# American Institute of Mathematical Sciences

March  2022, 12(1): 93-107. doi: 10.3934/naco.2021053

## Global optimality conditions and duality theorems for robust optimal solutions of optimization problems with data uncertainty, using underestimators

 1 Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand 2 Department of Mathematics, Faculty of Science, Research center for Academic Excellence in Mathematics, Naresuan University, Phitsanulok 65000, Thailand

* Corresponding author

Received  March 2020 Revised  October 2021 Published  March 2022 Early access  November 2021

Fund Project: This research was supported by Naresuan University, Grant No. R2563C024

In this paper, a robust optimization problem, which features a maximum function of continuously differentiable functions as its objective function, is investigated. Some new conditions for a robust KKT point, which is a robust feasible solution that satisfies the robust KKT condition, to be a global robust optimal solution of the uncertain optimization problem, which may have many local robust optimal solutions that are not global, are established. The obtained conditions make use of underestimators, which were first introduced by Jayakumar and Srisatkunarajah [1,2] of the Lagrangian associated with the problem at the robust KKT point. Furthermore, we also investigate the Wolfe type robust duality between the smooth uncertain optimization problem and its uncertain dual problem by proving the sufficient conditions for a weak duality and a strong duality between the deterministic robust counterpart of the primal model and the optimistic counterpart of its dual problem. The results on robust duality theorems are established in terms of underestimators. Additionally, to illustrate or support this study, some examples are presented.

