Strong Karush–Kuhn–Tucker optimality conditions for multiobjective semi-infinite programming via tangential subdifferential
RAIRO - Operations Research - Recherche Opérationnelle, Tome 52 (2018) no. 4-5, pp. 1019-1041.

The main aim of this paper is to study strong Karush–Kuhn–Tucker (KKT) optimality conditions for nonsmooth multiobjective semi-infinite programming (MSIP) problems. By using tangential subdifferential and suitable regularity conditions, we establish some strong necessary optimality conditions for some types of efficient solutions of nonsmooth MSIP problems. In addition to the theoretical results, some examples are provided to illustrate the advantages of our outcomes.

Reçu le :
Accepté le :
DOI : 10.1051/ro/2018020
Classification : 90C32, 90C29, 49K99
Mots-clés : Multiobjective semi-infinite programming, efficient solution, weakly efficient solution, strong Karush–Kuhn–Tucker optimality conditions, tangential subdifferential
Tung, Le Thanh 1

1 
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     title = {Strong {Karush{\textendash}Kuhn{\textendash}Tucker} optimality conditions for multiobjective semi-infinite programming via tangential subdifferential},
     journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
     pages = {1019--1041},
     publisher = {EDP-Sciences},
     volume = {52},
     number = {4-5},
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Tung, Le Thanh. Strong Karush–Kuhn–Tucker optimality conditions for multiobjective semi-infinite programming via tangential subdifferential. RAIRO - Operations Research - Recherche Opérationnelle, Tome 52 (2018) no. 4-5, pp. 1019-1041. doi : 10.1051/ro/2018020. http://www.numdam.org/articles/10.1051/ro/2018020/

[1] J.P. Aubin and H. Frankowska, Set-Valued Analysis. Birkhäuser, Boston (1990). | MR | Zbl

[2] G. Caristi and M. Ferrara, Necessary conditions for nonsmooth multiobjective semi-infinite problems using Michel-Penot subdifferential. Decisions Econ. Finan. 40 (2017) 103–113. | DOI | MR | Zbl

[3] S. Chandra, J. Dutta and C.S. Lalitha, Regularity conditions and optimality in vector optimization. Numer. Funct. Anal. Optim. 25 (2004) 479–501. | DOI | MR | Zbl

[4] T.D. Chuong and D.S. Kim, Nonsmooth semi-infinite multiobjective optimization problems. J. Optim. Theory Appl. 160 (2014) 748–762. | DOI | MR | Zbl

[5] T.D. Chuong and J.-C. Yao, Isolated and proper efficiencies in semi-infinite vector optimization problems. J. Optim. Theory Appl. 162 (2014) 447–462. | DOI | MR | Zbl

[6] F.H. Clarke, Optimization and Nonsmooth Analysis. Wiley Interscience, New York (1983). | MR | Zbl

[7] V.F. Demyanov and A.M. Rubinov, Constructive Nonsmooth Analysis. Peter Lang, Frankfurt (1995). | MR | Zbl

[8] J. Dutta and C.S. Lalitha, Optimality conditions in convex optimization revisited. Optim. Lett. 7 (2013) 221–229. | DOI | MR | Zbl

[9] M. Ehrgott, Multicriteria Optimization. Springer, Berlin (2005). | MR | Zbl

[10] A.M. Geoffrion, Proper efficiency and the theory of vector maximization. J. Math. Anal. Appl. 22 (1968) 618–630. | DOI | MR | Zbl

[11] P.G. Georgiev, D.T. Luc and P.M. Pardalos, Robust aspects of solutions in deterministic multiple objective linear programming. Eur. J. Oper. Res. 229 (2013) 29–36. | DOI | MR | Zbl

[12] G. Giorgi and S. Komlósi, Dini derivatives in optimization - Part I. Riv. Mat. Sci. Econ. Soc. 15 (1993) 3–30. | MR | Zbl

[13] G. Giorgi and C. Zuccotti, Again on regularity conditions in differentiable vector optimization. Ann. Univ. Buchar. Math. Ser. 2 (2011) 157–177. | MR | Zbl

[14] G. Giorgi, B. Jiménez and V. Novo, On constraint qualification in directionally differentiable multiobjective optimization problems. RAIRO: OR 38 (2004) 255–274. | DOI | Numdam | MR | Zbl

[15] G. Giorgi, B. Jiménez and V. Novo, Strong Kuhn-Tucker conditions and constraint qualifications in locally Lipschitz multiobjective optimization problems. TOP 17 (2009) 288–304. | DOI | MR | Zbl

[16] M.A. Goberna and M.A. López, Linear Semi-Infinite Optimization. Wiley, Chichester (1998). | MR | Zbl

[17] M.A. Goberna and N. Kanzi, Optimality conditions in convex multiobjective SIP. Math. Program. 164 (2017) 167–191. | DOI | MR | Zbl

[18] M.A. Goberna, F. Guerra-Vázquez and M.I. Todorov, Constraint qualifications in convex vector semi-infinite optimization. Eur. J. Oper. Res. 249 (2016) 32–40. | DOI | MR | Zbl

[19] J.B. Hiriart-Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms I. Springer, Berlin (1993). | MR

[20] V. Jeyakumar, D.T. Luc, Nonsmooth calculus, minimality, and monotonicity of convexificators. J. Optim. Theory Appl. 101 (1999) 599–621. | DOI | MR | Zbl

[21] B. Jiménez and V. Novo, Cualificaciones de restricciones en problemas de optimizacion vectorial diferenciables, in Actas XVI C.E.D.Y.A./VI C.M.A. Universidad de las Palmas de Gran Canaria (1999) Vol. 1, 727–734.

