doi: 10.3934/jimo.2018163

Some characterizations of robust solution sets for uncertain convex optimization problems with locally Lipschitz inequality constraints

1. 

Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand

2. 

Research center for Academic Excellence in Mathematics, Naresuan University, Phitsanulok 65000, Thailand

3. 

Department of Applied Mathematics, Pukyong National University, Busan 48513, Korea

* Corresponding author: Rabian Wangkeeree

Received  November 2017 Revised  February 2018 Published  October 2018

Fund Project: This research was supported by the Thailand Research Fund through the Royal Golden Jubilee Ph.D. Program (Grant No. PHD/0026/2555), the Thailand Research Fund, Grant No. RSA6080077 and Naresuan University, and the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (Grant No. 2017R1E1A1A03069931)

In this paper, we consider an uncertain convex optimization problem with a robust convex feasible set described by locally Lipschitz constraints. Using robust optimization approach, we give some new characterizations of robust solution sets of the problem. Such characterizations are expressed in terms of convex subdifferentails, Clarke subdifferentials, and Lagrange multipliers. In order to characterize the solution set, we first introduce the so-called pseudo Lagrangian function and establish constant pseudo Lagrangian-type property for the robust solution set. We then used to derive Lagrange multiplier-based characterizations of robust solution set. By means of linear scalarization, the results are applied to derive characterizations of weakly and properly robust efficient solution sets of convex multi-objective optimization problems with data uncertainty. Some examples are given to illustrate the significance of the results.

Citation: 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, doi: 10.3934/jimo.2018163
References:
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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.

[2]

A. Ben-Tal, L. E. Ghaoui and A. Nemirovski, Robust Optimization, Princeton University, Princeton, 2009. doi: 10.1515/9781400831050.

[3]

A. Ben-Tal and A. Nemirovski, Robust Optimization-methodology and applications, Math. Program. Ser. B, 92 (2002), 453-480. doi: 10.1007/s101070100286.

[4]

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

[5]

R. I. BoţV. Jeyakumar and G. Y. Li, Robust duality in parametric convex optimization, Set-Valued Var. Anal., 21 (2013), 177-189. doi: 10.1007/s11228-012-0219-y.

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J. V. Burke and M. Ferris, Characterization of solution sets of convex programs, Oper. Res. Lett., 10 (1991), 57-60. doi: 10.1016/0167-6377(91)90087-6.

[7]

J. V. Burke and M. C. Ferris, Weak sharp minima in mathematical programming, SIAM J. Control Optim., 31 (1993), 1340-1359. doi: 10.1137/0331063.

[8]

M. Castellani and M. Giuli, A characterization of the solution set of pseudoconvex extremum problems, J. Convex Anal., 19 (2012), 113-123.

[9]

F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983.

[10]

J. Dutta and C. S. Lalitha, Optimality conditions in convex optimization revisited, Optim. Lett., 7 (2013), 221-229. doi: 10.1007/s11590-011-0410-3.

[11]

M.A. GobernaV. JeyakumarG. Li and M. Lopez, Robust linear semi-infinite programming duality, Math. Program, Series B, 139 (2013), 185-203. doi: 10.1007/s10107-013-0668-6.

[12]

J. Jahn, Vector Optimization: Theory, Applications and Extensions, Series in Operations Research and Decision Theory, Springer, New York, 2004. doi: 10.1007/978-3-540-24828-6.

[13]

V. JeyakumarG. M. Lee and N. Dinh, Characterizations of solution sets of convex vector minimization problems, Eur. J. Oper. Res., 174 (2006), 1380-1395. doi: 10.1016/j.ejor.2005.05.007.

[14]

V. JeyakumarG. M. Lee and N. Dinh, Lagrange multiplier conditions characterizing optimal solution sets of convex programs, J. Optim. Theory Appl., 123 (2004), 83-103. doi: 10.1023/B:JOTA.0000043992.38554.c8.

[15]

V. JeyakumarG. M. Lee and G. Y. 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.

[16]

V. Jeyakumar and G. Li, Characterizing robust set containments and solutions of uncertain linear programs without qualifications, Oper. Res. Lett, 38 (2010), 188-194. doi: 10.1016/j.orl.2009.12.004.