Citation: 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 and Optimization, 2022, 12 (1) : 93-107. doi: 10.3934/naco.2021053
##### References:
 [1] V. Jeyakumar and S. Srisatkunarajah, New sufficiency for global optimality and duality of mathematical programming problems via underestimators, Journal of Optimization Theory and Applications, 140 (2009), 239-247.  doi: 10.1007/s10957-008-9438-7. [2] V. Jeyakumar and S. Srisatkunarajah, Geometric conditions for Kuhn-Tucker sufficiency of global optimality in mathematical programming, European Journal of Operational Research, 194 (2009), 363-367.  doi: 10.1016/j.ejor.2007.12.021. [3] A. Ben-Tal, L. E. Ghaoui and A. Nemirovski, Robust Optimization, Princeton University Press, Princeton, 2009.  doi: 10.1515/9781400831050. [4] D. Bertsimas, D. B. Brown and C. Caramanis, Theory and applications of robust optimization, SIAM Review, 53 (2011), 464-501.  doi: 10.1137/080734510. [5] A. Ben-Tal and A. Nemirovski, Robust convex optimization, Mathematics of Operations Research, 23 (1998), 769-805.  doi: 10.1287/moor.23.4.769. [6] V. Jeyakumar and G. Y. Li, Strong duality in robust convex programming: complete characterizations, SIAM Journal on Optimization, 20 (2010), 3384-3407.  doi: 10.1137/100791841. [7] R. I. Bot, V. Jeyakumar and G. Y. Li, Robust duality in parametric convex optimization, Journal of Nonlinear and Variational Analysis, 21 (2013), 177-189.  doi: 10.1007/s11228-012-0219-y. [8] A. Beck and A. Ben-Tal, Duality in robust optimization: primal worst equals dual best, Operations Research Letters, 37 (2009), 1-6.  doi: 10.1016/j.orl.2008.09.010. [9] D. Kuroiwa and G. M. Lee, On robust convex multiobjective optimization, Journal of Nonlinear and Convex Analysis, 15 (2014), 1125-1136. [10] A. Ben-Tal and A. Nemirovski, Robust optimization-methodology and applications, Mathematical Programming, Series B, 92 (2002), 453-480.  doi: 10.1007/s101070100286. [11] D. Bertsimas and D. B. Brown, Constructing uncertainty sets for robust linear optimization, Operational Research, 57 (2009), 1483-1495.  doi: 10.1287/opre.1080.0646. [12] V. Jeyakumar, G. M. Lee and G. Y. Li, Characterizing robust solution sets of convex programs under data uncertainty, Journal of Optimization Theory and Applications, 164 (2015), 407-435.  doi: 10.1007/s10957-014-0564-0. [13] X. K. Sun, Z. Y. Peng and X. L. Guo, Some characterizations of robust optimal solutions for uncertain convex optimization problems, Optimization Letters, 10 (2016), 1463-1478.  doi: 10.1007/s11590-015-0946-8. [14] X. B. Li and S. Wang, Characterizations of robust solution set of convex programs with uncertain data, Optimization Letters, 12 (2018), 1387-1402.  doi: 10.1007/s11590-017-1187-9. [15] C. A. Floudas, Deterministic Global Optimization: Theory, Methods and Applications, Kluwer Academic, Dordrecht, 2000. doi: 10.1007/978-1-4757-4949-6. [16] R. Horst and P. Pardalos, Handbook of Global Optimization, Kluwer Academic, Dordrecht, 1994. doi: 10.1007/978-1-4615-2025-2. [17] A. Migdalas, P. M. Pardalos and P. Varbrand, From Local to Global Optimization, Nonconvex Optimization and Its Applications, Kluwer Academic, Dordrecht, 2001. doi: 10.1007/978-1-4613-0307-7. [18] P. Pardalos and H. Romeijn, Handbook in Global Optimization-2, Kluwer Academic, Dordrecht, 2002. [19] N. Q. Huy, V. Jeyakumar and G. M. Lee, Sufficient global optimality conditions for multi-extremal smooth minimization problems with bounds and linear matrix inequality constraints, Asia-Pacific Journal of Operational Research, 47 (2006), 439-450.  doi: 10.1017/S1446181100010063. [20] V. Jeyakumar, A. M. Rubinov and Z. Y. Wu, Sufficient global optimality constrains for non-convex quadratic minimization problems with box constraints, Journal of Global Optimization, 36 (2006), 461-468.  doi: 10.1007/s10898-006-9022-3. [21] V. Jeyakumar, S. Srisatkunrajah and N. Q. Huy, Kuhn-Tucker sufficiency for global minimum of multiextremal mathematical programming problems, Journal of Mathematical Analysis and Applications, 335 (2007), 779-788.  doi: 10.1016/j.jmaa.2007.02.013. [22] M. S. Bazaraa, H. D. Sherali and C. M. Shetty, Nonlinear Programming: Theory and Algorithms, Wiley-Interscience, New York, 2006. doi: 10.1002/0471787779. [23] B. D. Craven, Control and Optimization, Chapman and Hall, London, 1995. doi: 10.1007/978-1-4899-7226-2. [24] B. D. Craven, Optimization with generalized invexity, Optimization, 54 (2005), 595-603.  doi: 10.1080/02331930500342716. [25] M. A. Hanson, On sufficiency of Kuhn-Tucker conditions, Journal of Mathematical Analysis and Applications, 80 (1991), 545-550.  doi: 10.1016/0022-247X(81)90123-2. [26] M. A. Hanson, A generalization of the Kuhn-Tucker sufficiency conditions, Journal of Mathematical Analysis and Applications, 184 (1994), 146-155.  doi: 10.1006/jmaa.1994.1190. [27] V. Jeyakumar and B. Mond, On generalized convex mathematical programming, Journal of the Australian Mathematical Society Series B, 34 (1992), 43-53.  doi: 10.1017/S0334270000007372. [28] O. L. Mangasarian, Nonlinear Programming, Classics in Applied Mathematics, SIAM, Philadelphia, 1994. doi: 10.1137/1.9781611971255. [29] B. S. Murdukhovich, Variational Analysis and Generalized Differentiation I: Basic Theory, Springer, Berlin, 2006. [30] J. B. Hiriart-Urruty, When is a point $x$ satisfying $\nabla f=0$ a global minimum of $f$?, The American Mathematical Monthly, 93 (1986), 556-558.  doi: 10.2307/2323035.

