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January  2021, 17(1): 485-500. doi: 10.3934/jimo.2020117

Some properties of nonconvex oriented distance function and applications to vector optimization problems

1. 

School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, China

2. 

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

3. 

College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China

* Corresponding author: Liping Tang

Received  August 2019 Revised  January 2020 Published  January 2021 Early access  June 2020

In this paper, we study some interesting properties of nonconvex oriented distance function. In particular, we present complete characterizations of monotonicity properties of oriented distance function. Moreover, the Clark subdifferentials of nonconvex oriented distance function are explored in the solid case. As applications, fuzzy necessary optimality conditions for approximate solutions to vector optimization problems are provided.

Citation: Liping Tang, Ying Gao. Some properties of nonconvex oriented distance function and applications to vector optimization problems. Journal of Industrial and Management Optimization, 2021, 17 (1) : 485-500. doi: 10.3934/jimo.2020117
References:
[1]

Q. H. AnsariE. Köbis and P. K. Sharma, Characterizations of multiobjective robustness via oriented distance function and image space analysis, J. Optim. Theory Appl., 181 (2019), 817-839.  doi: 10.1007/s10957-019-01505-y.

[2]

Y. Araya, Four types of nonlinear scalarizations and some applications in set optimization, Nonlinear Anal., 75 (2012), 3821-3835.  doi: 10.1016/j.na.2012.02.004.

[3]

R. I. BoţS.-M. Grad and G. Wanka, A general approach for studying duality in multiobjective optimization, Math. Methods Oper. Res., 65 (2007), 417-444.  doi: 10.1007/s00186-006-0125-x.

[4]

J. V. BurkeM. C. Ferris and M. Qian, On the Clarke subdifferential of the distance function of a closed set, J. Math. Anal. Appl., 166 (1992), 199-213.  doi: 10.1016/0022-247X(92)90336-C.

[5]

G.-Y. Chen, X. Huang and X. Yang, Vector Optimization. Set-Valued and Variational Analysis, Lecture Notes in Economics and Mathematical Systems, 541, Springer-Verlag, Berlin, 2005. doi: 10.1007/3-540-28445-1.

[6]

F. Clarke, Functional Analysis, Calculus of Variations and Optimal Control, Graduate Texts in Mathematics, 264, Springer, London, 2013. doi: 10.1007/978-1-4471-4820-3.

[7]

M.-O. Czarnecki and L. Thibault, Sublevel representations of epi-Lipschitz sets and other properties, Math. Program., 168 (2018), 555-569.  doi: 10.1007/s10107-016-1070-y.

[8]

M. C. Delfour and J.-P. Zolésio, Shape analysis via oriented distance functions, J. Funct. Anal., 123 (1994), 129-201.  doi: 10.1006/jfan.1994.1086.

[9]

M. DureaJ. Dutta and C. Tammer, Lagrange multipliers for $\epsilon$-Pareto solutions in vector optimization with nonsolid cones in Banach spaces, J. Optim. Theory Appl., 145 (2010), 196-211.  doi: 10.1007/s10957-009-9609-1.

[10]

M. DureaR. Strugariu and C. Tammer, Scalarization in geometric and functional vector optimization revisited, J. Optim. Theory Appl., 159 (2013), 635-655.  doi: 10.1007/s10957-013-0360-2.

[11]

J. Dutta and C. Tammer, Lagrangian conditions for vector optimization in Banach spaces, Math. Methods Oper. Res., 64 (2006), 521-540.  doi: 10.1007/s00186-006-0079-z.

[12]

Y. Gao and X.-M. Yang, Properties of the nonlinear scalar functional and its applications to vector optimization problems, J. Global Optim., 73 (2019), 869-889.  doi: 10.1007/s10898-018-0725-z.

[13]

Y. GaoX. Yang and K. L. Teo, Optimality conditions for approximate solutions of vector optimization problems, J. Ind. Manag. Optim., 7 (2011), 483-496.  doi: 10.3934/jimo.2011.7.483.