[22] B. Jiménez and V. Novo, Alternative theorems and necessary optimality conditions for directionally differentiable multiobjective programs. J. Convex Anal. 9 (2002) 97–116. | MR | Zbl

[23] B. Jiménez and V. Novo, Optimality conditions in directionally differentiable Pareto problems with a set constraint via tangent cones. Numer. Funct. Anal. Optim. 24 (2003) 557–574. | DOI | MR | Zbl

[24] A. Kabgani and M. Soleimani-Damaneh, Characterization of (weakly/properly/robust) efficient solutions in nonsmooth semi-infinite multiobjective optimization using convexificators. Optimization 67 (2017) 217–235. | DOI | MR | Zbl

[25] A. Kabgani, M. Soleimani-Damaneh and M. Zamani, Optimality conditions in optimization problems with convex feasible set using convexificators. Math. Methods Oper. Res. 86 (2017) 103–121. | DOI | MR | Zbl

[26] N. Kanzi,On strong KKT optimality conditions for multiobjective semi-infinite programming problems with Lipschitzian data. Optim. Lett. 9 (2015) 1121–1129. | DOI | MR | Zbl

[27] N. Kanzi and S. Nobakhtian, Optimality conditions for nonsmooth semi-infinite multiobjective programming. Optim. Lett. 8 (2014) 1517–1528. | DOI | MR | Zbl

[28] P.Q. Khanh and L.T. Tung, Optimality conditions and duality for nonsmooth semi-infinite programming via convexificators, in New trends in Optimization and Variational Analysis for Applications - International Conference. Quy Nhon, Binh Dinh, Vietnam (2016).

[29] D.S. Kim and T.Q. Son, Characterizations of solutions sets of a class of nonconvex semi-infinite programming problems. J. Nonlinear Convex Anal. 12 (2011) 429–440. | MR | Zbl

[30] D. Kuroiwa and G.M. Lee, On robust multiobjective optimization. Vietnam J. Math. 40 (2012) 305–317. | MR | Zbl

[31] J.B. Lasserre, On representations of the feasible set in convex optimization. Optim. Lett. 5 (2011) 549–556. | DOI | Zbl

[32] C. Lemaréchal, An introduction to the theory of nonsmooth optimization. Optimization 17 (1986) 827–858. | DOI | MR | Zbl

[33] X.F. Li, Constraint qualifications in nonsmooth multiobjective optimization. J. Optim. Theory Appl. 106 (2008) 373–398. | MR | Zbl

[34] W. Li, C. Nahak and I. Singer, Constraint qualifications for semi-infinite systems of convex inequalities. SIAM J. Optim. 11 (2000) 31–52. | DOI | MR | Zbl

[35] D.T. Luc, Theory of Vector Optimization. Springer, Berlin (1989). | DOI | MR

[36] T. Maeda, Constraint qualifications in multiobjective optimization problems: differentiable case. J. Optim. Theory Appl. 80 (1994) 483–500. | DOI | MR | Zbl

[37] J.E. Martínez-Legaz, Optimality conditions for pseudo-convex minimization over convex sets defined by tangentially convex constraints. Optim. Lett. 9 (2015) 1017–1023. | DOI | MR | Zbl

[38] P. Michel and J.-P. Penot, Calcus sous-differentiel pour des fonctions Lipschitziennes et non Lipschitziennes. C. R. Acad. Sci. Paris Ser. I Math. 12 (1984) 269–272. | MR | Zbl

[39] B.S. Mordukhovich and T.T.A. Nghia, Constraint qualifications and optimality conditions in semi-infinite and infinite programming. Math. Program. 139 (2013) 271–300. | DOI | MR | Zbl

[40] Y. Pandey and S.K. Mishra, Optimality conditions and duality for semi-infinite mathematical programming problems with equilibrium constraints, using convexificators. Ann. Oper. Res. 269 (2018) 549–564. | DOI | MR | Zbl

[41] V. Preda and I. Chitescu, On constraint qualification in multiobjective optimization problems: semidifferentiable case. J. Optim. Theory Appl. 100 (1999) 417–433. | DOI | MR | Zbl

[42] B.N. Pshenichnyi, Necessary Conditions for an Extremum. Marcel Dekker Inc, New York (1971). | MR | Zbl

[43] R. Reemtsen and J.-J. Rückmann, Nonconvex optimization and its applications, in Semi-Infinite Programming. Kluwer Academic, Boston (1998). | DOI | MR | Zbl

[44] R.T. Rockafellar, Convex Analysis. In Vol. 28 of Princeton Math. Ser. Princeton University Press, Princeton, New Jersey (1970). | MR | Zbl

[45] T.Q. Son and D.S. Kim, A new approach to characterize the solution set of a pseudoconvex programming problem. J. Comput. Appl. Math. 261 (2014) 333–340. | DOI | MR | Zbl

[46] L.T. Tung, Karush-Kuhn-Tucker optimality conditions and duality for multiobjective semi-infinite programming via tangential subdifferential. In preparation (2017). | Numdam | MR | Zbl

[47] S. Yamamoto and D. Kuroiwa, Constraint qualifications for KKT optimality condition in convex optimization with locally Lipschitz inequality constraints. Linear Nonlinear Anal. 2 (2016) 239–244. | MR | Zbl

[48] J.J. Ye, Nondifferentiable multiplier rules for optimization and bilevel optimization problems. SIAM J. Optim. 15 (2014) 252–274. | MR | Zbl

[49] M. Zamani, M. Soleimani-Damaneh and A. Kabgani, Robustness in nonsmooth nonlinear multi-objective programming. Eur. J. Oper. Res. 247 (2015) 370–378. | DOI | MR | Zbl

[50] K.Q. Zhao, Strong Kuhn-Tucker optimality in nonsmooth multiobjective optimization problems. Pac. J. Optim. 11 (2015) 483–494. | MR | Zbl

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