[17]

V. Jeyakumar and G. Li, Strong duality in robust convex programming: Complete characterizations, SIAM J. Optim., 20 (2010), 3384-3407. doi: 10.1137/100791841.

[18]

V. JeyakumarG. Li and J. H. Wang, Some robust convex programs without a duality gap, J. Convex Anal., 20 (2013), 377-394.

[19]

V. Jeyakumar and X. Q. Yang, On characterizing the solution sets of pseudolinear programs, J. Optim. Theory Appl., 87 (1995), 747-755. doi: 10.1007/BF02192142.

[20]

V. Jeyakumar and A. Zaffaroni, Asymptotic conditions for weak and proper optimality in infinite dimensional convex vector optimization, Numer Func Anal Opt., 17 (1996), 323-343. doi: 10.1080/01630569608816697.

[21]

A. KabganiM. Soleimani-damaneh and M. Zamani, Optimality conditions in optimization problems with convex feasible set using convexificators, Math Meth Oper Res., 86 (2017), 103-121. doi: 10.1007/s00186-017-0584-2.

[22]

D. Kuroiwa and G. M. Lee, On robust convex multiobjective optimization, J. Nonlinear Convex Anal., 15 (2014), 1125-1136.

[23]

C. S. Lalitha and M. Mehta, Characterizations of solution sets of mathematical programs in terms of Lagrange multipliers, Optimization, 58 (2009), 995-1007. doi: 10.1080/02331930701763272.

[24]

J. B. Lasserre, On representations of the feasible set in convex optimization, Optim. Lett., 4 (2010), 1-5. doi: 10.1007/s11590-009-0153-6.

[25]

G. M. Lee and J. H. Lee, On nonsmooth optimality theorems for robust multiobjective optimization problems, J. Nonlinear Convex Anal., 16 (2015), 2039-2052.

[26]

G. M. Lee and P. T. Son, On nonsmooth optimality theorems for robust optimization problems, Bull. Korean Math. Soc., 51 (2014), 287-301. doi: 10.4134/BKMS.2014.51.1.287.

[27]

X.-B. Li and S. Wang, Characterizations of robust solution set of convex programs with uncertain data, Optim. Lett., 12 (2018), 1387-1402. doi: 10.1007/s11590-017-1187-9.

[28]

D. T. Luc, Theory of Vector Optimization, Lecture Notes Econ. Math. Syst. 319. Springer, Berlin, 1989. doi: 10.1007/978-3-642-50280-4.

[29]

O. L. Mangasarian, A simple characterization of solution sets of convex programs, Oper. Res. Lett., 7 (1988), 21-26. doi: 10.1016/0167-6377(88)90047-8.

[30]

J. E. Martinez-Legaz, Optimality conditions for pseudoconvex minimization over convex sets defined by tangentially convex constraints, Optim. Lett., 9 (2015), 1017-1023. doi: 10.1007/s11590-014-0822-y.

[31]

Y. Sawaragi, H. Nakayama and T. Tanino, Theory of Multiobjective Optimization, Mathematics in Science and Engineering, vol. 176. Academic Press, Orlando, 1985.

[32]

T. Q. Son and N. Dinh, Characterizations of optimal solution sets of convex infinite programs, TOP., 16 (2008), 147-163. doi: 10.1007/s11750-008-0039-2.

[33]

T. Q. Son and D. S. Kim, A new approach to characterize the solution set of a pseudoconvex programming problem, J. Comput. Appl. Math., 261 (2014), 333-340. doi: 10.1016/j.cam.2013.11.004.

[34]

X. K. SunX. J. LongH. Y. Fu and X. B. Li, Some characterizations of robust optimal solutions for uncertain fractional optimization and applications, J. Ind. Manag. Optim., 13 (2017), 803-824. doi: 10.3934/jimo.2016047.

[35]

X. K. SunZ. Y. Peng and X. L. Guo, Some characterizations of robust optimal solutions for uncertain convex optimization problems, Optim. Lett., 10 (2016), 1463-1478. doi: 10.1007/s11590-015-0946-8.

[36]

Z. L. Wu and S. Y. Wu, Characterizations of the solution sets of convex programs and variational inequality problems, J. Optim. Theory Appl., 130 (2006), 339-358. doi: 10.1007/s10957-006-9108-6.

