# zbMATH — the first resource for mathematics

Second-order optimality conditions in locally Lipschitz inequality-constrained multiobjective optimization. (English) Zbl 07220275
Summary: The main goal of this paper is to give some primal and dual Karush-Kuhn-Tucker second-order necessary conditions for the existence of a strict local Pareto minimum of order two for an inequality-constrained multiobjective optimization problem. Dual Karush-Kuhn-Tucker second-order sufficient conditions are provided too. We suppose that the objective function and the active inequality constraints are only locally Lipschitz in the primal necessary conditions and only strictly differentiable in sense of Clarke at the extremum point in the dual conditions. Examples illustrate the applicability of the obtained results.
##### MSC:
 90C29 Multi-objective and goal programming 49K27 Optimality conditions for problems in abstract spaces 90C30 Nonlinear programming 90C48 Programming in abstract spaces
MODENA
Full Text:
##### References:
 [1] Constantin, E., First-order necessary conditions in locally Lipschitz multiobjective optimization, Optimization, 67, 9, 1447-1460 (2018) · Zbl 06987979 [2] Constantin, E., Necessary conditions for weak efficiency for nonsmooth degenerate multiobjective optimization problems, J. Glob. Optim., 75, 1, 111-129 (2019) · Zbl 1435.90125 [3] Lara, F., Optimality conditions for nonconvex nonsmooth optimization via global derivatives, J. Optim. Theory Appl., 185, 134-150 (2020) · Zbl 1448.90098 [4] Luu, DV, Second-order necessary efficiency conditions for nonsmooth vector equilibrium problems, J. Glob. Optim., 70, 437-453 (2018) · Zbl 1405.90120 [5] Xiao, Y-B; Tuyen, NV; Yao, J-C; Wen, C-F, Locally Lipschitz optimization problems: second-order constraint qualifications, regularity condition and KKT necessary optimality conditions, Positivity, 24, 313-337 (2020) · Zbl 1451.90147 [6] You, M.; Li, S., Separation functions and optimality conditions in vector optimization, J. Optim. Theory Appl., 175, 527-544 (2017) · Zbl 1391.90570 [7] Giannessi, F., Vector Variational Inequalities and Vector Equilibria, Mathematical Theories (2000), Dordrecht: Kluwer Academic Publishers, Dordrecht · Zbl 0952.00009 [8] Ansari, QH; Yao, JC, Recent Developments in Vector Optimization (2012), New York: Springer, New York [9] Burachik, RS; Rizvi, MM, On weak and strong Kuhn-Tucker conditions for smooth multiobjective optimization, J. Optim. Theory Appl., 155, 2, 477-491 (2012) · Zbl 1270.90058 [10] Feng, M.; Li, S., On second-order optimality conditions for continuously Fréchet differentiable vector optimization problems, Optimization, 67, 12, 2117-2137 (2018) · Zbl 1427.90248 [11] Ginchev, I.; Ivanov, VI, Second-order optimality conditions for problems with $$C^1$$ Data, J. Math. Anal. Appl., 340, 646-657 (2008) · Zbl 1190.90208 [12] Huy, NQ; Kim, DS; Tuyen, NV, New second-order Karush-Kuhn-Tucker optimality conditions for vector optimization, Appl. Math. Optim., 79, 279-307 (2019) · Zbl 1417.49029 [13] Ivanov, VI, Optimality conditions for an isolated minimum of order two in $$C^1$$ constrained optimization, J. Math. Anal. Appl., 356, 30-41 (2009) · Zbl 1176.90569 [14] Ivanov, VI, Second-order optimality conditions for vector problems with continuously Fréchet differentiable data and second-order constraint qualifications, J. Optim. Theory Appl., 166, 777-790 (2015) · Zbl 1327.90378 [15] Tuyen, NV; Huy, NQ; Kim, DS, Strong second-order Karush-Kuhn-Tucker optimality conditions for vector optimization, Appl. Anal., 99, 1, 103-120 (2020) · Zbl 1434.90185 [16] Maciel, MC; Santos, SA; Sottosanto, GN, Regularity conditions in differentiable vector optimization revisited, J. Optim. Theory Appl., 142, 385-398 (2009) · Zbl 1181.90239 [17] Maciel, MC; Santos, SA; Sottosanto, GN, On second-order optimality conditions for vector optimization, J. Optim. Theory Appl., 149, 332-351 (2011) · Zbl 1262.90156 [18] Chandra, S.; Dutta, J.; Lalitha, CS, Regularity conditions and optimality in vector optimization, Num. Funct. Anal. Optim., 25, 5-6, 479-501 (2004) · Zbl 1071.90040 [19] Giorgi, G.; Jimenez, B.; Novo, V., Strong Kuhn-Tucker conditions and constraint qualifications in locally Lipschitz multiobjective optimization problems, TOP, 17, 288-304 (2009) · Zbl 1198.90348 [20] Constantin, E., Second-order necessary conditions based on second-order tangent cones, Math. Sci. Res. J., 10, 2, 42-56 (2006) · Zbl 1106.49041 [21] Constantin, E., Second-order necessary conditions in locally Lipschitz optimization with inequality constraints, Optim. Lett., 9, 2, 245-261 (2015) · Zbl 1318.49026 [22] Constantin, E., Second-order necessary conditions for set constrained nonsmooth optimization problems via second-order projective tangent cones, Libertas Math., 36, 1, 1-24 (2016) · Zbl 1375.49034 [23] Ivanov, VI, Second-order optimality conditions for inequality constrained problems with locally Lipschitz data, Optim. Lett., 4, 597-608 (2010) · Zbl 1244.90242 [24] Kell, J.; Handl, DB; Knowles, J., Multiobjective optimization in bioinformatics and computational biology, IEEE/ACM Trans. Comput. Biol. Bioinform., 4, 2, 279-292 (2007) [25] Taneda, A., Multi-objective pairwise RNA sequence alignment, Bioinformatics, 26, 19, 2383-2390 (2010) [26] Taneda, A., MODenA: a multi-objective RNA inverse folding, Adv. Appl. Bioinform. Chem., 4, 1-12 (2011) [27] Liu, Y.; Ning, Z.; Guo, M.; Liu, Y.; Gali, NK; Duan, Y.; Cai, J.; Westerdahl, D.; Liu, X.; Xu, K.; Ho, K.; Kan, H.; Fu, Q.; Lan, K., Aerodynamic analysis of SARS-CoV-2 in two Wuhan hospitals, Nature (2020) [28] Ben-Tal, A., Second-order and related extremality conditions in nonlinear programming, J. Optim. Theory Appl., 31, 2, 143-165 (1980) · Zbl 0416.90062 [29] Jiménez, B., Strict efficiency in vector optimization, J. Math. Anal. Appl., 265, 264-284 (2002) · Zbl 1010.90075 [30] Clarke, FH, Optimization and Nonsmooth Analysis (1983), New York: Wiley, New York [31] Páles, Z.; Zeidan, VM, Nonsmooth optimum problems with constraints, SIAM J. Control Optim., 32, 5, 1476-1502 (1994) · Zbl 0821.49020 [32] Bigi, G., On sufficient second-order optimality conditions in multiobjective optimization, Math. Meth. Oper. Res., 63, 77-85 (2006) · Zbl 1102.90055
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.