Abstract
In this paper, the robust approach (the worst case approach) for nonsmooth nonconvex optimization problems with uncertainty data is studied. First various robust constraint qualifications are introduced based on the concept of tangential subdifferential. Further, robust necessary and sufficient optimality conditions are derived in the absence of the convexity of the uncertain sets and the concavity of the related functions with respect to the uncertain parameters. Finally, the results are applied to obtain the necessary and sufficient optimality conditions for robust weakly efficient solutions in multiobjective programming problems. In addition, several examples are provided to illustrate the advantages of the obtained outcomes.
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References
Apostol, T.: Mathematical Analysis, 2nd edn. Addison-Wesley, Reading (1974)
Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming: Theory and Algorithms, 3rd edn. Wiley, NewYork (2006)
Beck, A., Ben-Tal, A.: Duality in robust optimization: primal worst equals dual best. Oper. Res. Lett. 37, 1–6 (2009)
Beh, E.H., Zheng, F., Dandy, G.C., Maier, H.R., Kapelan, Z.: Robust optimization of water infrastructure planning under deep uncertainty using metamodels. Environ. Model. Softw. 93, 92–105 (2017)
Ben-Tal, A., Nemirovski, A.: Robust convex optimization. Math. Oper. Res. 23, 769–805 (1998)
Ben-Tal, A., Nemirovski, A.: Robust optimization-methodology and applications. Math. Program. Ser. B 92, 453–480 (2002)
Ben-Tal, A., Ghaoui, L.E., Nemirovski, A.: Robust optimization. In: Princeton Series in Applied Mathematics (2009)
Bertsimas, D., Brown, D.: Constructing uncertainty sets for robust linear optimization. Oper. Res. 57, 1483–1495 (2009)
Bertsimas, D., Brown, D.B., Caramanis, C.: Theory and applications of robust optimization. SIAM Rev. 53, 464–501 (2011)
Bot, R.I., Jeyakumar, V., Li, G.Y.: Robust duality in parametric convex optimization. Set-Valued Var. Anal. 21, 177–189 (2013)
Borwein, J.M., Lewis, A.S.: Convex Analysis and Nonlinear Optimization: Theory and Examples. Springer, New York (2010)
Cadarso, L., Marn, Á.: Rapid transit network design considering risk aversion. Electron. Notes Discrete Math. 52, 29–36 (2016)
Carrizosa, E., Goerigk, M., Schöbel, A.: A biobjective approach to recoverable robustness based on location planning. Eur. J. Oper. Res. 261(2), 421–435 (2017)
Chassein, A., Goerigk, M.: On the recoverable robust traveling salesman problem. Optim. Lett. 10(7), 1479–1492 (2016)
Chen, J.W., Köbis, E., Yao, J.C.: Optimality conditions and duality for robust nonsmooth multiobjective optimization problems with constraints. J. Optim. Theory Appl. 181, 411–436 (2019)
Cheref, A., Artigues, C., Billaut, J.-C.: A new robust approach for a production scheduling and delivery routing problem. IFAC-PapersOnLine 49(12), 886–891 (2016)
Chuong, T.D.: Optimality and duality for robust multiobjective optimization problems. Nonlinear Anal. 134, 127–143 (2016)
Chuong, T.D.: Linear matrix inequality conditions and duality for a class of robust multiobjective convex polynomial programs. SIAM J. Optim. 28, 2466–2488 (2018)
Chuong, T.D.: Robust optimality and duality in multiobjective optimization problems under data uncertainty. SIAM J. Optim. 30, 1501–1526 (2020)
Clarke, F.H.: Functional Analysis, Calculus of Variations and Optimal Control. Springer, London (2013)
Clarke, F.H.: Nonsmooth Analysis and Control Theory. Springer, NewYork (1998)
Golestani, M., Nobakhtian, S.: Convexificators and strong Kuhn–Tucker conditions. Comput. Math. Appl. 64, 550–557 (2012)
Golestani, M., Nobakhtian, S.: Nonsmooth multiobjective programming and constraint qualifications. Optimization 62, 783–795 (2013)
Goryashko, A.P., Nemirovski, A.S.: Robust energy cost optimization of water distribution system with uncertain demand. Autom. Remote Control 75(10), 1754–1769 (2014)
Jeyakumar, V., Li, G.Y.: Strong duality in robust convex programming: complete characterizations. SIAM J. Optim. 20, 3384–3407 (2010)
Jeyakumar, V., Wang, J.H., Li, G.Y.: Lagrange multiplier characterizations of robust best approximations under constraint data uncertainty. J. Math. Anal. Appl. 393, 285–297 (2012)