[14]

C. Gerth and P. Weidner, Nonconvex separation theorems and some applications in vector optimization, J. Optim. Theory Appl., 67 (1990), 297-320.  doi: 10.1007/BF00940478.

[15]

C. GutiérrezB. JiménezE. Miglierina and E. Molho, Scalarization in set optimization with solid and nonsolid ordering cones, J. Global Optim., 61 (2015), 525-552.  doi: 10.1007/s10898-014-0179-x.

[16]

C. GutiérrezB. Jiménez and V. Novo, A unified approach and optimality conditions for approximate solutions of vector optimization problems, SIAM J. Optim., 17 (2006), 688-710.  doi: 10.1137/05062648X.

[17]

C. GutiérrezB. Jiménez and V. Novo, Improvement sets and vector optimization, European J. Oper. Res., 223 (2012), 304-311.  doi: 10.1016/j.ejor.2012.05.050.

[18]

C. Gutiérrez, B. Jiménez and V. Novo, Nonlinear scalarizations of set optimization problems with set orderings, in Set Optimization and Applications - The State of the Art, Springer Proc. Math. Stat., 151, Springer, Heidelberg, 2015, 43–63. doi: 10.1007/978-3-662-48670-2_2.

[19]

C. GutiérrezE. MiglierinaE. Molho and V. Novo, Pointwise well-posedness in set optimization with cone proper sets, Nonlinear Anal., 75 (2012), 1822-1833.  doi: 10.1016/j.na.2011.09.028.

[20]

C. GutiérrezV. NovoJ. L. Ródenas-Pedregosa and T. Tanaka, Nonconvex separation functional in linear spaces with applications to vector equilibria, SIAM J. Optim., 26 (2016), 2677-2695.  doi: 10.1137/16M1063575.

[21]

J.-B. Hiriart-Urruty, Tangent cones, generalized gradients and mathematical programming in Banach spaces, Math. Oper. Res., 4 (1979), 79-97.  doi: 10.1287/moor.4.1.79.

[22]

J.-B. Hiriart-Urruty, New concepts in nondifferentiable programming. Analyse non convexe, Bull. Soc. Math. France Mém., (1979), 57–85.

[23]

B. JiménezV. Novo and A. Vílchez, A set scalarization function based on the oriented distance and relations with other set scalarizations, Optimization, 67 (2018), 2091-2116.  doi: 10.1080/02331934.2018.1533554.

[24]

B. JiménezV. Novo and A. Vílchez, Characterization of set relations through extensions of the oriented distance, Math. Methods Oper. Res., 91 (2020), 89-115.  doi: 10.1007/s00186-019-00661-1.

[25]

A. A. Khan, C. Tammer and C. Zălinescu, Set-Valued Optimization. An Introduction with Applications, Vector Optimization, Springer, Heidelberg, 2015. doi: 10.1007/978-3-642-54265-7.

[26]

C. S. Lalitha and P. Chatterjee, Stability and scalarization in vector optimization using improvement sets, J. Optim. Theory Appl., 166 (2015), 825-843.  doi: 10.1007/s10957-014-0686-4.

[27]

G. H. LiS. J. Li and M. X. You, Relationships between the oriented distance functional and a nonlinear separation functional, J. Math. Anal. Appl., 466 (2018), 1109-1117.  doi: 10.1016/j.jmaa.2018.06.046.

[28]

C. G. LiuK. F. Ng and W. H. Yang, Merit functions in vector optimization, Math. Program., 119 (2009), 215-237.  doi: 10.1007/s10107-008-0208-y.

[29]

H. LuoX. Wang and B. Lukens, Variational analysis on the signed distance functions, J. Optim. Theory Appl., 180 (2019), 751-774.  doi: 10.1007/s10957-018-1414-2.

