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

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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

Abstract
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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 and 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.
[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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