# American Institute of Mathematical Sciences

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 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.
 [1] Qilin Wang, Shengji Li. Lower semicontinuity of the solution mapping to a parametric generalized vector equilibrium problem. Journal of Industrial and Management Optimization, 2014, 10 (4) : 1225-1234. doi: 10.3934/jimo.2014.10.1225 [2] Jiawei Chen, Shengjie Li, Jen-Chih Yao. Vector-valued separation functions and constrained vector optimization problems: optimality and saddle points. Journal of Industrial and Management Optimization, 2020, 16 (2) : 707-724. doi: 10.3934/jimo.2018174 [3] Tadeusz Antczak, Najeeb Abdulaleem. Optimality conditions for $E$-differentiable vector optimization problems with the multiple interval-valued objective function. Journal of Industrial and Management Optimization, 2020, 16 (6) : 2971-2989. doi: 10.3934/jimo.2019089 [4] Ying Gao, Xinmin Yang, Kok Lay Teo. Optimality conditions for approximate solutions of vector optimization problems. Journal of Industrial and Management Optimization, 2011, 7 (2) : 483-496. doi: 10.3934/jimo.2011.7.483 [5] Gaoxi Li, Zhongping Wan, Jia-wei Chen, Xiaoke Zhao. Necessary optimality condition for trilevel optimization problem. Journal of Industrial and Management Optimization, 2020, 16 (1) : 55-70. doi: 10.3934/jimo.2018140 [6] Nikos Katzourakis. Nonuniqueness in vector-valued calculus of variations in $L^\infty$ and some Linear elliptic systems. Communications on Pure and Applied Analysis, 2015, 14 (1) : 313-327. doi: 10.3934/cpaa.2015.14.313 [7] Nikos Katzourakis. Corrigendum to the paper: Nonuniqueness in Vector-Valued Calculus of Variations in $L^\infty$ and some Linear Elliptic Systems. Communications on Pure and Applied Analysis, 2019, 18 (4) : 2197-2198. doi: 10.3934/cpaa.2019098 [8] M. M. Rao. Integration with vector valued measures. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5429-5440. doi: 10.3934/dcds.2013.33.5429 [9] Xinmin Yang. On symmetric and self duality in vector optimization problem. Journal of Industrial and Management Optimization, 2011, 7 (3) : 523-529. doi: 10.3934/jimo.2011.7.523 [10] Guolin Yu. Global proper efficiency and vector optimization with cone-arcwise connected set-valued maps. Numerical Algebra, Control and Optimization, 2016, 6 (1) : 35-44. doi: 10.3934/naco.2016.6.35 [11] Ying Gao, Xinmin Yang, Jin Yang, Hong Yan. Scalarizations and Lagrange multipliers for approximate solutions in the vector optimization problems with set-valued maps. Journal of Industrial and Management Optimization, 2015, 11 (2) : 673-683. doi: 10.3934/jimo.2015.11.673 [12] Tadeusz Antczak. The $F$-objective function method for differentiable interval-valued vector optimization problems. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2761-2782. doi: 10.3934/jimo.2020093 [13] Monica Motta, Caterina Sartori. Generalized solutions to nonlinear stochastic differential equations with vector--valued impulsive controls. Discrete and Continuous Dynamical Systems, 2011, 29 (2) : 595-613. doi: 10.3934/dcds.2011.29.595 [14] Xiao-Bing Li, Xian-Jun Long, Zhi Lin. Stability of solution mapping for parametric symmetric vector equilibrium problems. Journal of Industrial and Management Optimization, 2015, 11 (2) : 661-671. doi: 10.3934/jimo.2015.11.661 [15] Adela Capătă. Optimality conditions for vector equilibrium problems and their applications. Journal of Industrial and Management Optimization, 2013, 9 (3) : 659-669. doi: 10.3934/jimo.2013.9.659 [16] Qiu-Sheng Qiu. Optimality conditions for vector equilibrium problems with constraints. Journal of Industrial and Management Optimization, 2009, 5 (4) : 783-790. doi: 10.3934/jimo.2009.5.783 [17] Fatemeh Abtahi, Zeinab Kamali, Maryam Toutounchi. The BSE concepts for vector-valued Lipschitz algebras. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1171-1186. doi: 10.3934/cpaa.2021011 [18] Reuven Segev, Lior Falach. The co-divergence of vector valued currents. Discrete and Continuous Dynamical Systems - B, 2012, 17 (2) : 687-698. doi: 10.3934/dcdsb.2012.17.687 [19] Adela Capătă. Optimality conditions for strong vector equilibrium problems under a weak constraint qualification. Journal of Industrial and Management Optimization, 2015, 11 (2) : 563-574. doi: 10.3934/jimo.2015.11.563 [20] Pooja Louhan, S. K. Suneja. On fractional vector optimization over cones with support functions. Journal of Industrial and Management Optimization, 2017, 13 (2) : 549-572. doi: 10.3934/jimo.2016031

2021 Impact Factor: 1.411