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November  2020, 16(6): 2651-2673. doi: 10.3934/jimo.2019074

Characterizing robust weak sharp solution sets of convex optimization problems with uncertainty

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

* Corresponding author: R. Wangkeeree

Received  March 2018 Revised  March 2019 Published  July 2019

Fund Project: The first author was partially supported by the Science Achivement Scholarship of Thailand and the second author was partially supported by the Thailand Research Fund, Grant No. RSA6080077 and Naresuan University

We introduce robust weak sharp and robust sharp solution to a convex programming with the objective and constraint functions involved uncertainty. The characterizations of the sets of all the robust weak sharp solutions are obtained by means of subdiferentials of convex functions, DC fuctions, Fermat rule and the robust-type subdifferential constraint qualification, which was introduced in X.K. Sun, Z.Y. Peng and X. Le Guo, Some characterizations of robust optimal solutions for uncertain convex optimization problems, Optim Lett. 10. (2016), 1463-1478. In addition, some applications to the multi-objective case are presented.

Citation: 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
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.  Google Scholar

[2]

A. Ben-Tal and A. Nemirovski, Robust convex optimization, Math. Oper. Res., 23 (1998), 769-805.  doi: 10.1287/moor.23.4.769.  Google Scholar

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

[4]

A. Ben-Tal, L. E. Ghaoui and A. Nemirovski, Robust Optimization, Princeton Series in Applied Mathematics, 2009. doi: 10.1515/9781400831050.  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. V. BurkeA. Lewis and M. Overton, Optimization matrix stability, Proc. Am. Math. Soc., 129 (2000), 1635-1642.  doi: 10.1090/S0002-9939-00-05985-2.  Google Scholar

[7]

J. V. BurkeA. Lewis and M. Overton, Optimization stability and eigenvalue multiplicity, Found. Comput. Math., 1 (2001), 205-225.  doi: 10.1007/PL00021726.  Google Scholar

[8]

J. V. Burke and M. C. Ferris, A Gauss-Newton method for convex composite optimization, Math. Program., 71 (1995), 179-194.  doi: 10.1007/BF01585997.  Google Scholar

[9]

J. V. Burke and M. C. Ferris, Weak sharp minima in mathematical programming,, SIAM J. Control Optim., 36 (1993), 1340-1359.  doi: 10.1137/0331063.  Google Scholar

[10]

J. V. Burke and S. Deng, Weak sharp minima revisited, part Ⅰ: Basic theory, Control Cybern., 31 (2002), 439-469.   Google Scholar

[11]

J. V. Burke and S. Deng, Weak sharp minima revisited, part Ⅱ: Application to linear regularity and error bounds, Math. Program., Ser. B, 104 (2005), 235-261.  doi: 10.1007/s10107-005-0615-2.  Google Scholar

[12]

J. V. Burke and S. Deng, Weak sharp minima revisited, part Ⅲ: Error bounds for differentiable convex inclusions, Math. Program., Ser. B, 116 (2009), 37-56.  doi: 10.1007/s10107-007-0130-8.  Google Scholar

[13]

T. D. Chuong, Optimality and duality for proper and isolated efficiencies in multi-objective optimization, Nonlinear Anal., 76 (2013), 93-104.  doi: 10.1016/j.na.2012.08.005.  Google Scholar

[14]

T. D. Chuong and J. C. Yao, Isolated and proper efficiencies in semi-infinite vector optimization problems, J. Optim. Theor. Appl., 162 (2014), 447-462.  doi: 10.1007/s10957-013-0425-2.  Google Scholar

[15]

M. C. Ferris, Weak Sharp Minima and Penalty Functions in Mathematical Programming, Ph.D. thesis, Universiy of Cambridge, Cambridge, UK, 1988. Google Scholar

[16]

J. B. Hiriart-Urruty, $\varepsilon$-Subdifferential calculus, in Convex Analysis and Optimization (eds. J. P. Aubin and R. Vinter), Boston, Mass.-London, 1982, 43–92.  Google Scholar

[17]

V. Jeyakumar, Constraint qualifications characterizing lagrangian duality in convex optimization,, J. Optim. Theo. Appl., 136 (2008), 31-41.  doi: 10.1007/s10957-007-9294-x.  Google Scholar

[18]

