Second-order optimality conditions for strict local Pareto minima of cosntrained nonsmooth multiobjective optimization problems
Nội dung chính của bài viết
Tóm tắt
This paper presents primal and dual Fritz John secondorder necessary conditions for strict local Pareto minima of order two of nonsmooth vector optimization problems in terms of the P´ales–Zeidan
second-order directional derivatives without constraint qualifications. Dual second-order Karush–Kuhn–Tucker necessary and sufficient conditions for strict local Pareto minima of order two are established under a suitable constraint qualification.
Chi tiết bài viết
Từ khóa
.Primal and dual second-order necessary eciency conditions; Pales{ Zeidan second-order directional derivatives; First and second-order tangent vectors
Tài liệu tham khảo
[1] Aghezzaf, B., Hachimi, M.: Second-order optimality conditions in multiobjective optimization problems. J. Optim. Theory Appl. 102(1), 37-50 (1999)
[2] Ben-Tal, A.: Second-order and related extremality conditions in nonlinearprogramming. J. Optim. Theory Appl. 31(2), 143-165 (1980)
[3] Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 127-149 (1994)
[4] Constantin, E.: Second-order necessary conditions in locally Lipschitz optimization with inequality constraints. Optim. Lett. 9, 245-261 (2015)
[5] Constantin, E.: Higher order necessary conditions in smooth constrained optimization. Communicating Mathematics, AMS Contemporary Mathematics, 479, 41-49 (2009)
[6] Constantin, E.: Second-order optimality conditions in locally Lipschitz inequality constrained multiobjective optimization, J. Optim. Theory Appl. 186, 50-67(2020)
[7] Daniele, P.: Lagrange multipliers and infinite-dimensional equilibrium problems. J. Glob. Optim. 40, 65-70 (2008)
[8] Daniele, P.: Dynamic Networks and Evolutionary Variational Inequalities.Edward Elgar Publishing, UK (2006)
[9] Giannessi, F., Mastroeni, G., Pellegrini, L.: On the theory of vector optimization and variational inequalities, image space analysis and separation.In: Giannessi, F. (ed.): Vector Variational Inequalities and Vector Equilibria: Mathematical Theories, pp. 153-215. Kluwer, Dordrecht (2000)
[10] Ginchev, I., Ivanov, V.I.: Second-order optimality conditions for problems with C1 data. J. Math. Anal. Appl. 340, 646-657 (2008)
[11] Girsanov, I. Lectures on Mathematical Theory of Extremum Problems, Springer, Berlin (1972)
[12] Gong, X.H.: Optimality conditions for efficient solution to the vector equilibrium problems with constraints. Taiwanese J. Math. 16, 1453-1473(2012)
[13] Gong, X.H.: Optimality conditions for vector equilibrium problems. J. Math. Anal. Appl. 342, 1455-1466 (2008)
[14] Guti´errez, C., Jim´enez, B., Novo, V.: On second-order Fritz John type optimality conditions in nonsmooth multiobjective programming. Math. Program. Ser. B 123, 199-223 (2010)
[15] Ivanov, V.I.: Second-order optimality conditions for inequality constrained problems with locally Lipschitz data. Optim. Lett. 4, 597-608(2010)
[16] Jim´enez, B., Novo, V.: Second order necessary conditions in set constrained differentiable vector optimization. Math. Meth. Oper. Res. 58, 299-317 (2003)
[17] Jim´enez, B., Novo, V.: A finite dimensional extension of Lyusternik theorem with applications to multiobjective optimization. J. Math. Anal.Appl. 270, 340–356 (2002)
[18] Jim´enez. B.: Strict efficiency in vector optimization, J. Math. Anal. Appl. 265, 264-284 (2002)
[19] Luu, D.V.: Necessary and sufficient conditions for efficiency via convexificators. J. Optim. Theory Appl. 160, 510–526 (2014)
[20] Luu, D.V.: Higher-order efficiency conditions via higher-order tangent cones. Numer. Funct. Anal. Optim. 35, 68-84 (2014)
[21] Luu, D.V.: Higher-order necessary and sufficient conditions for strict local Pareto minima in terms of Studniarski’s derivatives. Optimization, 57, 593-605 (2008)
[22] Luu, D.V.: Second-order necessary efficiency conditions for nonsmooth vector equilibrium problems, J. Glob. Optim., 70(2018),437 -453
[23] Luu, D.V., Hang, D.D.: On optimality conditions for vector variational inequalities. J. Math. Anal. Appl. 412, 792-804 (2014)
[24] Luu, D.V., Hang, D.D.: Efficient solutions and optimality conditions for vector equilibrium problems. Math. Meth. Oper. Res. 79, 163-177(2014)
[25] Luu, D.V., Hang, D.D.: On efficiency conditions for nonsmooth vector equilibrium problems with equilibrium constraints. Numer. Funct. Anal. Optim. 36, 1622–1642 (2015)
[26] Ma, B.C., Gong, X.H.: Optimality conditions for vector equilibrium problems in normed spaces. Optimization 60, 1441-1455 (2011)
[27] Morgan, J., Romaniello, M.: Scalarization and Kuhn-Tucker-like conditions for weak vector generalized quasivariational inequalities. J. Optim. Theory Appl. 130, 309-316 (2006)
[28] Noor, M.A., Oettli, W., On general nonlinear complementarity problems and quasi-equilibria. Le Matematiche 49, 313-330 (1994)
