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doi: 10.3934/jimo.2022046
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Asymptotic analysis of scalarization functions and applications

1. 

Chongqing Key Laboratory of Social Economic and Applied Statistics, School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China

2. 

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

3. 

College of Mathematics and Information, China West Normal University, Nanchong 637009, Sichuan, China

*Corresponding author: Shengjie Li

Received  September 2021 Revised  January 2022 Early access March 2022

In this paper, we consider two common scalarization functions and their applications via asymptotic analysis. We mainly analyze the recession and asymptotic properties of translation invariant function and oriented distance function, and discuss their monotonicity and Lipschitz continuity in terms of recession functions. Finally, we apply these scalarization functions to the characterization of the nonemptiness and boundedness of the solution set for a general constrained nonconvex optimization problem.

Citation: Genghua Li, Shengjie Li, Manxue You. Asymptotic analysis of scalarization functions and applications. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022046
References:
[1]

A. Auslender and M. Teboulle, Asymptotic Cones and Functions in Optimization and Variational Inequalities, Springer Monographs in Mathematics. Springer-Verlag, New York, 2003. doi: 10.1007/b97594.

[2]

C. R. ChenL. L. Li and M. H. Li, Hölder continuity results for nonconvex parametric generalized vector quasiequilibrium problems via nonlinear scalarizing functions, Optimization, 65 (2016), 35-51.  doi: 10.1080/02331934.2014.984707.

[3]

M. ChiccoF. MignanegoL. Pusillo and S. Tijs, Vector optimization problems via improvement sets, J. Optim. Theory App., 150 (2011), 516-529.  doi: 10.1007/s10957-011-9851-1.

[4]

S. Deng, Boundedness and nonemptiness of the efficient solution sets in multiobjective optimization, J. Optim. Theory Appl., 144 (2010), 29-42.  doi: 10.1007/s10957-009-9589-1.

[5]

F. Flores-BazánF. Flores-Bazán and C. Vera, Maximizing and minimizing quasiconvex functions: Related properties, existence and optimality conditions via radial epiderivatives, J. Glob. Optim., 63 (2015), 99-123.  doi: 10.1007/s10898-015-0267-6.

[6]

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

[7]

M. S. Gowda, An analysis of zero set and global error bound properties of a piecewise affine function via its recession function, SIAM J. Matrix Anal. Appl., 17 (1996), 594-609.  doi: 10.1137/S0895479894278940.

[8]

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.

[9]

N. HadjisavvasF. Lara and J. E. Martínez-Legaz, A quasiconvex asymptotic function with applications in optimization, J. Optim. Theory App., 180 (2019), 170-186.  doi: 10.1007/s10957-018-1317-2.

[10]

N. HadjisavvasF. Lara and D. T. Luc, A general asymptotic function with applications in nonconvex optimization, J. Glob. Optim., 78 (2020), 49-68.  doi: 10.1007/s10898-020-00891-2.

[11]

J. B. Hiriart-Urruty, New concepts in nondifferentiable programming, Bull. Soc. Math. France., 60 (1979), 57-85.  doi: 10.24033/msmf.261.

[12]

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.

[13]

F. Lara, Generalized asymptotic functions in nonconvex multiobjective optimization problems, Optimization, 66 (2017), 1259-1272.  doi: 10.1080/02331934.2016.1235162.

[14]

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.

[15]

G. H. LiS. J. Li and M. X. You, Recession function and its applications in optimization, Optimization, 70 (2021), 2559-2578.  doi: 10.1080/02331934.2020.1786569.

[16]

D. T. Luc, Recession cones and the domination property in vector optimization, Math. Program., 49 (1990/91), 113-122.  doi: 10.1007/BF01588781.

[17] R. T. Rockafellar, Convex Analysis, Princeton University Press, 1970.  doi: 10.1515/9781400873173.
[18]

R. T. Rockafellar and R. J. B. Wets, Variational Analysis, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-642-02431-3.

[19]

C. Tammer and P. Weidner, Scalarization and Separation by Translation Invariant Functions, Vector Optimization. Springer, Cham, 2020. doi: 10.1007/978-3-030-44723-6.

[20]

C. Tammer and C. Zǎlinescu, Lipschitz properties of the scalarization function and applications, Optimization, 59 (2010), 305-319.  doi: 10.1080/02331930801951033.

[21]

P. Weidner, Lower semicontinuous functionals with uniform sublevel sets, Optimization, 66 (2017), 491-505.  doi: 10.1080/02331934.2017.1279161.

[22]

P. Weidner, Gerstewitz functionals on linear spaces and functionals with uniform sublevel sets, J. Optim. Theory Appl., 173 (2017), 812-827.  doi: 10.1007/s10957-017-1098-z.