show all references

##### References:
 [1] V. Jeyakumar and S. Srisatkunarajah, New sufficiency for global optimality and duality of mathematical programming problems via underestimators, Journal of Optimization Theory and Applications, 140 (2009), 239-247.  doi: 10.1007/s10957-008-9438-7. [2] V. Jeyakumar and S. Srisatkunarajah, Geometric conditions for Kuhn-Tucker sufficiency of global optimality in mathematical programming, European Journal of Operational Research, 194 (2009), 363-367.  doi: 10.1016/j.ejor.2007.12.021. [3] A. Ben-Tal, L. E. Ghaoui and A. Nemirovski, Robust Optimization, Princeton University Press, Princeton, 2009.  doi: 10.1515/9781400831050. [4] D. Bertsimas, D. B. Brown and C. Caramanis, Theory and applications of robust optimization, SIAM Review, 53 (2011), 464-501.  doi: 10.1137/080734510. [5] A. Ben-Tal and A. Nemirovski, Robust convex optimization, Mathematics of Operations Research, 23 (1998), 769-805.  doi: 10.1287/moor.23.4.769. [6] V. Jeyakumar and G. Y. Li, Strong duality in robust convex programming: complete characterizations, SIAM Journal on Optimization, 20 (2010), 3384-3407.  doi: 10.1137/100791841. [7] R. I. Bot, V. Jeyakumar and G. Y. Li, Robust duality in parametric convex optimization, Journal of Nonlinear and Variational Analysis, 21 (2013), 177-189.  doi: 10.1007/s11228-012-0219-y. [8] A. Beck and A. Ben-Tal, Duality in robust optimization: primal worst equals dual best, Operations Research Letters, 37 (2009), 1-6.  doi: 10.1016/j.orl.2008.09.010. [9] D. Kuroiwa and G. M. Lee, On robust convex multiobjective optimization, Journal of Nonlinear and Convex Analysis, 15 (2014), 1125-1136. [10] A. Ben-Tal and A. Nemirovski, Robust optimization-methodology and applications, Mathematical Programming, Series B, 92 (2002), 453-480.  doi: 10.1007/s101070100286. [11] D. Bertsimas and D. B. Brown, Constructing uncertainty sets for robust linear optimization, Operational Research, 57 (2009), 1483-1495.  doi: 10.1287/opre.1080.0646. [12] V. Jeyakumar, G. M. Lee and G. Y. Li, Characterizing robust solution sets of convex programs under data uncertainty, Journal of Optimization Theory and Applications, 164 (2015), 407-435.  doi: 10.1007/s10957-014-0564-0. [13] X. K. Sun, Z. Y. Peng and X. L. Guo, Some characterizations of robust optimal solutions for uncertain convex optimization problems, Optimization Letters, 10 (2016), 1463-1478.  doi: 10.1007/s11590-015-0946-8. [14] X. B. Li and S. Wang, Characterizations of robust solution set of convex programs with uncertain data, Optimization Letters, 12 (2018), 1387-1402.  doi: 10.1007/s11590-017-1187-9. [15] C. A. Floudas, Deterministic Global Optimization: Theory, Methods and Applications, Kluwer Academic, Dordrecht, 2000. doi: 10.1007/978-1-4757-4949-6. [16] R. Horst and P. Pardalos, Handbook of Global Optimization, Kluwer Academic, Dordrecht, 1994. doi: 10.1007/978-1-4615-2025-2. [17] A. Migdalas, P. M. Pardalos and P. Varbrand, From Local to Global Optimization, Nonconvex Optimization and Its Applications, Kluwer Academic, Dordrecht, 2001. doi: 10.1007/978-1-4613-0307-7. [18] P. Pardalos and H. Romeijn, Handbook in Global Optimization-2, Kluwer Academic, Dordrecht, 2002. [19] N. Q. Huy, V. Jeyakumar and G. M. Lee, Sufficient global optimality conditions for multi-extremal smooth minimization problems with bounds and linear matrix inequality constraints, Asia-Pacific Journal of Operational Research, 47 (2006), 439-450.  