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

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.
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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