[37]

S. Yamamoto and D. Kuroiwa, Constraint qualifications for KKT optimality condition in convex optimization with locally Lipschitz inequality constraints, Linear Nonlinear Anal., 2 (2016), 101-111.

[38]

X. M. Yang, On characterizing the solution sets of pseudoinvex extremum problems, J. Optim. Theory Appl., 140 (2009), 537-542. doi: 10.1007/s10957-008-9470-7.

[39]

K. Q. Zhao and X. M. Yang, Characterizations of the solution set for a class of nonsmooth optimization problems, Optim. Lett., 7 (2013), 685-694. doi: 10.1007/s11590-012-0471-y.

show all references

References:
[1]

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.

[2]

A. Ben-Tal, L. E. Ghaoui and A. Nemirovski, Robust Optimization, Princeton University, Princeton, 2009. doi: 10.1515/9781400831050.

[3]

A. Ben-Tal and A. Nemirovski, Robust Optimization-methodology and applications, Math. Program. Ser. B, 92 (2002), 453-480. doi: 10.1007/s101070100286.

[4]

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

[5]

R. I. BoţV. Jeyakumar and G. Y. Li, Robust duality in parametric convex optimization, Set-Valued Var. Anal., 21 (2013), 177-189. doi: 10.1007/s11228-012-0219-y.

[6]

J. V. Burke and M. Ferris, Characterization of solution sets of convex programs, Oper. Res. Lett., 10 (1991), 57-60. doi: 10.1016/0167-6377(91)90087-6.

[7]

J. V. Burke and M. C. Ferris, Weak sharp minima in mathematical programming, SIAM J. Control Optim., 31 (1993), 1340-1359. doi: 10.1137/0331063.

[8]

M. Castellani and M. Giuli, A characterization of the solution set of pseudoconvex extremum problems, J. Convex Anal., 19 (2012), 113-123.

[9]

F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983.

[10]

J. Dutta and C. S. Lalitha, Optimality conditions in convex optimization revisited, Optim. Lett., 7 (2013), 221-229. doi: 10.1007/s11590-011-0410-3.

[11]

M.A. GobernaV. JeyakumarG. Li and M. Lopez, Robust linear semi-infinite programming duality, Math. Program, Series B, 139 (2013), 185-203. doi: 10.1007/s10107-013-0668-6.

[12]

J. Jahn, Vector Optimization: Theory, Applications and Extensions, Series in Operations Research and Decision Theory, Springer, New York, 2004. doi: 10.1007/978-3-540-24828-6.

[13]

V. JeyakumarG. M. Lee and N. Dinh, Characterizations of solution sets of convex vector minimization problems, Eur. J. Oper. Res., 174 (2006), 1380-1395. doi: 10.1016/j.ejor.2005.05.007.

[14]

V. JeyakumarG. M. Lee and N. Dinh, Lagrange multiplier conditions characterizing optimal solution sets of convex programs, J. Optim. Theory Appl., 123 (2004), 83-103. doi: 10.1023/B:JOTA.0000043992.38554.c8.

[15]

V. JeyakumarG. M. Lee and G. Y. 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.

[16]

V. Jeyakumar and G. Li, Characterizing robust set containments and solutions of uncertain linear programs without qualifications, Oper. Res. Lett, 38 (2010), 188-194. doi: 10.1016/j.orl.2009.12.004.

[17]

V. Jeyakumar and G. Li, Strong duality in robust convex programming: Complete characterizations, SIAM J. Optim., 20 (2010), 3384-3407. doi: 10.1137/100791841.

[18]

V. JeyakumarG. Li and J. H. Wang, Some robust convex programs without a duality gap, J. Convex Anal., 20 (2013), 377-394.

[19]

V. Jeyakumar and X. Q. Yang, On characterizing the solution sets of pseudolinear programs, J. Optim. Theory Appl., 87 (1995), 747-755. doi: 10.1007/BF02192142.

[20]

V. Jeyakumar and A. Zaffaroni, Asymptotic conditions for weak and proper optimality in infinite dimensional convex vector optimization, Numer Func Anal Opt., 17 (1996), 323-343. doi: 10.1080/01630569608816697.

