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2014, 4(4): 295-307. doi: 10.3934/naco.2014.4.295

Nonlinear scalarization with applications to Hölder continuity of approximate solutions

1. 

College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

2. 

College of Mathematics and Statistics, Chongqing University, Chongqing, 401331

Received  May 2014 Revised  September 2014 Published  December 2014

In this article, firstly, some useful properties of the Gerstewitz scalarizing function are discussed, such as its globally Lipschitz property, concavity and monotonicity. Secondly, as an application of these properties, verifiable sufficient conditions for Hölder continuity of approximate solutions to parametric generalized vector equilibrium problems are established via Gerstewitz scalarizations. Moreover, some examples are provided to illustrate our main conclusions in the vector settings.
Citation: Lili Li, Chunrong Chen. Nonlinear scalarization with applications to Hölder continuity of approximate solutions. Numerical Algebra, Control & Optimization, 2014, 4 (4) : 295-307. doi: 10.3934/naco.2014.4.295
References:
[1]

L. Q. Anh and P. Q. Khanh, Semicontinuity of the approximate solution sets of multivalued quasiequilibrium problems,, Numer. Funct. Anal. Optim., 29 (2008), 24. doi: 10.1080/01630560701873068.

[2]

L. Q. Anh, P. Q. Khanh and T. N. Tam, On Hölder continuity of approximate solutions to parametric equilibrium problems,, Nonlinear Anal., 75 (2012), 2293. doi: 10.1016/j.na.2011.10.029.

[3]

L. Q. Anh and P. Q. Khanh, On the Hölder continuity of solutions to parametric multivalued vector equilibrium problems,, J. Math. Anal. Appl., 321 (2006), 308. doi: 10.1016/j.jmaa.2005.08.018.

[4]

L. Q. Anh and P. Q. Khanh, Uniqueness and Hölder continuity of the solution to multivalued equilibrium problems in metric spaces,, J. Global Optim., 37 (2007), 449. doi: 10.1007/s10898-006-9062-8.

[5]

L. Q. Anh and P. Q. Khanh, Sensitivity analysis for multivalued quasiequilibrium problems in metric spaces: Hölder continuity of solutions,, J. Global Optim., 42 (2008), 515. doi: 10.1007/s10898-007-9268-4.

[6]

Q. H. Ansari and J. C. Yao (eds.), Recent Developments in Vector Optimization,, Springer, (2012). doi: 10.1007/978-3-642-21114-0.

[7]

M. Bianchi and R. Pini, Sensitivity for parametric vector equilibria,, Optimization, 55 (2006), 221. doi: 10.1080/02331930600662732.

[8]

C. R. Chen, S. J. Li and K. L. Teo, Solution semicontinuity of parametric generalized vector equilibrium problems,, J. Global Optim., 45 (2009), 309. doi: 10.1007/s10898-008-9376-9.

[9]

C. R. Chen and S. J. Li, Semicontinuity results on parametric vector variational inequalities with polyhedral constraint sets,, J. Optim. Theory Appl., 158 (2013), 97. doi: 10.1007/s10957-012-0199-y.

[10]

C. R. Chen, Hölder continuity of the unique solution to parametric vector quasiequilibrium problems via nonlinear scalarization,, Positivity, 17 (2013), 133. doi: 10.1007/s11117-011-0153-5.

[11]

C. R. Chen and M. H. Li, Hölder continuity of solutions to parametric vector equilibrium problems with nonlinear scalarization,, Numer. Funct. Anal. Optim., 35 (2014), 685. doi: 10.1080/01630563.2013.818549.

[12]

G. Y. Chen, X. X. Huang and X. Q. Yang, Vector Optimization: Set-Valued and Variational Analysis,, Springer, (2005).

[13]

M. Durea and R. Strugariu, Scalarization of constraints system in some vector optimization problems and applications,, Optim. Lett., 8 (2014), 2021. doi: 10.1007/s11590-013-0690-x.

[14]

M. Durea and C. Tammer, Fuzzy necessary optimality conditions for vector optimization problems,, Optimization, 58 (2009), 449. doi: 10.1080/02331930701761615.

[15]

Chr. Gerstewitz (Tammer), Nichtkonvexe Dualität in der Vektoroptimierung,, Wiss. Z. TH Leuna-Merseburg, 25 (1983), 357.

[16]

X. H. Gong and J. C. Yao, Connectedness of the set of efficient solutions for generalized systems,, J. Optim. Theory Appl., 138 (2008), 189. doi: 10.1007/s10957-008-9378-2.