[23]

P. Weidner, A new tool for the investigation of extended real-valued functions, Optim. Lett., 13 (2019), 1651-1661.  doi: 10.1007/s11590-018-1370-7.

[24]

Y. D. Xu and S. J. Li, A new nonlinear scalarization function and applications, Optimization, 65 (2016), 207-231.  doi: 10.1080/02331934.2015.1014479.

[25]

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

show all references

References:
[1]

A. Auslender and M. Teboulle, Asymptotic Cones and Functions in Optimization and Variational Inequalities, Springer Monographs in Mathematics. Springer-Verlag, New York, 2003. doi: 10.1007/b97594.

[2]

C. R. ChenL. L. Li and M. H. Li, Hölder continuity results for nonconvex parametric generalized vector quasiequilibrium problems via nonlinear scalarizing functions, Optimization, 65 (2016), 35-51.  doi: 10.1080/02331934.2014.984707.

[3]

M. ChiccoF. MignanegoL. Pusillo and S. Tijs, Vector optimization problems via improvement sets, J. Optim. Theory App., 150 (2011), 516-529.  doi: 10.1007/s10957-011-9851-1.

[4]

S. Deng, Boundedness and nonemptiness of the efficient solution sets in multiobjective optimization, J. Optim. Theory Appl., 144 (2010), 29-42.  doi: 10.1007/s10957-009-9589-1.

[5]

F. Flores-BazánF. Flores-Bazán and C. Vera, Maximizing and minimizing quasiconvex functions: Related properties, existence and optimality conditions via radial epiderivatives, J. Glob. Optim., 63 (2015), 99-123.  doi: 10.1007/s10898-015-0267-6.

[6]

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

[7]

M. S. Gowda, An analysis of zero set and global error bound properties of a piecewise affine function via its recession function, SIAM J. Matrix Anal. Appl., 17 (1996), 594-609.  doi: 10.1137/S0895479894278940.

[8]

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.

[9]

N. HadjisavvasF. Lara and J. E. Martínez-Legaz, A quasiconvex asymptotic function with applications in optimization, J. Optim. Theory App., 180 (2019), 170-186.  doi: 10.1007/s10957-018-1317-2.

[10]

N. HadjisavvasF. Lara and D. T. Luc, A general asymptotic function with applications in nonconvex optimization, J. Glob. Optim., 78 (2020), 49-68.  doi: 10.1007/s10898-020-00891-2.

[11]

J. B. Hiriart-Urruty, New concepts in nondifferentiable programming, Bull. Soc. Math. France., 60 (1979), 57-85.  doi: 10.24033/msmf.261.

[12]

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.

[13]

F. Lara, Generalized asymptotic functions in nonconvex multiobjective optimization problems, Optimization, 66 (2017), 1259-1272.  doi: 10.1080/02331934.2016.1235162.

[14]

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.

[15]

G. H. LiS. J. Li and M. X. You, Recession function and its applications in optimization, Optimization, 70 (2021), 2559-2578.  doi: 10.1080/02331934.2020.1786569.

[16]

D. T. Luc, Recession cones and the domination property in vector optimization, Math. Program., 49 (1990/91), 113-122.  doi: 10.1007/BF01588781.

[17] R. T. Rockafellar, Convex Analysis, Princeton University Press, 1970.  doi: 10.1515/9781400873173.
[18]

R. T. Rockafellar and R. J. B. Wets, Variational Analysis, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-642-02431-3.

[19]

C. Tammer and P. Weidner, Scalarization and Separation by Translation Invariant Functions, Vector Optimization. Springer, Cham, 2020. doi: 10.1007/978-3-030-44723-6.

[20]

C. Tammer and C. Zǎlinescu, Lipschitz properties of the scalarization function and applications, Optimization, 59 (2010), 305-319.  doi: 10.1080/02331930801951033.

[21]

P. Weidner, Lower semicontinuous functionals with uniform sublevel sets, Optimization, 66 (2017), 491-505.  doi: 10.1080/02331934.2017.1279161.

[22]

P. Weidner, Gerstewitz functionals on linear spaces and functionals with uniform sublevel sets, J. Optim. Theory Appl., 173 (2017), 812-827.  doi: 10.1007/s10957-017-1098-z.

[23]

P. Weidner, A new tool for the investigation of extended real-valued functions, Optim. Lett., 13 (2019), 1651-1661.  doi: 10.1007/s11590-018-1370-7.

[24]

Y. D. Xu and S. J. Li, A new nonlinear scalarization function and applications, Optimization, 65 (2016), 207-231.  doi: 10.1080/02331934.2015.1014479.

[25]

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

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