doi: 10.1017/S1446181100010063. [20] V. Jeyakumar, A. M. Rubinov and Z. Y. Wu, Sufficient global optimality constrains for non-convex quadratic minimization problems with box constraints, Journal of Global Optimization, 36 (2006), 461-468.  doi: 10.1007/s10898-006-9022-3. [21] V. Jeyakumar, S. Srisatkunrajah and N. Q. Huy, Kuhn-Tucker sufficiency for global minimum of multiextremal mathematical programming problems, Journal of Mathematical Analysis and Applications, 335 (2007), 779-788.  doi: 10.1016/j.jmaa.2007.02.013. [22] M. S. Bazaraa, H. D. Sherali and C. M. Shetty, Nonlinear Programming: Theory and Algorithms, Wiley-Interscience, New York, 2006. doi: 10.1002/0471787779. [23] B. D. Craven, Control and Optimization, Chapman and Hall, London, 1995. doi: 10.1007/978-1-4899-7226-2. [24] B. D. Craven, Optimization with generalized invexity, Optimization, 54 (2005), 595-603.  doi: 10.1080/02331930500342716. [25] M. A. Hanson, On sufficiency of Kuhn-Tucker conditions, Journal of Mathematical Analysis and Applications, 80 (1991), 545-550.  doi: 10.1016/0022-247X(81)90123-2. [26] M. A. Hanson, A generalization of the Kuhn-Tucker sufficiency conditions, Journal of Mathematical Analysis and Applications, 184 (1994), 146-155.  doi: 10.1006/jmaa.1994.1190. [27] V. Jeyakumar and B. Mond, On generalized convex mathematical programming, Journal of the Australian Mathematical Society Series B, 34 (1992), 43-53.  doi: 10.1017/S0334270000007372. [28] O. L. Mangasarian, Nonlinear Programming, Classics in Applied Mathematics, SIAM, Philadelphia, 1994. doi: 10.1137/1.9781611971255. [29] B. S. Murdukhovich, Variational Analysis and Generalized Differentiation I: Basic Theory, Springer, Berlin, 2006. [30] J. B. Hiriart-Urruty, When is a point $x$ satisfying $\nabla f=0$ a global minimum of $f$?, The American Mathematical Monthly, 93 (1986), 556-558.  doi: 10.2307/2323035.
 [1] Xiaoqing Ou, Suliman Al-Homidan, Qamrul Hasan Ansari, Jiawei Chen. Image space analysis for uncertain multiobjective optimization problems: Robust optimality conditions. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021199 [2] Nazih Abderrazzak Gadhi, Fatima Zahra Rahou. Sufficient optimality conditions and Mond-Weir duality results for a fractional multiobjective optimization problem. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021216 [3] Heinz Schättler, Urszula Ledzewicz, Helmut Maurer. Sufficient conditions for strong local optimality in optimal control problems with $L_{2}$-type objectives and control constraints. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2657-2679. doi: 10.3934/dcdsb.2014.19.2657 [4] Yong Xia. New sufficient global optimality conditions for linearly constrained bivalent quadratic optimization problems. Journal of Industrial and Management Optimization, 2009, 5 (4) : 881-892. doi: 10.3934/jimo.2009.5.881 [5] Bernard Dacorogna. Necessary and sufficient conditions for strong ellipticity of isotropic functions in any dimension. Discrete and Continuous Dynamical Systems - B, 2001, 1 (2) : 257-263. doi: 10.3934/dcdsb.2001.1.257 [6] Miniak-Górecka Alicja, Nowakowski Andrzej. Sufficient optimality conditions for a class of epidemic problems with control on the boundary. Mathematical Biosciences & Engineering, 