Articles

ON APPROXIMATE EFFICIENCY FOR NONSMOOTH ROBUST VECTOR OPTIMIZATION PROBLEMS

  • Tadeusz ANTCZAK ,
  • Yogendra PANDEY ,
  • Vinay SINGH ,
  • Shashi Kant MISHRA
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  • 1 Faculty of Mathematics and Computer Science, University of Lódź, Banacha 22, 90-238 Lódź, Poland;
    2 Department of Mathematics, Satish Chandra College, Ballia 277001, India;
    3 Department of Mathematics, National Institute of Technology, Aizawl-796012, Mizoram, India;
    4 Department of Mathematics, Banaras Hindu University, Varanasi-221005, India

Received date: 2018-08-15

  Revised date: 2019-04-18

  Online published: 2020-07-17

Supported by

The research of Yogendra Pandey and Vinay Singh are supported by the Science and Engineering Research Board, a statutory body of the Department of Science and Technology (DST), Government of India, through file no. PDF/2016/001113 and SCIENCE & ENGINEERING RESEARCH BOARD (SERB-DST) through project reference no. EMR/2016/002756, respectively.

Abstract

In this article, we use the robust optimization approach (also called the worst-case approach) for finding ε-efficient solutions of the robust multiobjective optimization problem defined as a robust (worst-case) counterpart for the considered nonsmooth multiobjective programming problem with the uncertainty in both the objective and constraint functions. Namely, we establish both necessary and sufficient optimality conditions for a feasible solution to be an ε-efficient solution (an approximate efficient solution) of the considered robust multiobjective optimization problem. We also use a scalarizing method in proving these optimality conditions.

Cite this article

Tadeusz ANTCZAK , Yogendra PANDEY , Vinay SINGH , Shashi Kant MISHRA . ON APPROXIMATE EFFICIENCY FOR NONSMOOTH ROBUST VECTOR OPTIMIZATION PROBLEMS[J]. Acta mathematica scientia, Series B, 2020 , 40(3) : 887 -902 . DOI: 10.1007/s10473-020-0320-5

