal{C} $-robust efficient solution (respectively, optimistic $ \mathcal{C} $-robust efficient solution) of (UMOP).

Citation: Xiaoqing Ou, Suliman Al-Homidan, Qamrul Hasan Ansari, Jiawei Chen. Image space analysis for uncertain multiobjective optimization problems: Robust optimality conditions. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021199
References:
[1]

Q. H. AnsariE. Köbis and P. K. Sharma, Characterizations of multiobjective robustness via oriented distance function and image space analysis, J. Optim. Theory Appl., 181 (2019), 817-839.  doi: 10.1007/s10957-019-01505-y.

[2]

A. Beck and A. Ben-Tal, Duality in robust optimization: Primal worst equals dual best, Oper. Res. Lett., 37 (2009), 1-6.  doi: 10.1016/j.orl.2008.09.010.

[3] A. Ben-TalL. El Ghaoui and A. Nemirovski, Robust Optimization, Princeton University Press, Princeton, 2009.  doi: 10.1515/9781400831050.
[4]

A. Ben-Tal and A. Nemirovski, Robust solutions of linear programming problems contaminated with uncertain data, Math. Program. Ser. A, 88 (2000), 411-424.  doi: 10.1007/PL00011380.

[5]

D. BertsimasD. B. Brown and C. Caramanis, Theory and applications of robust optimization, SIAM Rev., 53 (2011), 464-501.  doi: 10.1137/080734510.

[6]

J. ChenY. J. Cho and Z. Wan, Optimality conditions for cone constrained nonsmooth multiobjective optimization, J. Nonlinear Convex Anal., 17 (2016), 1627-1642. 

[7]

J. ChenL. Huang and S. Li, Separations and optimality of constrained multiobjective optimization via improvement sets, J. Optim. Theory Appl., 178 (2018), 794-823.  doi: 10.1007/s10957-018-1325-2.

[8]

J. ChenE. KöbisM. Köbis and J.-C. Yao, Image space analysis for constrained inverse vector variational inequalities via multiobjective optimization, J. Optim. Theory Appl., 177 (2018), 816-834.  doi: 10.1007/s10957-017-1197-x.

[9]

J. ChenE. Köbis and J.-C. Yao, Optimality conditions and duality for robust nonsmooth multiobjective optimization problems with constraints, J. Optim. Theory Appl., 181 (2019), 411-436.  doi: 10.1007/s10957-018-1437-8.

[10]

J. ChenS. Li and J.-C. Yao, Vector-valued separation functions and constrained vector optimization problems: Optimality and saddle points, J. Ind. Manag. Optim., 16 (2020), 707-724.  doi: 10.3934/jimo.2018174.

[11]

T. D. Chuong, Optimality and duality for robust multiobjective optimization problems, Nonlinear Anal., 134 (2016), 127-143.  doi: 10.1016/j.na.2016.01.002.

[12]

C. Gerth and P. Weidner, Nonconvex separation theorems and some applications in vector optimization, J. Optim. Theory Appl., 67 (1990), 297-320.  doi: 10.1007/BF00940478.

[13]

F. Giannessi, Constrained Optimization and Image Space Analysis: Separation of Sets and Optimality Conditions, Vol. 1, Springer, Berlin, 2005.

[14]

J.-B. Hiriart-Urruty, Tangent cone, generalized gradients and mathematical programming in Banach spaces, Math. Oper. Res., 4 (1979), 79-97.  doi: 10.1287/moor.4.1.79.

[15]

L. Huang and J. Chen, Weighted robust optimality of convex optimization problems with data uncertainty, Optim. Lett., 14 (2020), 1089-1105.  doi: 10.1007/s11590-019-01406-z.

[16]

V. JeyakumarG. M. Lee and G. Li, Characterizing robust solution sets of convex programs under data uncertainty, J. Optim. Theory Appl., 164 (2015), 407-435.  doi: 10.1007/s10957-014-0564-0.

[17]

A. A. Khan, C. Tammer and C. Zǎlinescu, Set-Valued Optimization: An Introduction with Applications, Springer, Berlin, 2015. doi: 10.1007/978-3-642-54265-7.

[18]

K. KlamrothE. KöbisA. Schöbel and C. Tammer, A unified approach for different kinds of robustness and stochastic programming via nonlinear scalarizing functionals, Optim., 62 (2013), 649-671.  doi: 10.1080/02331934.2013.769104.

[19]

K. KlamrothE. KöbisA. Schöbel and C. Tammer, A unified approach to uncertain optimization, Eur. J. Oper. Res., 260 (2017), 403-420.  doi: 10.1016/j.ejor.2016.12.045.

