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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
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show all references

References:
[1]

Set-Valued Anal., 16 (2008), 413-472. doi: 10.1007/s11228-008-0085-9.  Google Scholar

[2]

Cybernet. Systems Anal., 38 (2002), 412-421. doi: 10.1023/A:1020316811823.  Google Scholar

[3]

J. Optim. Theory Appl., 100 (1999), 233-240. doi: 10.1023/A:1021733402240.  Google Scholar

[4]

Math. Scand., 48 (1981), 189-204.  Google Scholar

[5]

J. Glob. Optim., 50 (2011), 485-502. doi: 10.1007/s10898-010-9604-y.  Google Scholar

[6]

Math. Meth. Oper. Res., 48 (1998), 187-200. doi: 10.1007/s001860050021.  Google Scholar

[7]

J. Ind. Manag. Optm., 6 (2010), 401-410. doi: 10.3934/jimo.2010.6.401.  Google Scholar

[8]

Math. Oper. Res., 1 (1976), 165-174. doi: 10.1287/moor.1.2.165.  Google Scholar

[9]

SIAM J. Optim., 19 (2008), 1970-1994. doi: 10.1137/070704046.  Google Scholar

[10]

Nonlinear Anal., 8 (1984), 517-539. doi: 10.1016/0362-546X(84)90091-9.  Google Scholar

[11]

Optimization, 60 (2011), 537-552. doi: 10.1080/02331930903524670.  Google Scholar

[12]

(Chinese) Gaoxiao Yingyong Shuxue Xuebao Ser. A, 13 (1998), 463-472.  Google Scholar

[13]

Nonlinear Anal., 71 (2009), 5781-5789. doi: 10.1016/j.na.2009.04.065.  Google Scholar

[14]

J. Appl. Math. Mech., 40 (1976), 960-969. doi: 10.1016/0021-8928(76)90136-2.  Google Scholar

[15]

Optimization, 55 (2006), 685-708. doi: 10.1080/02331930600816395.  Google Scholar

[16]

J. Glob. Optim., 49 (2011), 505-519. doi: 10.1007/s10898-010-9615-8.  Google Scholar

[17]

Soviet Math. Dokl., 8 (1967), 910-912. Google Scholar

[18]

R & E, 1, Heldermann Verlag, Berlin, 1981.  Google Scholar

[19]

Math. Oper. Res., 6 (1981), 424-436. doi: 10.1287/moor.6.3.424.  Google Scholar

[20]

Optimization, 52 (2003), 263-276. doi: 10.1080/0233193031000120051.  Google Scholar

[21]

J. Math. Anal. Appl., 167 (1992), 84-97. doi: 10.1016/0022-247X(92)90237-8.  Google Scholar

[22]

Math. Meth. Oper. Res., 68 (2008), 493-508. doi: 10.1007/s00186-007-0193-6.  Google Scholar

[23]

J. Ind. Manag. Optm., 5 (2009), 851-866. doi: 10.3934/jimo.2009.5.851.  Google Scholar

[24]

Math. Scand., 34 (1974), 69-83.  Google Scholar

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