doi: 10.3934/jimo.2021199
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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 & 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.  Google Scholar

[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.  Google Scholar

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

[5]

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

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[30]

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

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.  Google Scholar

[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.  Google Scholar

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

[5]

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

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[30]

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

[1]

Hong-Zhi Wei, Chun-Rong Chen. Three concepts of robust efficiency for uncertain multiobjective optimization problems via set order relations. Journal of Industrial & Management Optimization, 2019, 15 (2) : 705-721. doi: 10.3934/jimo.2018066

[2]

Jiawei Chen, Shengjie Li, Jen-Chih Yao. Vector-valued separation functions and constrained vector optimization problems: optimality and saddle points. Journal of Industrial & Management Optimization, 2020, 16 (2) : 707-724. doi: 10.3934/jimo.2018174

[3]

Jutamas Kerdkaew, Rabian Wangkeeree, Rattanaporn Wangkeeree. Global optimality conditions and duality theorems for robust optimal solutions of optimization problems with data uncertainty, using underestimators. Numerical Algebra, Control & Optimization, 2022, 12 (1) : 93-107. doi: 10.3934/naco.2021053

[4]

Fengming Lin, Xiaolei Fang, Zheming Gao. Distributionally Robust Optimization: A review on theory and applications. Numerical Algebra, Control & Optimization, 2022, 12 (1) : 159-212. doi: 10.3934/naco.2021057

[5]

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

[6]

Ruotian Gao, Wenxun Xing. Robust sensitivity analysis for linear programming with ellipsoidal perturbation. Journal of Industrial & Management Optimization, 2020, 16 (4) : 2029-2044. doi: 10.3934/jimo.2019041

[7]

Matthew H. Henry, Yacov Y. Haimes. Robust multiobjective dynamic programming: Minimax envelopes for efficient decisionmaking under scenario uncertainty. Journal of Industrial & Management Optimization, 2009, 5 (4) : 791-824. doi: 10.3934/jimo.2009.5.791

[8]

Jutamas Kerdkaew, Rabian Wangkeeree. Characterizing robust weak sharp solution sets of convex optimization problems with uncertainty. Journal of Industrial & Management Optimization, 2020, 16 (6) : 2651-2673. doi: 10.3934/jimo.2019074

[9]

Xiang-Kai Sun, Xian-Jun Long, Hong-Yong Fu, Xiao-Bing Li. Some characterizations of robust optimal solutions for uncertain fractional optimization and applications. Journal of Industrial & Management Optimization, 2017, 13 (2) : 803-824. doi: 10.3934/jimo.2016047

[10]

Haodong Yu, Jie Sun. Robust stochastic optimization with convex risk measures: A discretized subgradient scheme. Journal of Industrial & Management Optimization, 2021, 17 (1) : 81-99. doi: 10.3934/jimo.2019100

[11]

Lingshuang Kong, Changjun Yu, Kok Lay Teo, Chunhua Yang. Robust real-time optimization for blending operation of alumina production. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1149-1167. doi: 10.3934/jimo.2016066

[12]

Ripeng Huang, Shaojian Qu, Xiaoguang Yang, Zhimin Liu. Multi-stage distributionally robust optimization with risk aversion. Journal of Industrial & Management Optimization, 2021, 17 (1) : 233-259. doi: 10.3934/jimo.2019109

[13]

Vadim Azhmyakov, Juan P. Fernández-Gutiérrez, Erik I. Verriest, Stefan W. Pickl. A separation based optimization approach to Dynamic Maximal Covering Location Problems with switched structure. Journal of Industrial & Management Optimization, 2021, 17 (2) : 669-686. doi: 10.3934/jimo.2019128

[14]

Nazih Abderrazzak Gadhi, Fatima Zahra Rahou. Sufficient optimality conditions and Mond-Weir duality results for a fractional multiobjective optimization problem. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021216

[15]

Ningyu Sha, Lei Shi, Ming Yan. Fast algorithms for robust principal component analysis with an upper bound on the rank. Inverse Problems & Imaging, 2021, 15 (1) : 109-128. doi: 10.3934/ipi.2020067

[16]

Hui Zhang, Jian-Feng Cai, Lizhi Cheng, Jubo Zhu. Strongly convex programming for exact matrix completion and robust principal component analysis. Inverse Problems & Imaging, 2012, 6 (2) : 357-372. doi: 10.3934/ipi.2012.6.357

[17]

G. Mastroeni, L. Pellegrini. On the image space analysis for vector variational inequalities. Journal of Industrial & Management Optimization, 2005, 1 (1) : 123-132. doi: 10.3934/jimo.2005.1.123

[18]

Shouhong Yang. Semidefinite programming via image space analysis. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1187-1197. doi: 10.3934/jimo.2016.12.1187

[19]

Alireza Goli, Hasan Khademi Zare, Reza Tavakkoli-Moghaddam, Ahmad Sadeghieh. Application of robust optimization for a product portfolio problem using an invasive weed optimization algorithm. Numerical Algebra, Control & Optimization, 2019, 9 (2) : 187-209. doi: 10.3934/naco.2019014

[20]

Nithirat Sisarat, Rabian Wangkeeree, Gue Myung Lee. Some characterizations of robust solution sets for uncertain convex optimization problems with locally Lipschitz inequality constraints. Journal of Industrial & Management Optimization, 2020, 16 (1) : 469-493. doi: 10.3934/jimo.2018163

2020 Impact Factor: 1.801

Metrics

  • PDF downloads (123)
  • HTML views (66)
  • Cited by (0)

[Back to Top]