[30]

E. MiglierinaE. Molho and M. Rocca, Well-posedness and scalarization in vector optimization, J. Optim. Theory Appl., 126 (2005), 391-409.  doi: 10.1007/s10957-005-4723-1.

[31]

B. S. Mordukhovich, Variational Analysis and Generalized Differentiation. I. Basic Theory, Fundamental Principles of Mathematical Sciences, 330, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-31247-1.

[32]

D. Ralph, A chain rule for nonsmooth composite functions via minimisation, Bull. Austral. Math. Soc., 49 (1994), 129-137.  doi: 10.1017/S0004972700016178.

[33]

R. T. Rockafellar, Generalized directional derivatives and subgradients of nonconvex functions, Canadian J. Math., 32 (1980), 257-280.  doi: 10.4153/CJM-1980-020-7.

[34]

L. Thibault, On compactly Lipschitzian mappings, in Recent Advances in Optimization, Lecture Notes in Econom. and Math. Systems, 452, Springer, Berlin, 1997,356–364. doi: 10.1007/978-3-642-59073-3_25.

[35]

H.-Z. WeiC.-R. Chen and S.-J. Li, Characterizations for optimality conditions of general robust optimization problems, J. Optim. Theory Appl., 177 (2018), 835-856.  doi: 10.1007/s10957-018-1256-y.

[36]

L. Zadeh, Optimality and non-scalar-valued performance criteria, IEEE Trans. Automatic Control, 8 (1963), 59-60.  doi: 10.1109/TAC.1963.1105511.

[37]

A. Zaffaroni, Degrees of efficiency and degrees of minimality, SIAM J. Control Optim., 42 (2003), 1071-1086.  doi: 10.1137/S0363012902411532.

[38]

C. Zălinescu, Convex Analysis in General Vector Spaces, World Scientific Publishing Co., Inc., River Edge, NJ, 2002. doi: 10.1142/9789812777096.

[39]

K. Q. Zhao and X. M. Yang, $E-$Benson proper efficiency in vector optimization, Optimization, 64 (2015), 739-752.  doi: 10.1080/02331934.2013.798321.

[40]

K.-Q. ZhaoX.-M. Yang and J.-W. Peng, Weak $E$-optimal solution in vector optimization, Taiwanese J. Math., 17 (2013), 1287-1302.  doi: 10.11650/tjm.17.2013.2721.

show all references

References:
[1]

Q. H. AnsariE. Köbis and P. K. Sharma, Characterizations of multiobjective robustness via oriented distance function and image space analysis, J. Optim. Theory Appl., 181 (2019), 817-839.  doi: 10.1007/s10957-019-01505-y.

[2]

Y. Araya, Four types of nonlinear scalarizations and some applications in set optimization, Nonlinear Anal., 75 (2012), 3821-3835.  doi: 10.1016/j.na.2012.02.004.

[3]

R. I. BoţS.-M. Grad and G. Wanka, A general approach for studying duality in multiobjective optimization, Math. Methods Oper. Res., 65 (2007), 417-444.  doi: 10.1007/s00186-006-0125-x.

[4]

J. V. BurkeM. C. Ferris and M. Qian, On the Clarke subdifferential of the distance function of a closed set, J. Math. Anal. Appl., 166 (1992), 199-213.  doi: 10.1016/0022-247X(92)90336-C.

[5]

G.-Y. Chen, X. Huang and X. Yang, Vector Optimization. Set-Valued and Variational Analysis, Lecture Notes in Economics and Mathematical Systems, 541, Springer-Verlag, Berlin, 2005. doi: 10.1007/3-540-28445-1.

[6]

F. Clarke, Functional Analysis, Calculus of Variations and Optimal Control, Graduate Texts in Mathematics, 264, Springer, London, 2013. doi: 10.1007/978-1-4471-4820-3.

[7]

M.-O. Czarnecki and L. Thibault, Sublevel representations of epi-Lipschitz sets and other properties, Math. Program., 168 (2018), 555-569.  doi: 10.1007/s10107-016-1070-y.

