In this paper, we introduce the concepts of second-order radial epiderivative and second-order generalized radial epiderivative for nonconvex set-valued maps. We also investigate some of their properties. We give existence theorems for the second-order generalized radial epiderivatives.
Aubin, J.P.,1981, Contingent Derivatives of Set-Valued Maps and Existence of Solutions to Nonlinear Inclusions and Differential Inclusions. In: Nachbin, L (ed.) Mathematics Analysis and Applications, part A, 160-229, Academic Press, New York.
Aubin, J.P., Frankowska, H., 1990, Set Valued Analysis,Birkhauser, Boston.
Aghezzaf, B. and Hachimi, M., 1999, Second Order Optimality Conditions in Multiobjective Optimization Problems,J. Optim. Theory Apply., 102,1,37-50.
Anh, N.L.H., and Khanh, P.Q., 2013, Higher-Order Optimality Conditions in Set-Valued optimization Using Radial Sets and Radial Derivatives. J. Glob Optim.,56,2,519-536.
Anh, N.L.H. and Khanh, P.Q., 2014, Higher-Order optimality Conditions for Proper Efficiency in Nonsmooth Vector Optimization Using Radial Sets and Radial Derivatives,J. Glob Optim., 58,4, 693-709.
Anh, N.L.H. Khanh, P.Q. and Tung, L.T., 2011, Higher-Order Radial Derivatives and Optimality Conditions in Nonsmooth Vector Optimization, Nonlinear Anal.Theory Meth.Appl.,74,7365-7379.
Bazan, F.F., 2001, Optimality Conditions in Nonconvex Set-Valued Optimization, Mathematical Methods of Operations Research,53, 403-417.
Bazan, F.F., 2003, Radial Epiderivatives and Asymptotic Functions in Nonconvex Vector Optimization, SIAM J. Optimization, 14,284-305.
Bigi, G. and Castellani, M.,2000, Second Order Optimality Conditions for Differentiable Multiobjective Problems, RARIO Operations Research, 34,411-426.
Chen, G.Y. and Jahn, J., 1998, Optimality Conditions for Set-Valued Optimization Problems,MathematicalMethods of Operations Research, 48,187-200.
Cambini, A. and Martein, L., 2002, First and Second Order Optimality Conditions in Vector Optimization,Journal of Statistics and Management Systems,5,295-319.
Cambini, A., Martein, L. and Vlach,M., 1999, Second Order Tangent Sets and Optimality Conditions, Matematica Japonica, 49,451-461.
Giorgi, G., Jimenez, B. and Novo, V., 2010, An Overview of Second Order Tangent Sets and Their Application to Vector Optimization, SeMA Journal,52, 1, 73-96.
Gutierrez,C., Jimenez,B. and Novo, V., 2009, New Second-Order Directional Derivative and Optimality Conditions in Scalar and Vector Optimization, J. Optim. Theory Appl., 142,85-106.
Ha,T.D.X., 2009, Optimality conditions for several types of efficient solutions of set-valued optimization problems,in: P. Pardolos, Th.M. Rassis, A.A. Khan (Eds.), Nnlinear Analysis and Variational Problems, Springer, p.305-324(Chapter 21).
Hachimi, M. and Aghezzaf, B., 2007, New Results on Second-Order Optimality Conditions in Vector Optimization Problems, J. Optim. Theory Appl.,135,117-133.
Jahn,J., 1986, Mathematical vector optimization in partially ordered linear space, Peter Lang, Frankfurt.
Jahn,J., Khan,A.A., and Zeillinger, P., 2005, Second Order Optimality Conditions in Set Optimization, J. Optim. Theory Apply., 125,2,331-347.
Jahn, J. and Rauh, R.,1997, Contingent Epiderivatives and Set-Valued Optimization Mathematical Methods of Operations Research, 46,193-211.
Jimenez, B. and Novo, V., 2003, Second Order Necessary Conditions in Set Constrained Differentiable Vector Optimization, Mathematical Methods of Operations Research, 58,299-317.
Jimenez, B. and Novo, V.,2004, Optimality Conditions in Differentiable Vector Optimization via Second-Order Tangent Sets, Appl. Math. Optim., 49,123-144.
Kasımbeyli,R.,2009 Radial Epiderivatives and Set-Valued Optimization, Optimization,58,5,519-532.
Kalashnikov, V., Jadamba, B. and Khan,A.A., 2006, First and Second- Order Optimality Condition in Set-Optimization, In Optimization with Multivalued Mappings, Edited by: Dempe, S and Kalashnikov, V. , Berlin, Heidelberg: Springer Verlag, 265-276.
Kasımbeyli,R. and ˙Inceo˘glu, G.,2010, Optimality Conditions viaGeneralized Radial Epiderivatives in Nonconvex Set-Valued Optimization, In: R. Kasımbeyli, C. Dinc¸er, S. ¨Ozpeynirci and L. Sakalauskas (Eds.) Selected papers. 24thMini EURO Conference on Continuous Optimization and Information-Based Technologies in the Financial Sector (24th MEC EurOPT 2010), June 23-26,2010, Izmir University of Economics, Izmir, Turkey, ISBN: 978-9955-28-597-7, Vilnius ”Technika”, p. 148-154.
Khan, A.A. and Tammer, C., 2013, Second Order Optimality Conditions in Set-Valued Optimization via Asymptotic Derivatives, Optimization, 62,6,743-758.