[20]

S. LiY. XuM. You and S. Zhu, Constrained extremum problems and image space analysis-Part I: optimality conditions and Part II: Duality and penalization, J. Optim. Theory Appl., 177 (2018), 637-659.  doi: 10.1007/s10957-018-1248-y.

[21]

G. Mastroeni, Nonlinear separation in the image space with applications to penalty methods, Appl. Anal., 91 (2012), 1901-1914.  doi: 10.1080/00036811.2011.614603.

[22]

A. L. Soyster, Convex programming with set-inclusive constraints and applications to inexact linear programming, Oper. Res., 21 (1973), 1154-1157. 

[23]

H.-Z. WeiC.-R. Chen and S.-J. Li, Characterizations for optimality conditions of general robust optimization problems, J. Optim. Theory Appl., 177 (2018), 835-856.  doi: 10.1007/s10957-018-1256-y.

[24]

H.-Z. WeiC.-R. Chen and S.-J. Li, A unified characterization of multiobjective robustness via separation, J. Optim. Theory Appl., 179 (2018), 86-102.  doi: 10.1007/s10957-017-1196-y.

[25]

H.-Z. WeiC.-R. Chen and S.-J. Li, Characterizations of multiobjective robustness on vectorization counterparts, Optim., 69 (2020), 493-518.  doi: 10.1080/02331934.2019.1625352.

[26]

H.-Z. WeiC.-R. Chen and S.-J. Li, A unified approach through image space analysis to robustness in uncertain optimization problems, J. Optim. Theory Appl., 184 (2020), 466-493.  doi: 10.1007/s10957-019-01609-5.

[27]

H.-Z. WeiC.-R. Chen and S.-J. Li, Robustness characterizations for uncertain optimization problems via image space analysis, J. Optim. Theory Appl., 186 (2020), 459-479.  doi: 10.1007/s10957-020-01709-7.

[28]

Y. D. Xu, Nonlinear separation functions, optimality conditions and error bounds for Ky Fan quasi-inequalities, Optim. Lett., 10 (2016), 527-542.  doi: 10.1007/s11590-015-0879-2.

[29]

Y. D. Xu, Nonlinear separation approach to inverse variational inequalities, Optim., 65 (2016), 1315-1335.  doi: 10.1080/02331934.2016.1149584.

[30]

A. Zaffaroni, Degrees of efficiency and degrees of minimality, SIAM J. Control Optim., 42 (2003), 1071-1086.  doi: 10.1137/S0363012902411532.

show all references

References:
[1]

Q. H. AnsariE. Köbis and P. K. Sharma, Characterizations of multiobjective robustness via oriented distance function and image space analysis, J. Optim. Theory Appl., 181 (2019), 817-839.  doi: 10.1007/s10957-019-01505-y.

[2]

A. Beck and A. Ben-Tal, Duality in robust optimization: Primal worst equals dual best, Oper. Res. Lett., 37 (2009), 1-6.  doi: 10.1016/j.orl.2008.09.010.

[3] A. Ben-TalL. El Ghaoui and A. Nemirovski, Robust Optimization, Princeton University Press, Princeton, 2009.  doi: 10.1515/9781400831050.
[4]

A. Ben-Tal and A. Nemirovski, Robust solutions of linear programming problems contaminated with uncertain data, Math. Program. Ser. A, 88 (2000), 411-424.  doi: 10.1007/PL00011380.

[5]

D. BertsimasD. B. Brown and C. Caramanis, Theory and applications of robust optimization, SIAM Rev., 53 (2011), 464-501.  doi: 10.1137/080734510.

[6]

J. ChenY. J. Cho and Z. Wan, Optimality conditions for cone constrained nonsmooth multiobjective optimization, J. Nonlinear Convex Anal., 17 (2016), 1627-1642. 

[7]

J. ChenL. Huang and S. Li, Separations and optimality of constrained multiobjective optimization via improvement sets, J. Optim. Theory Appl., 178 (2018), 794-823.  doi: 10.1007/s10957-018-1325-2.

[8]

J. ChenE. KöbisM. Köbis and J.-C. Yao, Image space analysis for constrained inverse vector variational inequalities via multiobjective optimization, J. Optim. Theory Appl., 177 (2018), 816-834.  doi: 10.1007/s10957-017-1197-x.

[9]

J. ChenE. Köbis and J.-C. Yao, Optimality conditions and duality for robust nonsmooth multiobjective optimization problems with constraints, J. Optim. Theory Appl., 181 (2019), 411-436.  doi: 10.1007/s10957-018-1437-8.

