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Calculus rules of generalized $\epsilon-$subdifferential for vector valued mappings and applications
1. | College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China |
References:
[1] |
T. Amahroq, J.-P. Penot and A. Syam, On the subdifferentiability of the difference of two functions and local minimization, Set-Valued Anal., 16 (2008), 413-472.
doi: 10.1007/s11228-008-0085-9. |
[2] |
A. Y. Azimov and R. N. Gasimov, Stability and duality of nonconvex problems via augmented Lagrangian, Cybernet. Systems Anal., 38 (2002), 412-421.
doi: 10.1023/A:1020316811823. |
[3] |
J. Baier and J. Jahn, On subdifferentials of set-valued maps, J. Optim. Theory Appl., 100 (1999), 233-240.
doi: 10.1023/A:1021733402240. |
[4] |
J. M. Borwein, A lagrange multiplier theorem and a sandwich theorem for convex relations, Math. Scand., 48 (1981), 189-204. |
[5] |
R. I. Boţ and D.-M. Nechita, On the Dini-Hadamard subdifferential of the difference of two functions, J. Glob. Optim., 50 (2011), 485-502.
doi: 10.1007/s10898-010-9604-y. |
[6] |
G. Y. Chen and J. Jahn, Optimality conditions for set-valued optimization problems. Set-valued optimization, Math. Meth. Oper. Res., 48 (1998), 187-200.
doi: 10.1007/s001860050021. |
[7] |
N. H. Chieu and J.-C. Yao, Subgradients of the optimal value function in a parametric discrete optimal control problem, J. Ind. Manag. Optm., 6 (2010), 401-410.
doi: 10.3934/jimo.2010.6.401. |
[8] |
F. H. Clarke, A new approach to Lagrange multipliers, Math. Oper. Res., 1 (1976), 165-174.
doi: 10.1287/moor.1.2.165. |
[9] |
M. El Maghri and M. Laghdir, Pareto subdifferential calculus for convex vector mappings and applications to vector optimization, SIAM J. Optim., 19 (2008), 1970-1994.
doi: 10.1137/070704046. |
[10] |
A. D. Ioffe, Calculus of Dini subdifferentials of functions and contingent coderivatives of set-valued maps, Nonlinear Anal., 8 (1984), 517-539.
doi: 10.1016/0362-546X(84)90091-9. |
[11] |
Y. Küçük, İ. Atasever and M. Küçük, Generalized weak subdiffrentials, Optimization, 60 (2011), 537-552.
doi: 10.1080/02331930903524670. |
[12] |
S. J. Li, Subgradient of S-convex set-valued mappings and weak efficient solutions, (Chinese) Gaoxiao Yingyong Shuxue Xuebao Ser. A, 13 (1998), 463-472. |
[13] |
S. J. Li and X. L. Guo, Weak subdifferential for set-valued mappings and its applications, Nonlinear Anal., 71 (2009), 5781-5789.
doi: 10.1016/j.na.2009.04.065. |
[14] |
B. Sh. Mordukhovich, Maximum principle in the problem of time-optimal control with nonsmooth constraints, J. Appl. Math. Mech., 40 (1976), 960-969.
doi: 10.1016/0021-8928(76)90136-2. |
[15] |
B. S. Mordukhovich, N. M. Nam and N. D. Yen, Fréchet subdifferential calculus and optimality conditions in nondifferentiable programming, Optimization, 55 (2006), 685-708.
doi: 10.1080/02331930600816395. |
[16] |
J.-P. Penot, The directional subdifferential of the differenceof two convex functions, J. Glob. Optim., 49 (2011), 505-519.
doi: 10.1007/s10898-010-9615-8. |
[17] |
L. S. Pontryagin, Linear differential games II, Soviet Math. Dokl., 8 (1967), 910-912. |
[18] |
R. T. Rockafellar, "The Theory of Subgradients and its Applications to Problems of Optimization-Convex and Nonconvex Functions," R & E, 1, Heldermann Verlag, Berlin, 1981. |
[19] |
R. T. Rockafellar, Proximal subgradients, marginal values, and augmented Lagrangians in nonconvex optimization, Math. Oper. Res., 6 (1981), 424-436.
doi: 10.1287/moor.6.3.424. |
[20] |
W. Song, Weak subdifferential of set-valued mappings, Optimization, 52 (2003), 263-276.
doi: 10.1080/0233193031000120051. |
[21] |
T. Tanino, Conjugate duality in vector optimization, J. Math. Anal. Appl., 167 (1992), 84-97.
doi: 10.1016/0022-247X(92)90237-8. |
[22] |
C. Zălinescu, Hahn-Banach extension theorems for multifunctions revisited, Math. Meth. Oper. Res., 68 (2008), 493-508.
doi: 10.1007/s00186-007-0193-6. |
[23] |
J. C. Zhou, C. Y. Wang, N. H. Xiu and S. Y. Wu, First-order optimality conditions for convex semi-infinite min-max programming with noncompact sets, J. Ind. Manag. Optm., 5 (2009), 851-866.
