×

The KKT optimality conditions for optimization problem with interval-valued objective function on Hadamard manifolds. (English) Zbl 1491.90156

Some of Karush-Kuhn-Tucker criteria under Hadamard’s manifolds are developed, by appropriate numerical applications.The results can be extended in the Infinite Ordered Vector Spaces, with real consequences.

MSC:

90C30 Nonlinear programming
90C26 Nonconvex programming, global optimization
Full Text: DOI

References:

[1] Aguirre-Cipe, I.; Lopez, R.; Mallea-Zepeda, E., A study of interval optimization problems, Optim Lett (2019) · Zbl 1471.90136 · doi:10.1007/s11590-019-01496-9
[2] Alefeld, G.; Mayer, G., Interval analysis: theory and applications, J Comput Appl Math, 121, 1-2, 421-464 (2000) · Zbl 0995.65056 · doi:10.1016/S0377-0427(00)00342-3
[3] Ding, K.; Huang, NJ., A new class of interval projection neural networks for solving interval quadratic program, Chaos Solitons Fract, 25, 718-725 (2008) · Zbl 1137.90020 · doi:10.1016/j.chaos.2006.05.037
[4] Jiang, C.; Han, X.; Liu, GR, The optimization of the variable binder force in U-shaped forming with uncertain friction coefficient, J Mater Process Technol, 182, 262-267 (2007) · doi:10.1016/j.jmatprotec.2006.08.002
[5] Ahmad, I.; Singh, D.; Dar, BA., Optimality conditions for invex interval valued nonlinear programming problems involving generalized H-derivative, Filomat, 30, 8, 2121-2138 (2016) · Zbl 1474.90391 · doi:10.2298/FIL1608121A
[6] Chalco-Cano, Y.; Lodwick, WA; Rufian-Lizana, A., Optimality conditions of type KKT for optimization problem with interval-valued objective function via generalized derivative, Fuzzy Optim Decis Ma, 12, 3, 305-322 (2013) · Zbl 1428.90189 · doi:10.1007/s10700-013-9156-y
[7] Ishibuchi, H.; Tanaka, H., Multiobjective programming in optimization of the interval objective function, Eur J Oper Res, 48, 219-225 (1990) · Zbl 0718.90079 · doi:10.1016/0377-2217(90)90375-L
[8] Jayswal, A.; Stancu-Minasian, IM; Ahmad, I., On sufficiency and duality for a class of interval-valued programming problems, Appl Math Comput, 218, 4119-4127 (2011) · Zbl 1268.90087
[9] Singh, D.; Dar, BA; Kim, DS., KKT optimality conditions in interval-valued multiobjective programming with generalized differentiable functions, Eur J Oper Res, 254, 1, 29-39 (2016) · Zbl 1346.90752 · doi:10.1016/j.ejor.2016.03.042
[10] Sun, YH; Wang, LS., Optimality conditions and duality in nondifferentiable interval-valued programming, J Ind Manag Optim, 9, 1, 131-142 (2017) · Zbl 1263.90122
[11] Wu, HC., The Karush-Kuhn-Tucker optimality conditions in an optimization problem with interval-valued objective function, Eur J Oper Res, 176, 46-59 (2007) · Zbl 1137.90712 · doi:10.1016/j.ejor.2005.09.007
[12] Wu, HC., Duality theory for optimization problems with interval-valued objective function, J Optim Theory Appl, 144, 3, 615-628 (2010) · Zbl 1198.90365 · doi:10.1007/s10957-009-9613-5
[13] Zhang, JK; Liu, SY; Li, LF., The KKT optimality conditions in a class of generalized convex optimization problems with an interval-valued objective function, Optim Lett, 8, 2, 607-631 (2014) · Zbl 1317.90240 · doi:10.1007/s11590-012-0601-6
[14] Zhou, HC, Wang, YJ.Optimality condition and mixed duality for interval-valued optimization. Fuzzy information and engineering. Vol. 2, Advances in Intelligent and Soft Computing, Vol. 62, Proceedings of the Third International Conference on Fuzzy Information and Engineering (ICFIE 2009); Springer; 2009. p. 1315-1323. · Zbl 1190.90226
[15] Bento, GC; Ferreira, OP; Oliveira, PR., Proximal point method for a special class of nonconvex functions on Hadamard manifolds, Optimization, 64, 2, 289-319 (2015) · Zbl 1310.49006 · doi:10.1080/02331934.2012.745531
[16] Bento, GC; Melo, JG., Subgradient method for convex feasibility on Riemannian manifolds, J Optim Theory Appl, 152, 773-785 (2012) · Zbl 1270.90101 · doi:10.1007/s10957-011-9921-4