[21]

A. KabganiM. Soleimani-damaneh and M. Zamani, Optimality conditions in optimization problems with convex feasible set using convexificators, Math Meth Oper Res., 86 (2017), 103-121. doi: 10.1007/s00186-017-0584-2.

[22]

D. Kuroiwa and G. M. Lee, On robust convex multiobjective optimization, J. Nonlinear Convex Anal., 15 (2014), 1125-1136.

[23]

C. S. Lalitha and M. Mehta, Characterizations of solution sets of mathematical programs in terms of Lagrange multipliers, Optimization, 58 (2009), 995-1007. doi: 10.1080/02331930701763272.

[24]

J. B. Lasserre, On representations of the feasible set in convex optimization, Optim. Lett., 4 (2010), 1-5. doi: 10.1007/s11590-009-0153-6.

[25]

G. M. Lee and J. H. Lee, On nonsmooth optimality theorems for robust multiobjective optimization problems, J. Nonlinear Convex Anal., 16 (2015), 2039-2052.

[26]

G. M. Lee and P. T. Son, On nonsmooth optimality theorems for robust optimization problems, Bull. Korean Math. Soc., 51 (2014), 287-301. doi: 10.4134/BKMS.2014.51.1.287.

[27]

X.-B. Li and S. Wang, Characterizations of robust solution set of convex programs with uncertain data, Optim. Lett., 12 (2018), 1387-1402. doi: 10.1007/s11590-017-1187-9.

[28]

D. T. Luc, Theory of Vector Optimization, Lecture Notes Econ. Math. Syst. 319. Springer, Berlin, 1989. doi: 10.1007/978-3-642-50280-4.

[29]

O. L. Mangasarian, A simple characterization of solution sets of convex programs, Oper. Res. Lett., 7 (1988), 21-26. doi: 10.1016/0167-6377(88)90047-8.

[30]

J. E. Martinez-Legaz, Optimality conditions for pseudoconvex minimization over convex sets defined by tangentially convex constraints, Optim. Lett., 9 (2015), 1017-1023. doi: 10.1007/s11590-014-0822-y.

[31]

Y. Sawaragi, H. Nakayama and T. Tanino, Theory of Multiobjective Optimization, Mathematics in Science and Engineering, vol. 176. Academic Press, Orlando, 1985.

[32]

T. Q. Son and N. Dinh, Characterizations of optimal solution sets of convex infinite programs, TOP., 16 (2008), 147-163. doi: 10.1007/s11750-008-0039-2.

[33]

T. Q. Son and D. S. Kim, A new approach to characterize the solution set of a pseudoconvex programming problem, J. Comput. Appl. Math., 261 (2014), 333-340. doi: 10.1016/j.cam.2013.11.004.

[34]

X. K. SunX. J. LongH. Y. Fu and X. B. Li, Some characterizations of robust optimal solutions for uncertain fractional optimization and applications, J. Ind. Manag. Optim., 13 (2017), 803-824. doi: 10.3934/jimo.2016047.

[35]

X. K. SunZ. Y. Peng and X. L. Guo, Some characterizations of robust optimal solutions for uncertain convex optimization problems, Optim. Lett., 10 (2016), 1463-1478. doi: 10.1007/s11590-015-0946-8.

[36]

Z. L. Wu and S. Y. Wu, Characterizations of the solution sets of convex programs and variational inequality problems, J. Optim. Theory Appl., 130 (2006), 339-358. doi: 10.1007/s10957-006-9108-6.

[37]

S. Yamamoto and D. Kuroiwa, Constraint qualifications for KKT optimality condition in convex optimization with locally Lipschitz inequality constraints, Linear Nonlinear Anal., 2 (2016), 101-111.

[38]

X. M. Yang, On characterizing the solution sets of pseudoinvex extremum problems, J. Optim. Theory Appl., 140 (2009), 537-542. doi: 10.1007/s10957-008-9470-7.

[39]

K. Q. Zhao and X. M. Yang, Characterizations of the solution set for a class of nonsmooth optimization problems, Optim. Lett., 7 (2013), 685-694. doi: 10.1007/s11590-012-0471-y.

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