[29] Noor, M.A., Fundamentals of equilibrium problems. Math. Inequal.Appl. 9, 529-566 (2006)
[30] Noor, M.A., Invexequilibrium problems.J. Math. Anal. Appl. 302, 463-475 (2005)
[31] P´ales, Z., Zeidan, V.M.: Nonsmooth optimum problems with constraints. SIAM J. Control Optim. 32(5), 1476-1502 (1994)
[32] Pavel, N.H., Huang, J.K., Kim, J.K.: Higher order necessary conditions for optimization. Libertas Math. 14, 41-50 (1994)
[2] Ben-Tal, A.: Second-order and related extremality conditions in nonlinearprogramming. J. Optim. Theory Appl. 31(2), 143-165 (1980)
[3] Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 127-149 (1994)
[4] Constantin, E.: Second-order necessary conditions in locally Lipschitz optimization with inequality constraints. Optim. Lett. 9, 245-261 (2015)
[5] Constantin, E.: Higher order necessary conditions in smooth constrained optimization. Communicating Mathematics, AMS Contemporary Mathematics, 479, 41-49 (2009)
[6] Constantin, E.: Second-order optimality conditions in locally Lipschitz inequality constrained multiobjective optimization, J. Optim. Theory Appl. 186, 50-67(2020)
[7] Daniele, P.: Lagrange multipliers and infinite-dimensional equilibrium problems. J. Glob. Optim. 40, 65-70 (2008)
[8] Daniele, P.: Dynamic Networks and Evolutionary Variational Inequalities.Edward Elgar Publishing, UK (2006)
[9] Giannessi, F., Mastroeni, G., Pellegrini, L.: On the theory of vector optimization and variational inequalities, image space analysis and separation.In: Giannessi, F. (ed.): Vector Variational Inequalities and Vector Equilibria: Mathematical Theories, pp. 153-215. Kluwer, Dordrecht (2000)
[10] Ginchev, I., Ivanov, V.I.: Second-order optimality conditions for problems with C1 data. J. Math. Anal. Appl. 340, 646-657 (2008)
[11] Girsanov, I. Lectures on Mathematical Theory of Extremum Problems, Springer, Berlin (1972)
[12] Gong, X.H.: Optimality conditions for efficient solution to the vector equilibrium problems with constraints. Taiwanese J. Math. 16, 1453-1473(2012)
[13] Gong, X.H.: Optimality conditions for vector equilibrium problems. J. Math. Anal. Appl. 342, 1455-1466 (2008)
[14] Guti´errez, C., Jim´enez, B., Novo, V.: On second-order Fritz John type optimality conditions in nonsmooth multiobjective programming. Math. Program. Ser. B 123, 199-223 (2010)
[15] Ivanov, V.I.: Second-order optimality conditions for inequality constrained problems with locally Lipschitz data. Optim. Lett. 4, 597-608(2010)
[16] Jim´enez, B., Novo, V.: Second order necessary conditions in set constrained differentiable vector optimization. Math. Meth. Oper. Res. 58, 299-317 (2003)
[17] Jim´enez, B., Novo, V.: A finite dimensional extension of Lyusternik theorem with applications to multiobjective optimization. J. Math. Anal.Appl. 270, 340–356 (2002)
[18] Jim´enez. B.: Strict efficiency in vector optimization, J. Math. Anal. Appl. 265, 264-284 (2002)
[19] Luu, D.V.: Necessary and sufficient conditions for efficiency via convexificators. J. Optim. Theory Appl. 160, 510–526 (2014)
[20] Luu, D.V.: Higher-order efficiency conditions via higher-order tangent cones. Numer. Funct. Anal. Optim. 35, 68-84 (2014)
[21] Luu, D.V.: Higher-order necessary and sufficient conditions for strict local Pareto minima in terms of Studniarski’s derivatives. Optimization, 57, 593-605 (2008)
[22] Luu, D.V.: Second-order necessary efficiency conditions for nonsmooth vector equilibrium problems, J. Glob. Optim., 70(2018),437 -453
[23] Luu, D.V., Hang, D.D.: On optimality conditions for vector variational inequalities. J. Math. Anal. Appl. 412, 792-804 (2014)
[24] Luu, D.V., Hang, D.D.: Efficient solutions and optimality conditions for vector equilibrium problems. Math. Meth. Oper. Res. 79, 163-177(2014)
[25] Luu, D.V., Hang, D.D.: On efficiency conditions for nonsmooth vector equilibrium problems with equilibrium constraints. Numer. Funct. Anal. Optim. 36, 1622–1642 (2015)
[26] Ma, B.C., Gong, X.H.: Optimality conditions for vector equilibrium problems in normed spaces. Optimization 60, 1441-1455 (2011)
[27] Morgan, J., Romaniello, M.: Scalarization and Kuhn-Tucker-like conditions for weak vector generalized quasivariational inequalities. J. Optim. Theory Appl. 130, 309-316 (2006)
[28] Noor, M.A., Oettli, W., On general nonlinear complementarity problems and quasi-equilibria. Le Matematiche 49, 313-330 (1994)
[29] Noor, M.A., Fundamentals of equilibrium problems. Math. Inequal.Appl. 9, 529-566 (2006)
[30] Noor, M.A., Invexequilibrium problems.J. Math. Anal. Appl. 302, 463-475 (2005)
[31] P´ales, Z., Zeidan, V.M.: Nonsmooth optimum problems with constraints. SIAM J. Control Optim. 32(5), 1476-1502 (1994)
[32] Pavel, N.H., Huang, J.K., Kim, J.K.: Higher order necessary conditions for optimization. Libertas Math. 14, 41-50 (1994)