[10]

J. ChenS. Li and J.-C. Yao, Vector-valued separation functions and constrained vector optimization problems: Optimality and saddle points, J. Ind. Manag. Optim., 16 (2020), 707-724.  doi: 10.3934/jimo.2018174.

[11]

T. D. Chuong, Optimality and duality for robust multiobjective optimization problems, Nonlinear Anal., 134 (2016), 127-143.  doi: 10.1016/j.na.2016.01.002.

[12]

C. Gerth and P. Weidner, Nonconvex separation theorems and some applications in vector optimization, J. Optim. Theory Appl., 67 (1990), 297-320.  doi: 10.1007/BF00940478.

[13]

F. Giannessi, Constrained Optimization and Image Space Analysis: Separation of Sets and Optimality Conditions, Vol. 1, Springer, Berlin, 2005.

[14]

J.-B. Hiriart-Urruty, Tangent cone, generalized gradients and mathematical programming in Banach spaces, Math. Oper. Res., 4 (1979), 79-97.  doi: 10.1287/moor.4.1.79.

[15]

L. Huang and J. Chen, Weighted robust optimality of convex optimization problems with data uncertainty, Optim. Lett., 14 (2020), 1089-1105.  doi: 10.1007/s11590-019-01406-z.

[16]

V. JeyakumarG. M. Lee and G. Li, Characterizing robust solution sets of convex programs under data uncertainty, J. Optim. Theory Appl., 164 (2015), 407-435.  doi: 10.1007/s10957-014-0564-0.

[17]

A. A. Khan, C. Tammer and C. Zǎlinescu, Set-Valued Optimization: An Introduction with Applications, Springer, Berlin, 2015. doi: 10.1007/978-3-642-54265-7.

[18]

K. KlamrothE. KöbisA. Schöbel and C. Tammer, A unified approach for different kinds of robustness and stochastic programming via nonlinear scalarizing functionals, Optim., 62 (2013), 649-671.  doi: 10.1080/02331934.2013.769104.

[19]

K. KlamrothE. KöbisA. Schöbel and C. Tammer, A unified approach to uncertain optimization, Eur. J. Oper. Res., 260 (2017), 403-420.  doi: 10.1016/j.ejor.2016.12.045.

[20]

S. LiY. XuM. You and S. Zhu, Constrained extremum problems and image space analysis-Part I: optimality conditions and Part II: Duality and penalization, J. Optim. Theory Appl., 177 (2018), 637-659.  doi: 10.1007/s10957-018-1248-y.

[21]

G. Mastroeni, Nonlinear separation in the image space with applications to penalty methods, Appl. Anal., 91 (2012), 1901-1914.  doi: 10.1080/00036811.2011.614603.

[22]

A. L. Soyster, Convex programming with set-inclusive constraints and applications to inexact linear programming, Oper. Res., 21 (1973), 1154-1157. 

[23]

H.-Z. WeiC.-R. Chen and S.-J. Li, Characterizations for optimality conditions of general robust optimization problems, J. Optim. Theory Appl., 177 (2018), 835-856.  doi: 10.1007/s10957-018-1256-y.

[24]

H.-Z. WeiC.-R. Chen and S.-J. Li, A unified characterization of multiobjective robustness via separation, J. Optim. Theory Appl., 179 (2018), 86-102.  doi: 10.1007/s10957-017-1196-y.

[25]

H.-Z. WeiC.-R. Chen and S.-J. Li, Characterizations of multiobjective robustness on vectorization counterparts, Optim., 69 (2020), 493-518.  doi: 10.1080/02331934.2019.1625352.

[26]

H.-Z. WeiC.-R. Chen and S.-J. Li, A unified approach through image space analysis to robustness in uncertain optimization problems, J. Optim. Theory Appl., 184 (2020), 466-493.  doi: 10.1007/s10957-019-01609-5.

[27]

H.-Z. WeiC.-R. Chen and S.-J. Li, Robustness characterizations for uncertain optimization problems via image space analysis, J. Optim. Theory Appl., 186 (2020), 459-479.  doi: 10.1007/s10957-020-01709-7.

[28]

Y. D. Xu, Nonlinear separation functions, optimality conditions and error bounds for Ky Fan quasi-inequalities, Optim. Lett., 10 (2016), 527-542.  doi: 10.1007/s11590-015-0879-2.

[29]

Y. D. Xu, Nonlinear separation approach to inverse variational inequalities, Optim., 65 (2016), 1315-1335.  doi: 10.1080/02331934.2016.1149584.

[30]

A. Zaffaroni, Degrees of efficiency and degrees of minimality, SIAM J. Control Optim., 42 (2003), 1071-1086.  doi: 10.1137/S0363012902411532.

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