Existence theorems for second-order radial epiderivatives

Existence theorems for second-order radial epiderivatives

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.

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  • 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.
  • N´emeth, A.B., 1986, A Nonconvex Vector Minimization Problem, Nonlinear Anal.Theory Meth. Appl., 10,669-678.
  • Penot, J.P., 1999,Second-Order Conditions for Optimization Problems Constraints, SIAM Journal on Optimization, 37,303-318.
  • Sonntag, Y. and Zalinescu, C., 2000, Comparision of Existence Result for Efficient Points, J. Optim. Theory Appl.,105,161-188.
  • Taa, A., 1998, Set-Valued Derivatives of Multifunctions and Optimality conditions, Numerical Functional Analysis Optimization, 19,121-140.
  • Ward, D., 1993, Calculus for Parabolic Second-Order Derivatives. Set Valued Analysis, 1,213-246.