American Institute of Mathematical Sciences

Advanced
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

[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

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

[1]

Qilin Wang, Shengji Li. Lower semicontinuity of the solution mapping to a parametric generalized vector equilibrium problem. Journal of Industrial & 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 & 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 & 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 & 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 & 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 & 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 & Applied Analysis, 2019, 18 (4) : 2197-2198. doi: 10.3934/cpaa.2019098
[8]

M. M. Rao. Integration with vector valued measures. Discrete & 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 & 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 & 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 & 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 & 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 & Continuous Dynamical Systems, 2011, 29 (2) : 595-613. doi: 10.3934/dcds.2011.29.595
[14]

Adela Capătă. Optimality conditions for vector equilibrium problems and their applications. Journal of Industrial & Management Optimization, 2013, 9 (3) : 659-669. doi: 10.3934/jimo.2013.9.659
[15]

Qiu-Sheng Qiu. Optimality conditions for vector equilibrium problems with constraints. Journal of Industrial & Management Optimization, 2009, 5 (4) : 783-790. doi: 10.3934/jimo.2009.5.783
[16]

Xiao-Bing Li, Xian-Jun Long, Zhi Lin. Stability of solution mapping for parametric symmetric vector equilibrium problems. Journal of Industrial & Management Optimization, 2015, 11 (2) : 661-671. doi: 10.3934/jimo.2015.11.661
[17]

Fatemeh Abtahi, Zeinab Kamali, Maryam Toutounchi. The BSE concepts for vector-valued Lipschitz algebras. Communications on Pure & 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 & 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 & 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 & Management Optimization, 2017, 13 (2) : 549-572. doi: 10.3934/jimo.2016031

2020 Impact Factor: 1.801

Tools

Metrics

  • PDF downloads (94)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences