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doi: 10.3934/jimo.2021066
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Painlevé-Kuratowski convergences of the solution sets for vector optimization problems with free disposal sets

1. 

Faculty of Mathematical Economics, Banking University of Ho Chi Minh City, Ho Chi Minh City, Vietnam

2. 

Department of Mathematics and Computing, University of Science, Vietnam National University Ho Chi Minh City, Ho Chi Minh City, Vietnam

* Corresponding author: Nguyen Minh Tung

Dedicated to Professor Phan Quoc Khanh on the occasion of his 75th birthday

Received  April 2020 Revised  January 2021 Early access April 2021

This paper aims to present results on the Painlev$ \mathrm{\acute{e}} $-Kuratowski set-convergence of the sets of both infimal and minimal points of a sequence of perturbed vector optimization problems through free disposal sets. By assumptions of sequential compactness of the feasible sets or the uniform coerciveness of objective functions, this convergence is obtained both in the image and given spaces under the perturbations of objective functions and feasible sets. Besides, we also establish set-convergences of sequences of approximation solution sets. Applications to the stability of conic, quadratic and linear vector optimization problems are given. Some examples are provided to illustrate our results.

Citation: Nguyen Minh Tung, Mai Van Duy. Painlevé-Kuratowski convergences of the solution sets for vector optimization problems with free disposal sets. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021066
References:
[1]

H. Attouch and H. Riahi, Stability results for Ekeland's -variational principle and cone extremal solutions, Math. Oper. Res., 18 (1993), 173-201.  doi: 10.1287/moor.18.1.173.

[2]

K. J. Arrow, An extension of the basic theorems of classical welfare economics, in Proceedings of the Second Berkeley Symposium, University of California Press, Berkeley, CA, 1951,507-532.

[3]

J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhauser, Boston, 1990.

[4]

A. O. CarusoA. A. Khan and F. Raciti, Continuity results for a class of variational inequalities with applications to time-dependent network problems, Numer. Funct. Anal. Optim., 30 (2009), 1272-1288.  doi: 10.1080/01630560903381696.

[5]

J. M. Bonnisseau and B. Cornet, Existence of marginal cost pricing equilibria in an economy with several nonconvex firms, Econometrica., 58 (1990), 661-682.  doi: 10.2307/2938195.

[6]

J. M. Bonnisseau and B. Cornet, On the characterization of efficient production vectors, Econom. Theory., 31 (2007), 213-223.  doi: 10.1007/s00199-006-0096-4.

[7]

N. DinhM. A. GobernaD. H. Long and M. A. López-Cerdá, New Farkas-type results for vector-valued functions: A non-abstract approach, J. Optim. Theory Appl., 182 (2019), 4-29.  doi: 10.1007/s10957-018-1352-z.

[8]

M. Ehrgott, Multicriteria Optimization, Springer, Berlin Heidelberg, 2005.

[9]

S. D. Flam and A. Jourani, Prices and Pareto Optima, Optimization, 55 (2006), 611-625.  doi: 10.1080/02331930600808434.

[10]

M. FlorenzanoP. Gourdel and A. Jofre, Supporting weakly Pareto optimal allocations in infinite dimensional nonconvex economies, Econom. Theory., 29 (2006), 549-564.  doi: 10.1007/s00199-005-0033-y.

[11]

R. Guesnerie, Pareto optimality in nonconvex economies, Econometrica., 43 (1975), 1-29.  doi: 10.2307/1913410.

[12]

X. X. Huang, Stability in vector-valued and set-valued optimization, Math. Methods Oper. Res., 52 (2000), 185-193.  doi: 10.1007/s001860000085.

[13]

A. Jofré, A second-welfare theorem in nonconvex economies, in Constructive, Experimental, and Nonlinear Analysis, CMS Conference Proceedings, 27, AMS, Providence, RI, 2000,175–184.

[14]

A. Jofré and J. Rivera, An intrinsic characterization of free disposal hypothesis, Econom. Lett., 92 (2006), 423-427.  doi: 10.1016/j.econlet.2006.03.025.

[15]

M. A. Khan and R. Vohra, An extension of the second welfare theorem to economies with nonconvexities and public goods, Quart. J. Econom., 102 (1987), 223-241.  doi: 10.2307/1885061.

[16]

A. A. Khan, C. Tammer and C. Z$\mathrm{\breve{a}}$linescu, Set-Valued Optimization: An Introduction with Applications, Springer, Berlin, 2015. doi: 10.1007/978-3-642-54265-7.

[17]

S. Kapoor and C. S. Lalitha, Stability and scalarization for a unified vector optimization problem, J. Optim. Theory Appl., 182 (2019), 1050-1067.  doi: 10.1007/s10957-019-01514-x.

[18]

C. S. Lalitha and P. Chatterjee, Stability and scalarization of weak efficient, efficient and Henig proper efficient sets using generalized quasiconvexities, J. Optim. Theory Appl., 155 (2012), 941-961.  doi: 10.1007/s10957-012-0106-6.

