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April  2012, 8(2): 411-427. doi: 10.3934/jimo.2012.8.411

Calculus rules of generalized $\epsilon-$subdifferential for vector valued mappings and applications

Shengji Li 1, and  Xiaole Guo 1,

1. 

College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China

Received  March 2011 Revised  October 2011 Published  April 2012

In this paper, a generalized $\epsilon-$subdifferential, which was defined by a norm, is first introduced for a vector valued mapping. Some existence theorems and the properties of the generalized $\epsilon-$subdifferential are discussed. A relationship between the generalized $\epsilon-$subdifferential and a directional derivative is investigated for a vector valued mapping. Then, the calculus rules of the generalized $\epsilon-$subdifferential for the sum and the difference of two vector valued mappings were given. The positive homogeneity of the generalized $\epsilon-$subdifferential is also provided. Finally, as applications, necessary and sufficient optimality conditions are established for vector optimization problems.
Keywords: vector optimization problem., Generalized $\epsilon-$subdifferential, calculus rule, optimality condition, vector valued mapping.
Mathematics Subject Classification: Primary: 49J52, 90C29; Secondary: 58C2.
Citation: Shengji Li, Xiaole Guo. Calculus rules of generalized $\epsilon-$subdifferential for vector valued mappings and applications. Journal of Industrial & Management Optimization, 2012, 8 (2) : 411-427. doi: 10.3934/jimo.2012.8.411
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.  Google Scholar

[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.  Google Scholar

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J. Baier and J. Jahn, On subdifferentials of set-valued maps, J. Optim. Theory Appl., 100 (1999), 233-240. doi: 10.1023/A:1021733402240.  Google Scholar

[4]

J. M. Borwein, A lagrange multiplier theorem and a sandwich theorem for convex relations, Math. Scand., 48 (1981), 189-204.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

F. H. Clarke, A new approach to Lagrange multipliers, Math. Oper. Res., 1 (1976), 165-174. doi: 10.1287/moor.1.2.165.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

Y. Küçük, İ. Atasever and M. Küçük, Generalized weak subdiffrentials, Optimization, 60 (2011), 537-552. doi: 10.1080/02331930903524670.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

L. S. Pontryagin, Linear differential games II, Soviet Math. Dokl., 8 (1967), 910-912. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

W. Song, Weak subdifferential of set-valued mappings, Optimization, 52 (2003), 263-276. doi: 10.1080/0233193031000120051.  Google Scholar

[21]

T. Tanino, Conjugate duality in vector optimization, J. Math. Anal. Appl., 167 (1992), 84-97. doi: 10.1016/0022-247X(92)90237-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

J. Zowe, Subdifferentiability of convex functions with values in an ordered vector space, Math. Scand., 34 (1974), 69-83.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

J. M. Borwein, A lagrange multiplier theorem and a sandwich theorem for convex relations, Math. Scand., 48 (1981), 189-204.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

F. H. Clarke, A new approach to Lagrange multipliers, Math. Oper. Res., 1 (1976), 165-174. doi: 10.1287/moor.1.2.165.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

Y. Küçük, İ. Atasever and M. Küçük, Generalized weak subdiffrentials, Optimization, 60 (2011), 537-552. doi: 10.1080/02331930903524670.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

L. S. Pontryagin, Linear differential games II, Soviet Math. Dokl., 8 (1967), 910-912. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

W. Song, Weak subdifferential of set-valued mappings, Optimization, 52 (2003), 263-276. doi: 10.1080/0233193031000120051.  Google Scholar

[21]

T. Tanino, Conjugate duality in vector optimization, J. Math. Anal. Appl., 167 (1992), 84-97. doi: 10.1016/0022-247X(92)90237-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

J. Zowe, Subdifferentiability of convex functions with values in an ordered vector space, Math. Scand., 34 (1974), 69-83.  Google Scholar

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