doi: 10.3934/jimo.2009.5.851. |
[24] |
J. Zowe, Subdifferentiability of convex functions with values in an ordered vector space, Math. Scand., 34 (1974), 69-83. |
show all references
References:
[1] |
T. Amahroq, J.-P. Penot and A. Syam, On the subdifferentiability of the difference of two functions and local minimization, Set-Valued Anal., 16 (2008), 413-472.
doi: 10.1007/s11228-008-0085-9. |
[2] |
A. Y. Azimov and R. N. Gasimov, Stability and duality of nonconvex problems via augmented Lagrangian, Cybernet. Systems Anal., 38 (2002), 412-421.
doi: 10.1023/A:1020316811823. |
[3] |
J. Baier and J. Jahn, On subdifferentials of set-valued maps, J. Optim. Theory Appl., 100 (1999), 233-240.
doi: 10.1023/A:1021733402240. |
[4] |
J. M. Borwein, A lagrange multiplier theorem and a sandwich theorem for convex relations, Math. Scand., 48 (1981), 189-204. |
[5] |
R. I. Boţ and D.-M. Nechita, On the Dini-Hadamard subdifferential of the difference of two functions, J. Glob. Optim., 50 (2011), 485-502.
doi: 10.1007/s10898-010-9604-y. |
[6] |
G. Y. Chen and J. Jahn, Optimality conditions for set-valued optimization problems. Set-valued optimization, Math. Meth. Oper. Res., 48 (1998), 187-200.
doi: 10.1007/s001860050021. |
[7] |
N. H. Chieu and J.-C. Yao, Subgradients of the optimal value function in a parametric discrete optimal control problem, J. Ind. Manag. Optm., 6 (2010), 401-410.
doi: 10.3934/jimo.2010.6.401. |
[8] |
F. H. Clarke, A new approach to Lagrange multipliers, Math. Oper. Res., 1 (1976), 165-174.
doi: 10.1287/moor.1.2.165. |
[9] |
M. El Maghri and M. Laghdir, Pareto subdifferential calculus for convex vector mappings and applications to vector optimization, SIAM J. Optim., 19 (2008), 1970-1994.
doi: 10.1137/070704046. |
[10] |
A. D. Ioffe, Calculus of Dini subdifferentials of functions and contingent coderivatives of set-valued maps, Nonlinear Anal., 8 (1984), 517-539.
doi: 10.1016/0362-546X(84)90091-9. |
[11] |
Y. Küçük, İ. Atasever and M. Küçük, Generalized weak subdiffrentials, Optimization, 60 (2011), 537-552.
doi: 10.1080/02331930903524670. |
[12] |
S. J. Li, Subgradient of S-convex set-valued mappings and weak efficient solutions, (Chinese) Gaoxiao Yingyong Shuxue Xuebao Ser. A, 13 (1998), 463-472. |
[13] |
S. J. Li and X. L. Guo, Weak subdifferential for set-valued mappings and its applications, Nonlinear Anal., 71 (2009), 5781-5789.
doi: 10.1016/j.na.2009.04.065. |
[14] |
B. Sh. Mordukhovich, Maximum principle in the problem of time-optimal control with nonsmooth constraints, J. Appl. Math. Mech., 40 (1976), 960-969.
doi: 10.1016/0021-8928(76)90136-2. |
[15] |
B. S. Mordukhovich, N. M. Nam and N. D. Yen, Fréchet subdifferential calculus and optimality conditions in nondifferentiable programming, Optimization, 55 (2006), 685-708.
doi: 10.1080/02331930600816395. |
[16] |
J.-P. Penot, The directional subdifferential of the differenceof two convex functions, J. Glob. Optim., 49 (2011), 505-519.
doi: 10.1007/s10898-010-9615-8. |
[17] |
L. S. Pontryagin, Linear differential games II, Soviet Math. Dokl., 8 (1967), 910-912. |
[18] |
R. T. Rockafellar, "The Theory of Subgradients and its Applications to Problems of Optimization-Convex and Nonconvex Functions," R & E, 1, Heldermann Verlag, Berlin, 1981. |
[19] |
R. T. Rockafellar, Proximal subgradients, marginal values, and augmented Lagrangians in nonconvex optimization, Math. Oper. Res., 6 (1981), 424-436.
doi: 10.1287/moor.6.3.424. |
[20] |
W. Song, Weak subdifferential of set-valued mappings, Optimization, 52 (2003), 263-276.
doi: 10.1080/0233193031000120051. |
[21] |
T. Tanino, Conjugate duality in vector optimization, J. Math. Anal. Appl., 167 (1992), 84-97.
doi: 10.1016/0022-247X(92)90237-8. |
[22] |
C. Zălinescu, Hahn-Banach extension theorems for multifunctions revisited, Math. Meth. Oper. Res., 68 (2008), 493-508.
doi: 10.1007/s00186-007-0193-6. |
[23] |
J. C. Zhou, C. Y. Wang, N. H. Xiu and S. Y. Wu, First-order optimality conditions for convex semi-infinite min-max programming with noncompact sets, J. Ind. Manag. Optm., 5 (2009), 851-866.
doi: 10.3934/jimo.2009.5.851. |
[24] |
J. Zowe, Subdifferentiability of convex functions with values in an ordered vector space, Math. Scand., 34 (1974), 69-83. |
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