[17] Ferreira, OP; Lucambio Perez, LR; Nemeth, SZ., Singularities of monotone vector fields and an extragradient-type algorithm, J Global Optim, 31, 133-151 (2005) · Zbl 1229.58007 · doi:10.1007/s10898-003-3780-y
[18] Hosseini, S.; Pouryayevali, MR., Generalized gradients and characterization of epi-Lipschitz sets in Riemannian manifolds, Nonlinear Anal, 74, 3884-3895 (2011) · Zbl 1225.49046 · doi:10.1016/j.na.2011.02.023
[19] Hosseini, S.; Pouryayevali, MR., Nonsmooth optimization techniques on Riemannian manifolds, J Optim Theory Appl, 158, 328-342 (2013) · Zbl 1273.90110 · doi:10.1007/s10957-012-0250-z
[20] Li, C.; Yao, JC., Weak sharp minima on Riemannian manifolds, SIAM J Optim, 21, 4, 1523-1560 (2011) · Zbl 1236.49089 · doi:10.1137/09075367X
[21] Nemeth, SZ., Five kinds of monotone vector fields, Pure Appl Math, 9, 417-428 (1999) · Zbl 0937.58010
[22] Rapcsak, T., Geodesic convexity in nonlinear optimization, J Optim Theory and Appl, 69, 169-183 (1991) · Zbl 0702.90066 · doi:10.1007/BF00940467
[23] Rapcsak, T.Smooth nonlinear optimization in \(####\). Nonconvex optimization and applications. Vol.19. Dordrecht: Kluwer Academic Publishers; 1997. · Zbl 1009.90109
[24] Ruiz-Garzon, G.; Osuna-Gmez, R.; Rufin-Lizana, A., Optimality and duality on Riemannian manifolds, Taiwan J Math, 22, 5, 1245-1259 (2018) · Zbl 1402.49031 · doi:10.11650/tjm/180501
[25] Udriste, C.Convex functions and optimization methods on Riemannian manifolds. Mathematics and its Applications. Vol. 297. Dordrecht: Kluwer Academic Publishers; 1994. · Zbl 0932.53003
[26] Wang, JM; Lopez, G.; Martin-Marquez, V., Monotone and accretive vector fields on Riemannian manifolds, J Optim Theory Appl, 146, 3, 691-708 (2010) · Zbl 1208.53049 · doi:10.1007/s10957-010-9688-z
[27] Chavel, I., Riemannian geometry-A modern introduction (1993), Cambridge: Cambridge University Press, Cambridge · Zbl 0810.53001
[28] Klingenberg, W., A course in differential geometry (1978), Berlin: Springer-verlag, Berlin · Zbl 0366.53001
[29] Sakai, T.Riemannian geometry. Translations of mathematical monographs. Vol. 149. Providence (RI): American Mathematical Society; 1996. · Zbl 0886.53002
[30] Alefeld, G.; Herzberger, J., Introduction to interval computations (1983), NY: Academic Press, NY · Zbl 0552.65041
[31] Moore, RE., Interval analysis (1966), Englewood Cliffs (NJ): Prentice-Hall, Englewood Cliffs (NJ) · Zbl 0176.13301
[32] Moore, RE., Method and applications of interval analysis (1979), Philadelphia: SIAM, Philadelphia · Zbl 0417.65022
[33] Stefanini, L., Generalized Hukuhara differentiability of interval-valued functions and interval differential equations, Nonlinear Anal., 71, 1311-1328 (2009) · Zbl 1188.28002 · doi:10.1016/j.na.2008.12.005
[34] Bazaraa, MS; Sherali, HD; Shetty, CM., Nonlinear programming: theory and algorithms (1993), New Jersey: John Wiley and Sons, New Jersey · Zbl 0774.90075
[35] Mangasarian, OL., Nonlinear programming (1969), New York: MacGraw-Hill, New York · Zbl 0194.20201
[36] Bhattacharyya, R.; Kar, S.; Majumder, DD., Fuzzy mean-variance-skewness portfolio selection models by interval analysis, Comput Math Appl, 61, 126-137 (2011) · Zbl 1207.91059 · doi:10.1016/j.camwa.2010.10.039
[37] Giove, S.; Funari, S.; Nardelli, C., An interval portfolio selection problems based on regret function, Eur J Oper Res, 170, 253-264 (2006) · Zbl 1079.91030 · doi:10.1016/j.ejor.2004.05.030
[38] Li, J.; Xu, JP., A class of possibilistic portfolio selection model with interval coefficients and its application, Fuzzy Optim Decis Mak, 6, 123-137 (2007) · Zbl 1154.91459 · doi:10.1007/s10700-007-9005-y
[39] Cesarone, F.; Scozzari, A.; Tardella, F., A new method for mean-variance portfolio optimization with cardinality constraints, Ann Oper Res, 205, 213-234 (2013) · Zbl 1269.91069 · doi:10.1007/s10479-012-1165-7
[40] Le Thi, HA; Moeini, M., Long-short portfolio optimization under cardinality constraints by difference of convex functions algorithm, J Optim Appl, 161, 199-224 (2014) · Zbl 1300.91046 · doi:10.1007/s10957-012-0197-0
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.