References

[1] Ben-Tal A, Nemirovski A. Robust convex optimization. Mathematics of Operations Research, 1998, 23:769-805
[2] Ben-Tal A, Nemirovski A. Robust solutions to uncertain linear programs. Operations Research Letters, 1999, 25:1-13
[3] Ben-Tal A, Nemirovski A. Robust optimization-methodology and application. Mathematical Programming, 2002, 92B:453-480
[4] Ben-Tal A, Nemirovski A. A selected topic in robust convex optimization. Mathematical Programming, 2008, 112B:125-158
[5] Ben-Tal A, Ghaoui L E, Nemirovski A. Robust Optimization, Princeton Series in Applied Mathematics. Princeton:Princeton University Press, 2009
[6] Bertsimas D, Brown D, Sim M. Robust linear optimization under general norm. Operations Research Letters, 2004, 32:510-516
[7] Bokrantz R, Fredriksson A. Necessary and sufficient conditions for Pareto efficiency in robust multiobjective optimization. European Journal of Operational Research, 2017, 262:682-692
[8] Chen W, Unkelbach J, Trofimov A, et al. Including robustness in multi-criteria optimization for intensity-modulated proton therapy. Physics in Medicine and Biology, 2012, 57:591-608
[9] Chuong T D. Optimality and duality for robust multiobjective optimization problems. Nonlinear Analysis, 2016, 134:127-143
[10] Doolittle E K, Kerivin H L M, Wiecek M M. Robust multiobjective optimization with application to Internet routing. Annals of Operations Research, 2018, 271:487-525
[11] Doumpos M, Zopounidis C, Grigoroudis E. Robustness Analysisin Decision Aiding, Optimization, and Analytics. International Series in Operations Research & Management Science Vol 241. Switzerland:Springer International Publishing, 2016
[12] Ehrgott M, Ide J, Schöbel A. Minmax robustness for multi-objective optimization problems. European Journal of Operational Research, 2014, 239:17-31
[13] Engau A, Wiecek M M. Generating ε-efficient solutions in multiobjective programming. European Journal of Operational Research, 2007, 177:1566-1579
[14] Fabozzi F, Kolm P, Pachamanova D, Focardi S. Robust Portfolio Optimization and Management. Wiley:Frank J Fabozzi Series, 2007
[15] Fakhar M, Mahyarinia M R, Zafarani J. On nonsmooth robust multiobjective optimization under generalized convexity with applications to portfolio optimization. European Journal of Operational Research, 2018, 265:39-48
[16] Fliege J, Werner R. Robust multiobjective optimization & applications in portfolio optimization. European Journal of Operational Research, 2014, 234:422-433
[17] Gabrel V, Murat C, Thiele A. Recent advances in robust optimization:An overview. European Journal of Operational Research, 2014, 235:471-483
[18] Govil M G, Mehra A. ε-optimality for multiobjective programming on a Banach space. European Journal of Operational Research, 2004, 157:106-112
[19] Hamel A. An ε-Lagrange multiplier rule for a mathematical programming problem on Banach spaces. Optimization, 2001, 49:137-149
[20] Ide J, Schöbel A. Robustness for uncertain multiobjective optimization:A survey and analysis of different concepts. Journal of OR Spectrum, 2016, 38:235-271
[21] Jeyakumar V, Lee G M, Dinh N. New sequential Lagrange multiplier conditions characterizing optimality without constraint qualification for convex programs. SIAM Journal on Optimization, 2003, 14:534-547
[22] Jeyakumar V, Lee G M, Dinh N. Characterization of solution sets of convex vector minimization problems. European Journal of Operational Research, 2006, 174:1380-1395
[23] Jeyakumar V, Li G. Characterizing robust set containments and solutions of uncertain linear programs without qualification. Operations Research Letters, 2010, 38:188-194
[24] Jeyakumar V, Li G. Robust Farkas lemma for uncertain linear systems with applications. Positivity, 2011, 15:331-342
[25] Jeyakumar V, Li G. Strong duality in robust convex programming:complete characterizations. SIAM Journal on Optimization, 2010, 20:3384-3407
[26] Jeyakumar V, Li G, Lee G M. Robust duality for generalized convex programming problems under data uncertainty. Nonlinear Analysis, 2012, 75:1362-1373
[27] Kang J-S, Lee T-Y, Lee D-Y. Robust optimization for engineering design. Engineering Optimization, 2012, 44:175-194
[28] Kim M H. Duality theorem and vector saddle point theorem for robust multiobjective optimization problems. Communications Korean Mathematical Society, 2013, 28:597-602
[29] Köbis E. On robust optimization. Relations between scalar robust optimization and unconstrained multicriteria optimization. Journal of Optimization Theory and Applications, 2015, 167:969-984
[30] Krüger C, Castellani F, Geldermann J, Schöbel A. Peat and pots:An application of robust multiobjective optimization to a mixing problem in agriculture. Computers and Electronics in Agriculture, 2018, 154:265-275
[31] Kuroiwa D, Lee G M. On robust multiobjective optimization. Vietnam Journal of Mathematics, 2012, 40:305-317
[32] Lee J H, Lee G M. On ε-solutions for convex optimization problems with uncertainty data. Positivity, 2012, 16:509-526
[33] Li Z, Wang S. ε-approximate solutions in multiobjective optimization. Optimization, 1998, 44:161-174
[34] Liu J C. ε-Pareto optimality for nondifferentiable multiobjective programming via penalty function. Journal of Mathematical Analysis and Applications, 1996, 198:248-261
[35] Loridan P. Necessary conditions for ε-optimality. Mathematical Programming Studies, 1982, 19:140-152
[36] Strodiot J J, Nguyen V H, Heukemes N. ε-optimal solutions in nondifferentiable convex programming and some related questions. Mathematical Programming, 1983, 25:307-328
[37] Wang L, Li Q, Zhang B, Ding R, Sun M. Robust multi-objective optimization for energy production scheduling in microgrids. Engineering Optimization, 2019, 51:332-351
[38] Wang F, Liu S, Chai Y. Robust counterparts and robust efficient solutions in vector optimization under uncertainty. Operations Research Letters, 2015, 43:293-298
[39] White D J. Epsilon efficiency. Journal of Optimization Theory and Applications, 1986, 49:319-337
[40] Yokoyama K. Epsilon approximate solutions for multiobjective programming problems. Journal of Mathematical Analysis and Applications, 1996, 203:142-149
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