[17]

X. H. Gong and J. C. Yao, Lower semicontinuity of the set of efficient solutions for generalized systems,, J. Optim. Theory Appl., 138 (2008), 197. doi: 10.1007/s10957-008-9379-1.

[18]

A. Göpfert, H. Riahi, C. Tammer and C. Zălinescu, Variational Methods in Partially Ordered Spaces,, Springer-Verlag, (2003).

[19]

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

[20]

K. Kimura and J. C. Yao, Semicontinuity of solution mappings of parametric generalized vector equilibrium problems,, J. Optim. Theory Appl., 138 (2008), 429. doi: 10.1007/s10957-008-9386-2.

[21]

M. H. Li, S. J. Li and C. R. Chen, Hölder-likeness and contingent derivative of solutions to parametric weak vector equilibrium problems (in Chinese),, Sci. Sin. Math., 43 (2013), 61.

[22]

S. J. Li, X. B. Li, L. N. Wang and K. L. Teo, The Hölder continuity of solutions to generalized vector equilibrium problems,, European J. Oper. Res., 199 (2009), 334. doi: 10.1016/j.ejor.2008.12.024.

[23]

S. J. Li, C. R. Chen, X. B. Li and K. L. Teo, Hölder continuity and upper estimates of solutions to vector quasiequilibrium problems,, European J. Oper. Res., 210 (2011), 148. doi: 10.1016/j.ejor.2010.10.005.

[24]

S. J. Li and X. B. Li, Hölder continuity of solutions to parametric weak generalized Ky Fan inequality,, J. Optim. Theory Appl., 149 (2011), 540. doi: 10.1007/s10957-011-9803-9.

[25]

X. B. Li, S. J. Li and C. R. Chen, Lipschitz continuity of an approximate solution mapping to equilibrium problems,, Taiwanese J. Math., 16 (2012), 1027.

[26]

F. Lu and C. R. Chen, Notes on Lipschitz properties of nonlinear scalarization functions with applications,, Abstr. Appl. Anal., 2014 (2014). doi: 10.1155/2014/792364.

[27]

F. Lu and C. R. Chen, Newton-like methods for solving vector optimization problems,, Appl. Anal., 93 (2014), 1567. doi: 10.1080/00036811.2013.839781.

[28]

N. M. Nam and C. Zălinescu, Variational analysis of directional minimal time functions and applications to location problems,, Set-Valued Var. Anal., 21 (2013), 405. doi: 10.1007/s11228-013-0232-9.

[29]

C. Tammer and C. Zălinescu, Lipschitz properties of the scalarization function and applications,, Optimization, 59 (2010), 305. doi: 10.1080/02331930801951033.

[30]

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

show all references

References:
[1]

L. Q. Anh and P. Q. Khanh, Semicontinuity of the approximate solution sets of multivalued quasiequilibrium problems,, Numer. Funct. Anal. Optim., 29 (2008), 24. doi: 10.1080/01630560701873068.

[2]

L. Q. Anh, P. Q. Khanh and T. N. Tam, On Hölder continuity of approximate solutions to parametric equilibrium problems,, Nonlinear Anal., 75 (2012), 2293. doi: 10.1016/j.na.2011.10.029.

[3]

L. Q. Anh and P. Q. Khanh, On the Hölder continuity of solutions to parametric multivalued vector equilibrium problems,, J. Math. Anal. Appl., 321 (2006), 308. doi: 10.1016/j.jmaa.2005.08.018.

[4]

L. Q. Anh and P. Q. Khanh, Uniqueness and Hölder continuity of the solution to multivalued equilibrium problems in metric spaces,, J. Global Optim., 37 (2007), 449. doi: 10.1007/s10898-006-9062-8.

[5]

L. Q. Anh and P. Q. Khanh, Sensitivity analysis for multivalued quasiequilibrium problems in metric spaces: Hölder continuity of solutions,, J. Global Optim., 42 (2008), 515. doi: 10.1007/s10898-007-9268-4.

[6]

Q. H. Ansari and J. C. Yao (eds.), Recent Developments in Vector Optimization,, Springer, (2012). doi: 10.1007/978-3-642-21114-0.

[7]

M. Bianchi and R. Pini, Sensitivity for parametric vector equilibria,, Optimization, 55 (2006), 221. doi: 10.1080/02331930600662732.