Luc, D.T., 1991, Theory of Vector Optimization, Springer, Berlin.
Aubin, J.P.,1981, Contingent Derivatives of Set-Valued Maps and Existence of Solutions to Nonlinear Inclusions and Differential Inclusions. In: Nachbin, L (ed.) Mathematics Analysis and Applications, part A, 160-229, Academic Press, New York.
Aubin, J.P., Frankowska, H., 1990, Set Valued Analysis,Birkhauser, Boston.
Aghezzaf, B. and Hachimi, M., 1999, Second Order Optimality Conditions in Multiobjective Optimization Problems,J. Optim. Theory Apply., 102,1,37-50.
Anh, N.L.H., and Khanh, P.Q., 2013, Higher-Order Optimality Conditions in Set-Valued optimization Using Radial Sets and Radial Derivatives. J. Glob Optim.,56,2,519-536.
Anh, N.L.H. and Khanh, P.Q., 2014, Higher-Order optimality Conditions for Proper Efficiency in Nonsmooth Vector Optimization Using Radial Sets and Radial Derivatives,J. Glob Optim., 58,4, 693-709.
Anh, N.L.H. Khanh, P.Q. and Tung, L.T., 2011, Higher-Order Radial Derivatives and Optimality Conditions in Nonsmooth Vector Optimization, Nonlinear Anal.Theory Meth.Appl.,74,7365-7379.
Bazan, F.F., 2001, Optimality Conditions in Nonconvex Set-Valued Optimization, Mathematical Methods of Operations Research,53, 403-417.
Bazan, F.F., 2003, Radial Epiderivatives and Asymptotic Functions in Nonconvex Vector Optimization, SIAM J. Optimization, 14,284-305.
Bigi, G. and Castellani, M.,2000, Second Order Optimality Conditions for Differentiable Multiobjective Problems, RARIO Operations Research, 34,411-426.
Chen, G.Y. and Jahn, J., 1998, Optimality Conditions for Set-Valued Optimization Problems,MathematicalMethods of Operations Research, 48,187-200.
Cambini, A. and Martein, L., 2002, First and Second Order Optimality Conditions in Vector Optimization,Journal of Statistics and Management Systems,5,295-319.
Cambini, A., Martein, L. and Vlach,M., 1999, Second Order Tangent Sets and Optimality Conditions, Matematica Japonica, 49,451-461.
Giorgi, G., Jimenez, B. and Novo, V., 2010, An Overview of Second Order Tangent Sets and Their Application to Vector Optimization, SeMA Journal,52, 1, 73-96.
Gutierrez,C., Jimenez,B. and Novo, V., 2009, New Second-Order Directional Derivative and Optimality Conditions in Scalar and Vector Optimization, J. Optim. Theory Appl., 142,85-106.
Ha,T.D.X., 2009, Optimality conditions for several types of efficient solutions of set-valued optimization problems,in: P. Pardolos, Th.M. Rassis, A.A. Khan (Eds.), Nnlinear Analysis and Variational Problems, Springer, p.305-324(Chapter 21).
Hachimi, M. and Aghezzaf, B., 2007, New Results on Second-Order Optimality Conditions in Vector Optimization Problems, J. Optim. Theory Appl.,135,117-133.
Jahn,J., 1986, Mathematical vector optimization in partially ordered linear space, Peter Lang, Frankfurt.
Jahn,J., Khan,A.A., and Zeillinger, P., 2005, Second Order Optimality Conditions in Set Optimization, J. Optim. Theory Apply., 125,2,331-347.
Jahn, J. and Rauh, R.,1997, Contingent Epiderivatives and Set-Valued Optimization Mathematical Methods of Operations Research, 46,193-211.
Jimenez, B. and Novo, V., 2003, Second Order Necessary Conditions in Set Constrained Differentiable Vector Optimization, Mathematical Methods of Operations Research, 58,299-317.
Jimenez, B. and Novo, V.,2004, Optimality Conditions in Differentiable Vector Optimization via Second-Order Tangent Sets, Appl. Math. Optim., 49,123-144.
Kasımbeyli,R.,2009 Radial Epiderivatives and Set-Valued Optimization, Optimization,58,5,519-532.
Kalashnikov, V., Jadamba, B. and Khan,A.A., 2006, First and Second- Order Optimality Condition in Set-Optimization, In Optimization with Multivalued Mappings, Edited by: Dempe, S and Kalashnikov, V. , Berlin, Heidelberg: Springer Verlag, 265-276.
Kasımbeyli,R. and ˙Inceo˘glu, G.,2010, Optimality Conditions viaGeneralized Radial Epiderivatives in Nonconvex Set-Valued Optimization, In: R. Kasımbeyli, C. Dinc¸er, S. ¨Ozpeynirci and L. Sakalauskas (Eds.) Selected papers. 24thMini EURO Conference on Continuous Optimization and Information-Based Technologies in the Financial Sector (24th MEC EurOPT 2010), June 23-26,2010, Izmir University of Economics, Izmir, Turkey, ISBN: 978-9955-28-597-7, Vilnius ”Technika”, p. 148-154.
Khan, A.A. and Tammer, C., 2013, Second Order Optimality Conditions in Set-Valued Optimization via Asymptotic Derivatives, Optimization, 62,6,743-758.
Luc, D.T., 1991, Theory of Vector Optimization, Springer, Berlin.