[8]

M. C. Delfour and J.-P. Zolésio, Shape analysis via oriented distance functions, J. Funct. Anal., 123 (1994), 129-201.  doi: 10.1006/jfan.1994.1086.

[9]

M. DureaJ. Dutta and C. Tammer, Lagrange multipliers for $\epsilon$-Pareto solutions in vector optimization with nonsolid cones in Banach spaces, J. Optim. Theory Appl., 145 (2010), 196-211.  doi: 10.1007/s10957-009-9609-1.

[10]

M. DureaR. Strugariu and C. Tammer, Scalarization in geometric and functional vector optimization revisited, J. Optim. Theory Appl., 159 (2013), 635-655.  doi: 10.1007/s10957-013-0360-2.

[11]

J. Dutta and C. Tammer, Lagrangian conditions for vector optimization in Banach spaces, Math. Methods Oper. Res., 64 (2006), 521-540.  doi: 10.1007/s00186-006-0079-z.

[12]

Y. Gao and X.-M. Yang, Properties of the nonlinear scalar functional and its applications to vector optimization problems, J. Global Optim., 73 (2019), 869-889.  doi: 10.1007/s10898-018-0725-z.

[13]

Y. GaoX. Yang and K. L. Teo, Optimality conditions for approximate solutions of vector optimization problems, J. Ind. Manag. Optim., 7 (2011), 483-496.  doi: 10.3934/jimo.2011.7.483.

[14]

C. Gerth and P. Weidner, Nonconvex separation theorems and some applications in vector optimization, J. Optim. Theory Appl., 67 (1990), 297-320.  doi: 10.1007/BF00940478.

[15]

C. GutiérrezB. JiménezE. Miglierina and E. Molho, Scalarization in set optimization with solid and nonsolid ordering cones, J. Global Optim., 61 (2015), 525-552.  doi: 10.1007/s10898-014-0179-x.

[16]

C. GutiérrezB. Jiménez and V. Novo, A unified approach and optimality conditions for approximate solutions of vector optimization problems, SIAM J. Optim., 17 (2006), 688-710.  doi: 10.1137/05062648X.

[17]

C. GutiérrezB. Jiménez and V. Novo, Improvement sets and vector optimization, European J. Oper. Res., 223 (2012), 304-311.  doi: 10.1016/j.ejor.2012.05.050.

[18]

C. Gutiérrez, B. Jiménez and V. Novo, Nonlinear scalarizations of set optimization problems with set orderings, in Set Optimization and Applications - The State of the Art, Springer Proc. Math. Stat., 151, Springer, Heidelberg, 2015, 43–63. doi: 10.1007/978-3-662-48670-2_2.

[19]

C. GutiérrezE. MiglierinaE. Molho and V. Novo, Pointwise well-posedness in set optimization with cone proper sets, Nonlinear Anal., 75 (2012), 1822-1833.  doi: 10.1016/j.na.2011.09.028.

[20]

C. GutiérrezV. NovoJ. L. Ródenas-Pedregosa and T. Tanaka, Nonconvex separation functional in linear spaces with applications to vector equilibria, SIAM J. Optim., 26 (2016), 2677-2695.  doi: 10.1137/16M1063575.

[21]

J.-B. Hiriart-Urruty, Tangent cones, generalized gradients and mathematical programming in Banach spaces, Math. Oper. Res., 4 (1979), 79-97.  doi: 10.1287/moor.4.1.79.

[22]

J.-B. Hiriart-Urruty, New concepts in nondifferentiable programming. Analyse non convexe, Bull. Soc. Math. France Mém., (1979), 57–85.

[23]

B. JiménezV. Novo and A. Vílchez, A set scalarization function based on the oriented distance and relations with other set scalarizations, Optimization, 67 (2018), 2091-2116.  doi: 10.1080/02331934.2018.1533554.