[8]

C. R. Chen, S. J. Li and K. L. Teo, Solution semicontinuity of parametric generalized vector equilibrium problems,, J. Global Optim., 45 (2009), 309. doi: 10.1007/s10898-008-9376-9.

[9]

C. R. Chen and S. J. Li, Semicontinuity results on parametric vector variational inequalities with polyhedral constraint sets,, J. Optim. Theory Appl., 158 (2013), 97. doi: 10.1007/s10957-012-0199-y.

[10]

C. R. Chen, Hölder continuity of the unique solution to parametric vector quasiequilibrium problems via nonlinear scalarization,, Positivity, 17 (2013), 133. doi: 10.1007/s11117-011-0153-5.

[11]

C. R. Chen and M. H. Li, Hölder continuity of solutions to parametric vector equilibrium problems with nonlinear scalarization,, Numer. Funct. Anal. Optim., 35 (2014), 685. doi: 10.1080/01630563.2013.818549.

[12]

G. Y. Chen, X. X. Huang and X. Q. Yang, Vector Optimization: Set-Valued and Variational Analysis,, Springer, (2005).

[13]

M. Durea and R. Strugariu, Scalarization of constraints system in some vector optimization problems and applications,, Optim. Lett., 8 (2014), 2021. doi: 10.1007/s11590-013-0690-x.

[14]

M. Durea and C. Tammer, Fuzzy necessary optimality conditions for vector optimization problems,, Optimization, 58 (2009), 449. doi: 10.1080/02331930701761615.

[15]

Chr. Gerstewitz (Tammer), Nichtkonvexe Dualität in der Vektoroptimierung,, Wiss. Z. TH Leuna-Merseburg, 25 (1983), 357.

[16]

X. H. Gong and J. C. Yao, Connectedness of the set of efficient solutions for generalized systems,, J. Optim. Theory Appl., 138 (2008), 189. doi: 10.1007/s10957-008-9378-2.

[17]

X. H. Gong and J. C. Yao, Lower semicontinuity of the set of efficient solutions for generalized systems,, J. Optim. Theory Appl., 138 (2008), 197. doi: 10.1007/s10957-008-9379-1.

[18]

A. Göpfert, H. Riahi, C. Tammer and C. Zălinescu, Variational Methods in Partially Ordered Spaces,, Springer-Verlag, (2003).

[19]

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

[20]

K. Kimura and J. C. Yao, Semicontinuity of solution mappings of parametric generalized vector equilibrium problems,, J. Optim. Theory Appl., 138 (2008), 429. doi: 10.1007/s10957-008-9386-2.

[21]

M. H. Li, S. J. Li and C. R. Chen, Hölder-likeness and contingent derivative of solutions to parametric weak vector equilibrium problems (in Chinese),, Sci. Sin. Math., 43 (2013), 61.

[22]

S. J. Li, X. B. Li, L. N. Wang and K. L. Teo, The Hölder continuity of solutions to generalized vector equilibrium problems,, European J. Oper. Res., 199 (2009), 334. doi: 10.1016/j.ejor.2008.12.024.

[23]

S. J. Li, C. R. Chen, X. B. Li and K. L. Teo, Hölder continuity and upper estimates of solutions to vector quasiequilibrium problems,, European J. Oper. Res., 210 (2011), 148. doi: 10.1016/j.ejor.2010.10.005.

[24]

S. J. Li and X. B. Li, Hölder continuity of solutions to parametric weak generalized Ky Fan inequality,, J. Optim. Theory Appl., 149 (2011), 540. doi: 10.1007/s10957-011-9803-9.

[25]

X. B. Li, S. J. Li and C. R. Chen, Lipschitz continuity of an approximate solution mapping to equilibrium problems,, Taiwanese J. Math., 16 (2012), 1027.

[26]

F. Lu and C. R. Chen, Notes on Lipschitz properties of nonlinear scalarization functions with applications,, Abstr. Appl. Anal., 2014 (2014). doi: 10.1155/2014/792364.

[27]

F. Lu and C. R. Chen, Newton-like methods for solving vector optimization problems,, Appl. Anal., 93 (2014), 1567. doi: 10.1080/00036811.2013.839781.

[28]

N. M. Nam and C. Zălinescu, Variational analysis of directional minimal time functions and applications to location problems,, Set-Valued Var. Anal., 21 (2013), 405. doi: 10.1007/s11228-013-0232-9.

[29]

C. Tammer and C. Zălinescu, Lipschitz properties of the scalarization function and applications,, Optimization, 59 (2010), 305. doi: 10.1080/02331930801951033.

[30]

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

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