2017, 14 (1) : 263-275. doi: 10.3934/mbe.2017017 [7] Lihua Li, Yan Gao, Hongjie Wang. Second order sufficient optimality conditions for hybrid control problems with state jump. Journal of Industrial and Management Optimization, 2015, 11 (1) : 329-343. doi: 10.3934/jimo.2015.11.329 [8] Ying Gao, Xinmin Yang, Kok Lay Teo. Optimality conditions for approximate solutions of vector optimization problems. Journal of Industrial and Management Optimization, 2011, 7 (2) : 483-496. doi: 10.3934/jimo.2011.7.483 [9] Xiao-Bing Li, Qi-Lin Wang, Zhi Lin. Optimality conditions and duality for minimax fractional programming problems with data uncertainty. Journal of Industrial and Management Optimization, 2019, 15 (3) : 1133-1151. doi: 10.3934/jimo.2018089 [10] Adela Capătă. Optimality conditions for strong vector equilibrium problems under a weak constraint qualification. Journal of Industrial and Management Optimization, 2015, 11 (2) : 563-574. doi: 10.3934/jimo.2015.11.563 [11] M. Soledad Aronna. Second order necessary and sufficient optimality conditions for singular solutions of partially-affine control problems. Discrete and Continuous Dynamical Systems - S, 2018, 11 (6) : 1233-1258. doi: 10.3934/dcdss.2018070 [12] J.-P. Raymond, F. Tröltzsch. Second order sufficient optimality conditions for nonlinear parabolic control problems with state constraints. Discrete and Continuous Dynamical Systems, 2000, 6 (2) : 431-450. doi: 10.3934/dcds.2000.6.431 [13] Xiuhong Chen, Zhihua Li. On optimality conditions and duality for non-differentiable interval-valued programming problems with the generalized (F, ρ)-convexity. Journal of Industrial and Management Optimization, 2018, 14 (3) : 895-912. doi: 10.3934/jimo.2017081 [14] Samir Hodžić, Enes Pasalic. Generalized bent functions -sufficient conditions and related constructions. Advances in Mathematics of Communications, 2017, 11 (3) : 549-566. doi: 10.3934/amc.2017043 [15] David Yang Gao. Sufficient conditions and perfect duality in nonconvex minimization with inequality constraints. Journal of Industrial and Management Optimization, 2005, 1 (1) : 53-63. doi: 10.3934/jimo.2005.1.53 [16] Vladimir Srochko, Vladimir Antonik, Elena Aksenyushkina. Sufficient optimality conditions for extremal controls based on functional increment formulas. Numerical Algebra, Control and Optimization, 2017, 7 (2) : 191-199. doi: 10.3934/naco.2017013 [17] Bhawna Kohli. Sufficient optimality conditions using convexifactors for optimistic bilevel programming problem. Journal of Industrial and Management Optimization, 2021, 17 (6) : 3209-3221. doi: 10.3934/jimo.2020114 [18] Yuhua Sun, Laisheng Wang. Optimality conditions and duality in nondifferentiable interval-valued programming. Journal of Industrial and Management Optimization, 2013, 9 (1) : 131-142. doi: 10.3934/jimo.2013.9.131 [19] Xian-Jun Long, Nan-Jing Huang, Zhi-Bin Liu. Optimality conditions, duality and saddle points for nondifferentiable multiobjective fractional programs. Journal of Industrial and Management Optimization, 2008, 4 (2) : 287-298. doi: 10.3934/jimo.2008.4.287 [20] Xian-Jun Long, Jing Quan. Optimality conditions and duality for minimax fractional programming involving nonsmooth generalized univexity. Numerical Algebra, Control and Optimization, 2011, 1 (3) : 361-370. doi: 10.3934/naco.2011.1.361

Impact Factor: