# 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 & 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.  Google Scholar [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.  Google Scholar [3] A. Ben-Tal, L. E. Ghaoui and A. Nemirovski, Robust Optimization, Princeton University Press, Princeton, 2009.  doi: 10.1515/9781400831050.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [9] D. Kuroiwa and G. M. Lee, On robust convex multiobjective optimization, Journal of Nonlinear and Convex Analysis, 15 (2014), 1125-1136.   Google Scholar [10] A. Ben-Tal and A. Nemirovski, Robust optimization-methodology and applications, Mathematical Programming, Series B, 92 (2002), 453-480.  doi: 10.1007/s101070100286.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [15] C. A. Floudas, Deterministic Global Optimization: Theory, Methods and Applications, Kluwer Academic, Dordrecht, 2000. doi: 10.1007/978-1-4757-4949-6.  Google Scholar [16] R. Horst and P. Pardalos, Handbook of Global Optimization, Kluwer Academic, Dordrecht, 1994. doi: 10.1007/978-1-4615-2025-2.  Google Scholar [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.  Google Scholar [18] P. Pardalos and H. Romeijn, Handbook in Global Optimization-2, Kluwer Academic, Dordrecht, 2002. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [23] B. D. 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S. Murdukhovich, Variational Analysis and Generalized Differentiation I: Basic Theory, Springer, Berlin, 2006.  Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [3] A. Ben-Tal, L. E. Ghaoui and A. Nemirovski, Robust Optimization, Princeton University Press, Princeton, 2009.  doi: 10.1515/9781400831050.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [9] D. Kuroiwa and G. M. Lee, On robust convex multiobjective optimization, Journal of Nonlinear and Convex Analysis, 15 (2014), 1125-1136.   Google Scholar [10] A. Ben-Tal and A. Nemirovski, Robust optimization-methodology and applications, Mathematical Programming, Series B, 92 (2002), 453-480.  doi: 10.1007/s101070100286.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [15] C. A. Floudas, Deterministic Global Optimization: Theory, Methods and Applications, Kluwer Academic, Dordrecht, 2000. doi: 10.1007/978-1-4757-4949-6.  Google Scholar [16] R. Horst and P. Pardalos, Handbook of Global Optimization, Kluwer Academic, Dordrecht, 1994. doi: 10.1007/978-1-4615-2025-2.  Google Scholar [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.  Google Scholar [18] P. Pardalos and H. Romeijn, Handbook in Global Optimization-2, Kluwer Academic, Dordrecht, 2002. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [23] B. D. Craven, Control and Optimization, Chapman and Hall, London, 1995. doi: 10.1007/978-1-4899-7226-2.  Google Scholar [24] B. D. Craven, Optimization with generalized invexity, Optimization, 54 (2005), 595-603.  doi: 10.1080/02331930500342716.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [28] O. L. Mangasarian, Nonlinear Programming, Classics in Applied Mathematics, SIAM, Philadelphia, 1994. doi: 10.1137/1.9781611971255.  Google Scholar [29] B. S. Murdukhovich, Variational Analysis and Generalized Differentiation I: Basic Theory, Springer, Berlin, 2006.  Google Scholar [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.  Google Scholar
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