Jeyakumar, V., Lee, G.M., Li, G.Y.: Robust duality for generalized convex programming problems with data uncertainty. Nonlinear Anal. 75, 1362–1373 (2012)
Jeyakumar, V., Lee, G.M., Li, G.Y.: Characterizing robust solution sets of convex programs under data uncertainty. J. Optim. Theory Appl. 164, 407–435 (2015)
Kuroiwa, D., Lee, G.M.: On robust multiobjective optimization. J. Nonlinear Convex Anal. 15, 305–317 (2012)
Kuroiwa, D., Lee, G.M.: On robust convex multiobjective optimization. J. Nonlinear Convex Anal. 15, 1125–1136 (2014)
Kutschka M.: Robustness concepts for knapsack and network design problems under data uncertainty. In: Operations Research Proceedings 2014. Springer, Cham, pp. 341–347 (2016)
Lee, J.H., Lee, G.M.: On \(\epsilon \)-solutions for convex optimization problems with uncertainty data. Positivity 16, 509–526 (2012)
Lee, G.M., Son, P.T.: On nonsmooth optimality theorems for robust optimization problems. Bull. Korean Math. Soc. 51, 287–301 (2014)
Lee, G.M., Lee, J.H.: On nonsmooth optimality theorems for robust multiobjective optimization problems. J. Nonlinear Convex Anal. 16, 2039–2052 (2015)
Lee, J.H., Lee, G.M.: On optimality conditions and duality theorems for robust semi-infinite multiobjective optimization problems. Ann. Oper. Res. 269, 419–438 (2018)
Lemaréchal, M.: An introduction to the theory of nonsmooth optimization. Optimization 17, 827–858 (1986)
Li, X.F., Zhang, J.Z.: Stronger Kuhn–Tucker type conditions in nonsmooth multiobjective optimization: locally Lipschitz case. J. Optim. Theory Appl. 127, 367–388 (2005)
Li, G.Y., Jeyakumar, V., Lee, G.M.: Robust conjugate duality for convex optimization under uncertainty with application to data classification. Nonlinear Anal. 74, 2327–2341 (2011)
Maeda, T.: Constraint qualifications in multiobjective optimization problems: differentiable case. J. Optim. Theory Appl. 80(3), 483–500 (1994). https://doi.org/10.1007/BF02207776
Martínez-Legaz, J.E.: Optimality conditions for pseudoconvex minimization over convex sets defined by tangentially convex constraints. Optim. Lett. 9, 1017–1023 (2015)
Mashkoorzadeh, F., Movahedian, N., Nobakhtian, S.: Optimality conditions for nonconvex constrained optimization problems. Numer. Funct. Anal. Optim. 40, 1918–1938 (2019)
Mordukhovich, B.S., Nghia, T.T.A.: Subdifferentials of nonconvex supremum functions and their applications to semi-infinite and infinite programming with Lipschitzian data. SIAM J. Optim. 23, 406–431 (2013)
Mordukhovich, B.S., Nghia, T.T.A.: Nonsmooth cone-constrained optimization with applications to semi-infinite programming. Math. Oper. Res. 39, 301–337 (2014)
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation II, Applications. Springer, New York (2018)
Perez-Aros, P.: Subdifferential formulae for the supremum of an arbitrary family of functions. SIAM. J. Optim. 29, 1714–1743 (2019)
Pshenichnyi, B.N.: Necessary Conditions for an Extremum. Marcel Dekker, New York (1971)
Shapiro, A., Dentcheva, D., Ruszczynski, A.: Lectures on Stochastic Programming: Modeling and Theory. SIAM, Philadelphia (2009)
Soyster, A.L.: Convex programming with set-inclusive constraints and applications to inexact linear programming. Oper. Res. 1973, 1154–1157 (1973)
Sun, X.K., Peng, Z.Y., Guo, X.L.: Some characterizations of robust optimal solution for uncertain convex optimization problems. Optim. Lett. 10, 1463–1478 (2016)
Tung, L.T.: Karush-Kuhn-Tucker optimality conditions and duality for multiobjective semi-infinite programming via tangential subdifferentials. Numer. Funct. Anal. Optim. 41, 659–684 (2020)
Xiong, P., Singh, C.: Distributionally robust optimization for energy and reserve toward a low-carbon electricity market. Electr. Power Syst. Res. 149, 137–145 (2017)
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The third-named author was partially supported by a Grant from IPM (No. 99900416).
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Mashkoorzadeh, F., Movahedian, N. & Nobakhtian, S. Robustness in Nonsmooth Nonconvex Optimization Problems. Positivity 25, 701–729 (2021). https://doi.org/10.1007/s11117-020-00783-5
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DOI: https://doi.org/10.1007/s11117-020-00783-5
Keywords
- Nonconvex optimization
- Nonsmooth optimization
- Robustness
- Optimality condition
- Tangential subdifferential