Second-order Dini set-valued directional derivative in C^1,1 vector optimization

References

• Aubin , J. P. and Frankowska , H. 1990 . Set-valued Analysis , Boston, MA : Birkhäuser .
   Google Scholar
• Auslender , A. 1984 . Stability in mathematical programming with nondifferentiable data . SIAM J. Control Optim. , 22 : 239 – 254 .
   Web of Science ®Google Scholar
• Bao , T. Q. and Mordukhovich , B. S. in press . Relative Pareto minimizers for multiobjective problems: existence and optimality conditions . Math. Program. Ser. A , DOI 10.1007/s10107-008-0249-2
   Google Scholar
• Demyanov , V. F. and Rubinov , A. M. 1995 . Constructive Nonsmooth Analysis , Frankfurt am Main : Peter Lang .
   Google Scholar
• Georgiev , P. G. and Zlateva , N. 1996 . Second-order subdifferentials of C1,1 functions and optimality conditions . Set-Valued Anal. , 4 : 101 – 117 .
   Google Scholar
• Ginchev , I. 2002 . Higher order optimality conditions in nonsmooth vector optimization . J. Statist. Manage. Syst. , 5 : 321 – 339 .
   Google Scholar
• Ginchev , I. and Hoffmann , A. 2002 . Approximation of set-valued functions by single-valued one . Discuss. Math. Differ. Incl. Control Optim. , 22 : 33 – 66 .
   Google Scholar
• Ginchev , I. , Guerraggio , A. and Rocca , M. 2004 . C1,1 vector optimization problems and Riemann derivatives . Control Cybernet. , 34 ( 2 ) : 259 – 273 .
   Google Scholar
• Ginchev , I. , Guerraggio , A. and Rocca , M. First-order conditions for C0,1 constrained vector optimization, in Variational Analysis and Applications . Edited by: Giannessi , F. and Maugeri , A. Vol. New York, 2005, , pp. 427 – 450 . Springer . Nonconvex Optimization and its Applications, Vol. 79,
   Google Scholar
• Ginchev , I. , Guerraggio , A. and Rocca , M. 2005 . Second-order conditions in C1,1 constrained vector optimization . Math. Program. Ser. B , 104 : 389 – 405 .
   Web of Science ®Google Scholar
• Ginchev , I. , Guerraggio , A. and Rocca , M. 2006 . From scalar to vector optimization . Appl. Math. , 51 : 5 – 36 .
   Google Scholar
• Ginchev , I. , Guerraggio , A. and Rocca , M. Second-order conditions in C1,1 vector optimization with inequality and equality constraints in Recent Advances in Optimizations . Edited by: Seeger , A. Vol. Berlin, Heidelberg, 2006, , pp. 29 – 44 . Springer . Lecture Notes in Economics and Mathematical Systems, Vol. 563,
   Google Scholar
• Ginchev , I. , Guerraggio , A. and Rocca , M. in press . Dini directional derivatives in locally Lipschitz vector optimization . J. Optim. Theory Appl. , DOI 10.1007/s10957-009-9551-2
   Google Scholar
• Guerraggio , A. and Luc , D. T. 2001 . Optimality conditions for C1,1 vector optimization problems . J. Optim. Theory Appl. , 109 : 615 – 629 .
   Web of Science ®Google Scholar
• Guerraggio , A. and Luc , D. T. 2003 . Optimality conditions for C1,1 constrained multiobjective problems . J. Optim. Theory Appl. , 116 : 117 – 129 .
   Web of Science ®Google Scholar
• Hiriart-Urruty , J. B. 1979 . New concepts in nondifferentiable programming . Analyse non convexe, Bull. Soc. Math. France , 60 : 57 – 85 .
   Google Scholar
• Hiriart-Urruty , J. B. 1979 . Tangent cones, generalized gradients and mathematical programming in Banach spaces . Math. Oper. Res. , 4 : 79 – 97 .
   Google Scholar
• Hiriart-Urruty , J. B. , Strodiot , J. J. and Hien Nguyen , V. 1984 . Generalized Hessian matrix and second order optimality conditions for problems with C1,1 data . Appl. Math. Optim. , 11 : 169 – 180 .
   Web of Science ®Google Scholar
• Jeyakumar , V. and Luc , D. T. 2008 . Nonsmooth vector functions and continuous optimization , New York : Springer Optimization and its Applications. Vol. 10, Springer .
   Google Scholar
• Jiménez , B. 2001 . Strict efficiency in vector optimization . J. Math. Anal. Appl. , 265 : 264 – 284 .
   Google Scholar
• Jiménez , B. and Novo , V. 2003 . First and second order conditions for strict minimality in nonsmooth vector optimization . J. Math. Anal. Appl. , 284 : 496 – 510 .
   Web of Science ®Google Scholar
• Klatte , D. and Tammer , K. 1988 . On the second order sufficient conditions to perturbed C1,1 optimization problems . Optimization , 19 : 169 – 179 .
   Google Scholar
• Krantz , S. G. and Parks , H. R. 2002 . The implicit function theorem. History, Theory, and Applications , Boston, MA : Birkhäuser .
   Google Scholar
• Kummer , B. 1991 . An implicit function theorem for C0,1 equations and parametric C1,1 optimization . J. Math. Anal. Appl. , 158 : 35 – 46 .
   Web of Science ®Google Scholar
• Liu , L. 1991 . The second-order conditions of nondominated solutions for C1,1 generalized multiobjective mathematical programming . J. Syst. Sci. Math. Sci. , 4 : 128 – 138 .
   Google Scholar
• Liu , L. and Křířek , M. 1997 . The second-order optimality conditions for nonlinear mathematical programming with C1,1 data . Appl. Math. , 42 : 311 – 320 .
   Google Scholar
• Liu , L. , Neittaanmäki , P. and Křířek , M. 2000 . Second-order optimality conditions for nondominated solutions of multiobjective programming with C1,1data . Appl. Math. , 45 : 381 – 397 .
   Google Scholar
• Mordukhovich , B. S. 2006 . Variational analysis and generalized differentiation. I. Basic theory , Vol. 330 , Berlin : Grundlehren der Mathematischen Wissenschaften. Vol. 330, Springer-Verlag .
   Google Scholar
• Mordukhovich , B. S. 2006 . Variational analysis and generalized differentiation. II. Applications , Berlin : Grundlehren der Mathematischen Wissenschaften. Vol. 331, Springer-Verlag .
   Google Scholar
• Rockafellar , R. T. 1970 . Convex Analysis , Princeton : Princeton University Press .
   Google Scholar
• Rudin , W. 1964 . Principles of Mathematical Analysis , New York : McGraw-Hill .
   Google Scholar
• Yang , X. Q. 1993 . Second-order conditions in C1,1 optimization with applications . Numer. Funct. Anal. Optim. , 14 : 621 – 632 .
   Web of Science ®Google Scholar
• Yang , X. Q. and Jeyakumar , V. 1992 . Generalized second-order directional derivatives and optimization with C1,1 functions . Optimization , 26 : 165 – 185 .
   Google Scholar
• Zaffaroni , A. 2003 . Degrees of efficiency and degrees of minimality . SIAM J. Control Optim. , 42 : 1071 – 1086 .
   Web of Science ®Google Scholar