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.  Google Scholar

[19]

A. Jourani, Hoffman's error bounds, local controbility and sensitivity analysis, SIAM J. Control Optim., 38 (2000), 947-970.  doi: 10.1137/S0363012998339216.  Google Scholar

[20]

D. Kuroiwa and G. M. Lee, On robust multiobjective optimization, Vietnam Journal of Mathematics, 40 (2012), 305-317.   Google Scholar

[21]

A. S. Lewis and J. S. Pang, Error bounds for convex inequality systems, in Proceedings of the Fifth Symposium on Generalized Convexity (ed. J. P. Crouzeix), Luminy-Marseille, 1996. doi: 10.1007/978-1-4613-3341-8_3.  Google Scholar

[22]

B. T. Polyak, Sharp Minima, Institute of Control Sciences Lecture Notes, Moscow, USSR, 1979; Presented at the IIASA Workshop on Generalized Lagrangians and Their Applications, IIASA, Laxenburg, Austria, 1979. Google Scholar

[23]

X. K. Sun, Regularity conditions characterizing Fenchel-Lagrange duality and Farkas-type results in DC infinite programming,, J. Math. Anal. Appl., 414 (2014), 590-611.  doi: 10.1016/j.jmaa.2014.01.033.  Google Scholar

[24]

X. K. SunZ. Y. Peng and X. Le 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.  Google Scholar

[25]

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.  Google Scholar

[26]

W. Y. ZhangS. Xu and S. J. Li, Necessary conditions for weak sharp minima in cone-constrained optimization problems, Abstract and Applied Analysis, 11 (2012), 1-11.  doi: 10.1155/2012/909520.  Google Scholar

[27]

X. Y. Zheng and K. F. Ng, Strong KKT conditions and weak sharp minima in convex-composite optimization, Math. Program., 126 (2009), 259-279.  doi: 10.1007/s10107-009-0277-6.  Google Scholar

[28]

X. Y. Zheng and X. Q. Yang, Weak sharp minima for semi-infinite optimization problems with applications, SIAM J. Optim., 18 (2004), 573-588.  doi: 10.1137/060670213.  Google Scholar

[29]

J. C. ZhouB. S. Mordukhovich and N. H. Xiu, Complete characterizations of local weak sharp minima with applications to semi-infinite optimization and complementarity, Nonlinear Anal., 75 (2012), 1700-1718.  doi: 10.1016/j.na.2011.05.084.  Google Scholar

[30]

S. K. Zhu, Weak sharp efficiency in multi-objective optimization, Optim Lett., 10 (2016), 1287-1301.  doi: 10.1007/s11590-015-0925-0.  Google Scholar

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.  Google Scholar

[2]

A. Ben-Tal and A. Nemirovski, Robust convex optimization, Math. Oper. Res., 23 (1998), 769-805.  doi: 10.1287/moor.23.4.769.  Google Scholar

[3]

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

[4]

A. Ben-Tal, L. E. Ghaoui and A. Nemirovski, Robust Optimization, Princeton Series in Applied Mathematics, 2009. doi: 10.1515/9781400831050.  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. V. BurkeA. Lewis and M. Overton, Optimization matrix stability, Proc. Am. Math. Soc., 129 (2000), 1635-1642.  doi: 10.1090/S0002-9939-00-05985-2.  Google Scholar

[7]

J. V. BurkeA. Lewis and M. Overton, Optimization stability and eigenvalue multiplicity, Found. Comput. Math., 1 (2001), 205-225.  doi: 10.1007/PL00021726.  Google Scholar

[8]

J. V. Burke and M. C. Ferris, A Gauss-Newton method for convex composite optimization, Math. Program., 71 (1995), 179-194.  doi: 10.1007/BF01585997.  Google Scholar

[9]

J. V. Burke and M. C. Ferris, Weak sharp minima in mathematical programming,, SIAM J. Control Optim., 36 (1993), 1340-1359.  doi: 10.1137/0331063.  Google Scholar

[10]

J. V. Burke and S. Deng, Weak sharp minima revisited, part Ⅰ: Basic theory, Control Cybern., 31 (2002), 439-469.   Google Scholar

[11]

J. V. Burke and S. Deng, Weak sharp minima revisited, part Ⅱ: Application to linear regularity and error bounds, Math. Program., Ser. B, 104 (2005), 235-261.  doi: 10.1007/s10107-005-0615-2.  Google Scholar

[12]

J. V. Burke and S. Deng, Weak sharp minima revisited, part Ⅲ: Error bounds for differentiable convex inclusions, Math. Program., Ser. B, 116 (2009), 37-56.  doi: 10.1007/s10107-007-0130-8.  Google Scholar

[13]

T. D. Chuong, Optimality and duality for proper and isolated efficiencies in multi-objective optimization, Nonlinear Anal., 76 (2013), 93-104.  doi: 10.1016/j.na.2012.08.005.  Google Scholar

[14]

T. D. Chuong and J. C. Yao, Isolated and proper efficiencies in semi-infinite vector optimization problems, J. Optim. Theor. Appl., 162 (2014), 447-462.  doi: 10.1007/s10957-013-0425-2.  Google Scholar

[15]

M. C. Ferris, Weak Sharp Minima and Penalty Functions in Mathematical Programming, Ph.D. thesis, Universiy of Cambridge, Cambridge, UK, 1988. Google Scholar

[16]

J. B. Hiriart-Urruty, $\varepsilon$-Subdifferential calculus, in Convex Analysis and Optimization (eds. J. P. Aubin and R. Vinter), Boston, Mass.-London, 1982, 43–92.  Google Scholar

[17]

V. Jeyakumar, Constraint qualifications characterizing lagrangian duality in convex optimization,, J. Optim. Theo. Appl., 136 (2008), 31-41.  doi: 10.1007/s10957-007-9294-x.  Google Scholar

[18]

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.  Google Scholar

[19]

A. Jourani, Hoffman's error bounds, local controbility and sensitivity analysis, SIAM J. Control Optim., 38 (2000), 947-970.  doi: 10.1137/S0363012998339216.  Google Scholar

[20]

D. Kuroiwa and G. M. Lee, On robust multiobjective optimization, Vietnam Journal of Mathematics, 40 (2012), 305-317.   Google Scholar

[21]

A. S. Lewis and J. S. Pang, Error bounds for convex inequality systems, in Proceedings of the Fifth Symposium on Generalized Convexity (ed. J. P. Crouzeix), Luminy-Marseille, 1996. doi: 10.1007/978-1-4613-3341-8_3.  Google Scholar

[22]

B. T. Polyak, Sharp Minima, Institute of Control Sciences Lecture Notes, Moscow, USSR, 1979; Presented at the IIASA Workshop on Generalized Lagrangians and Their Applications, IIASA, Laxenburg, Austria, 1979. Google Scholar

[23]

X. K. Sun, Regularity conditions characterizing Fenchel-Lagrange duality and Farkas-type results in DC infinite programming,, J. Math. Anal. Appl., 414 (2014), 590-611.  doi: 10.1016/j.jmaa.2014.01.033.  Google Scholar

[24]

X. K. SunZ. Y. Peng and X. Le 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.  Google Scholar

[25]

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.  Google Scholar

[26]

W. Y. ZhangS. Xu and S. J. Li, Necessary conditions for weak sharp minima in cone-constrained optimization problems, Abstract and Applied Analysis, 11 (2012), 1-11.  doi: 10.1155/2012/909520.  Google Scholar

[27]

X. Y. Zheng and K. F. Ng, Strong KKT conditions and weak sharp minima in convex-composite optimization, Math. Program., 126 (2009), 259-279.  doi: 10.1007/s10107-009-0277-6.  Google Scholar

[28]

X. Y. Zheng and X. Q. Yang, Weak sharp minima for semi-infinite optimization problems with applications, SIAM J. Optim., 18 (2004), 573-588.  doi: 10.1137/060670213.  Google Scholar

[29]

J. C. ZhouB. S. Mordukhovich and N. H. Xiu, Complete characterizations of local weak sharp minima with applications to semi-infinite optimization and complementarity, Nonlinear Anal., 75 (2012), 1700-1718.  doi: 10.1016/j.na.2011.05.084.  Google Scholar

[30]

S. K. Zhu, Weak sharp efficiency in multi-objective optimization, Optim Lett., 10 (2016), 1287-1301.  doi: 10.1007/s11590-015-0925-0.  Google Scholar

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