[24]

B. JiménezV. Novo and A. Vílchez, Characterization of set relations through extensions of the oriented distance, Math. Methods Oper. Res., 91 (2020), 89-115.  doi: 10.1007/s00186-019-00661-1.

[25]

A. A. Khan, C. Tammer and C. Zălinescu, Set-Valued Optimization. An Introduction with Applications, Vector Optimization, Springer, Heidelberg, 2015. doi: 10.1007/978-3-642-54265-7.

[26]

C. S. Lalitha and P. Chatterjee, Stability and scalarization in vector optimization using improvement sets, J. Optim. Theory Appl., 166 (2015), 825-843.  doi: 10.1007/s10957-014-0686-4.

[27]

G. H. LiS. J. Li and M. X. You, Relationships between the oriented distance functional and a nonlinear separation functional, J. Math. Anal. Appl., 466 (2018), 1109-1117.  doi: 10.1016/j.jmaa.2018.06.046.

[28]

C. G. LiuK. F. Ng and W. H. Yang, Merit functions in vector optimization, Math. Program., 119 (2009), 215-237.  doi: 10.1007/s10107-008-0208-y.

[29]

H. LuoX. Wang and B. Lukens, Variational analysis on the signed distance functions, J. Optim. Theory Appl., 180 (2019), 751-774.  doi: 10.1007/s10957-018-1414-2.

[30]

E. MiglierinaE. Molho and M. Rocca, Well-posedness and scalarization in vector optimization, J. Optim. Theory Appl., 126 (2005), 391-409.  doi: 10.1007/s10957-005-4723-1.

[31]

B. S. Mordukhovich, Variational Analysis and Generalized Differentiation. I. Basic Theory, Fundamental Principles of Mathematical Sciences, 330, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-31247-1.

[32]

D. Ralph, A chain rule for nonsmooth composite functions via minimisation, Bull. Austral. Math. Soc., 49 (1994), 129-137.  doi: 10.1017/S0004972700016178.

[33]

R. T. Rockafellar, Generalized directional derivatives and subgradients of nonconvex functions, Canadian J. Math., 32 (1980), 257-280.  doi: 10.4153/CJM-1980-020-7.

[34]

L. Thibault, On compactly Lipschitzian mappings, in Recent Advances in Optimization, Lecture Notes in Econom. and Math. Systems, 452, Springer, Berlin, 1997,356–364. doi: 10.1007/978-3-642-59073-3_25.

[35]

H.-Z. WeiC.-R. Chen and S.-J. Li, Characterizations for optimality conditions of general robust optimization problems, J. Optim. Theory Appl., 177 (2018), 835-856.  doi: 10.1007/s10957-018-1256-y.

[36]

L. Zadeh, Optimality and non-scalar-valued performance criteria, IEEE Trans. Automatic Control, 8 (1963), 59-60.  doi: 10.1109/TAC.1963.1105511.

[37]

A. Zaffaroni, Degrees of efficiency and degrees of minimality, SIAM J. Control Optim., 42 (2003), 1071-1086.  doi: 10.1137/S0363012902411532.

[38]

C. Zălinescu, Convex Analysis in General Vector Spaces, World Scientific Publishing Co., Inc., River Edge, NJ, 2002. doi: 10.1142/9789812777096.

[39]

K. Q. Zhao and X. M. Yang, $E-$Benson proper efficiency in vector optimization, Optimization, 64 (2015), 739-752.  doi: 10.1080/02331934.2013.798321.

[40]

K.-Q. ZhaoX.-M. Yang and J.-W. Peng, Weak $E$-optimal solution in vector optimization, Taiwanese J. Math., 17 (2013), 1287-1302.  doi: 10.11650/tjm.17.2013.2721.

Figure 2.1.  $ \bar{y}\in bdA $ with $ \Delta_{A}(\bar{y}) = 0 $, $ \Delta_{cl A}(\bar{y}) = -2 $ and $ \Delta_{int A}(\bar{y}) = 1 $ in Example 2.1
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