[19]

C. S. Lalitha and P. Chatterjee, Stability for properly quasiconvex vector optimization problem, J.Optim. Theory Appl., 155 (2012), 492-506.  doi: 10.1007/s10957-012-0079-5.

[20]

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.

[21]

X. B. LiQ. L. Wang and Z. Lin, Stability results for properly quasiconvex vector optimization problems, Optimization, 64 (2015), 1329-1347.  doi: 10.1080/02331934.2013.860529.

[22]

X. B. LiZ. Lin and Z. Y. Peng, Convergence for vector optimization problems with variable ordering structure, Optimization, 65 (2016), 1615-1627.  doi: 10.1080/02331934.2016.1157879.

[23]

D. T. Luc, Theory of Vector Optimization, Lecture Notes in Economics and Mathematical Systems, 319, Springer, Berlin, 1989.

[24]

D. T. LucR. Lucchetti and C. Maliverti, Convergence of the efficient sets, Set-Valued Anal., 2 (1994), 207-218.  doi: 10.1007/BF01027102.

[25]

R. E. Lucchetti and E. Miglierina, Stability for convex vector optimization problems, Optimization, 53 (2004), 517-528.  doi: 10.1080/02331930412331327166.

[26]

A. Mas-Colell, The price equilibrium existence problem in topological vector lattices, Econometrica., 54 (1986), 1039-1055.  doi: 10.2307/1912321.

[27]

E. Miglierina and E. Molho, Scalarization and stability in vector optimization, J. Optim. Theory Appl., 114 (2002), 657-670.  doi: 10.1023/A:1016031214488.

[28]

B. S. Mordukhovich, Abstract extremal principle with applications to welfare economics, J. Math. Anal. Appl., 161 (2000), 187-212.  doi: 10.1006/jmaa.2000.7041.

[29]

H. Nakayama, Y. Sawaragi and T. Tanino, Theory of Multiobjective Optimization, Mathematics in science and engineering, 176, Academic Press Inc., London, 1985.

[30]

P. Oppezzi and A. M. Rossi, A convergence for vector valued functions, Optimization, 57 (2008), 435-448.  doi: 10.1080/02331930601129624.

[31]

P. Oppezzi and A. M. Rossi, A convergence for infinite dimensional vector valued functions, J. Glob. Optim., 42 (2008), 577-586.  doi: 10.1007/s10898-008-9284-z.

[32]

Z. Y. PengJ. M. PengX. J. Long and J. C. Yao, On the stability of solutions for semi-infinite vector optimization problems, J. Global Optim., 70 (2018), 55-69.  doi: 10.1007/s10898-017-0553-6.

[33]

Z. Y. PengZ. Y. Wang and X. M. Yang, Connectedness of solution sets for weak generalized symmetric Ky Fan inequality problems via addition-invariant sets, J. Optim. Theory Appl., 185 (2020), 188-206.  doi: 10.1007/s10957-020-01633-w.

[34]

Z. Y. PengX. F. Wang and X. M. Yang, Connectedness of approximate efficient solutions for generalized semi-infinite vector optimization problems, Set-Valued Var. Anal., 27 (2019), 103-118.  doi: 10.1007/s11228-017-0423-x.

[35]

G. Salinetti and R. J. B. Wets, On the convergence of sequences of convex sets in finite dimensions, SIAM Rev., 21 (1979), 18-33.  doi: 10.1137/1021002.

[36]

T. Tanaka, Generalized quasiconvexities, cone saddle points, and minimax theorem for vector-valued functions, J. Optim. Theory Appl., 81 (1994), 355-377.  doi: 10.1007/BF02191669.

[37]

H. Tuy, Minimax theorems revisited, Acta Math. Vietnam., 29 (2004), 217-229. 

[38]

N. V. Tuyen, Convergence of the relative Pareto efficient sets, Taiwanese J. Math., 20 (2016), 1149-1173.  doi: 10.11650/tjm.20.2016.6229.

[39]

J. J. WangZ. Y. PengZ. Lin and D. Q. Zhou, On the stability of solutions for the generalized vector quasi-equilibrium problems via free-disposal set, J. Ind. Manag. Optim., 17 (2021), 869-887.  doi: 10.3934/jimo.2020002.

show all references

References:
[1]

H. Attouch and H. Riahi, Stability results for Ekeland's -variational principle and cone extremal solutions, Math. Oper. Res., 18 (1993), 173-201.  doi: 10.1287/moor.18.1.173.

[2]

K. J. Arrow, An extension of the basic theorems of classical welfare economics, in Proceedings of the Second Berkeley Symposium, University of California Press, Berkeley, CA, 1951,507-532.

[3]

J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhauser, Boston, 1990.

[4]

A. O. CarusoA. A. Khan and F. Raciti, Continuity results for a class of variational inequalities with applications to time-dependent network problems, Numer. Funct. Anal. Optim., 30 (2009), 1272-1288.  doi: 10.1080/01630560903381696.

[5]

J. M. Bonnisseau and B. Cornet, Existence of marginal cost pricing equilibria in an economy with several nonconvex firms, Econometrica., 58 (1990), 661-682.  doi: 10.2307/2938195.

[6]

J. M. Bonnisseau and B. Cornet, On the characterization of efficient production vectors, Econom. Theory., 31 (2007), 213-223.  doi: 10.1007/s00199-006-0096-4.

[7]

N. DinhM. A. GobernaD. H. Long and M. A. López-Cerdá, New Farkas-type results for vector-valued functions: A non-abstract approach, J. Optim. Theory Appl., 182 (2019), 4-29.  doi: 10.1007/s10957-018-1352-z.

[8]

M. Ehrgott, Multicriteria Optimization, Springer, Berlin Heidelberg, 2005.

[9]

S. D. Flam and A. Jourani, Prices and Pareto Optima, Optimization, 55 (2006), 611-625.  doi: 10.1080/02331930600808434.

[10]

M. FlorenzanoP. Gourdel and A. Jofre, Supporting weakly Pareto optimal allocations in infinite dimensional nonconvex economies, Econom. Theory., 29 (2006), 549-564.  doi: 10.1007/s00199-005-0033-y.

[11]

R. Guesnerie, Pareto optimality in nonconvex economies, Econometrica., 43 (1975), 1-29.  doi: 10.2307/1913410.

[12]

X. X. Huang, Stability in vector-valued and set-valued optimization, Math. Methods Oper. Res., 52 (2000), 185-193.  doi: 10.1007/s001860000085.

[13]

A. Jofré, A second-welfare theorem in nonconvex economies, in Constructive, Experimental, and Nonlinear Analysis, CMS Conference Proceedings, 27, AMS, Providence, RI, 2000,175–184.

[14]

A. Jofré and J. Rivera, An intrinsic characterization of free disposal hypothesis, Econom. Lett., 92 (2006), 423-427.  doi: 10.1016/j.econlet.2006.03.025.

[15]

M. A. Khan and R. Vohra, An extension of the second welfare theorem to economies with nonconvexities and public goods, Quart. J. Econom., 102 (1987), 223-241.  doi: 10.2307/1885061.

[16]

A. A. Khan, C. Tammer and C. Z$\mathrm{\breve{a}}$linescu, Set-Valued Optimization: An Introduction with Applications, Springer, Berlin, 2015. doi: 10.1007/978-3-642-54265-7.

[17]

S. Kapoor and C. S. Lalitha, Stability and scalarization for a unified vector optimization problem, J. Optim. Theory Appl., 182 (2019), 1050-1067.  doi: 10.1007/s10957-019-01514-x.

[18]

C. S. Lalitha and P. Chatterjee, Stability and scalarization of weak efficient, efficient and Henig proper efficient sets using generalized quasiconvexities, J. Optim. Theory Appl., 155 (2012), 941-961.  doi: 10.1007/s10957-012-0106-6.

[19]

C. S. Lalitha and P. Chatterjee, Stability for properly quasiconvex vector optimization problem, J.Optim. Theory Appl., 155 (2012), 492-506.  doi: 10.1007/s10957-012-0079-5.

[20]

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.

[21]

X. B. LiQ. L. Wang and Z. Lin, Stability results for properly quasiconvex vector optimization problems, Optimization, 64 (2015), 1329-1347.  doi: 10.1080/02331934.2013.860529.

[22]

X. B. LiZ. Lin and Z. Y. Peng, Convergence for vector optimization problems with variable ordering structure, Optimization, 65 (2016), 1615-1627.  doi: 10.1080/02331934.2016.1157879.

[23]

D. T. Luc, Theory of Vector Optimization, Lecture Notes in Economics and Mathematical Systems, 319, Springer, Berlin, 1989.

[24]

D. T. LucR. Lucchetti and C. Maliverti, Convergence of the efficient sets, Set-Valued Anal., 2 (1994), 207-218.  doi: 10.1007/BF01027102.

[25]

R. E. Lucchetti and E. Miglierina, Stability for convex vector optimization problems, Optimization, 53 (2004), 517-528.  doi: 10.1080/02331930412331327166.

[26]

A. Mas-Colell, The price equilibrium existence problem in topological vector lattices, Econometrica., 54 (1986), 1039-1055.  doi: 10.2307/1912321.

[27]

E. Miglierina and E. Molho, Scalarization and stability in vector optimization, J. Optim. Theory Appl., 114 (2002), 657-670.  doi: 10.1023/A:1016031214488.

[28]

B. S. Mordukhovich, Abstract extremal principle with applications to welfare economics, J. Math. Anal. Appl., 161 (2000), 187-212.  doi: 10.1006/jmaa.2000.7041.

[29]

H. Nakayama, Y. Sawaragi and T. Tanino, Theory of Multiobjective Optimization, Mathematics in science and engineering, 176, Academic Press Inc., London, 1985.

[30]

P. Oppezzi and A. M. Rossi, A convergence for vector valued functions, Optimization, 57 (2008), 435-448.  doi: 10.1080/02331930601129624.

[31]

P. Oppezzi and A. M. Rossi, A convergence for infinite dimensional vector valued functions, J. Glob. Optim., 42 (2008), 577-586.  doi: 10.1007/s10898-008-9284-z.

[32]

Z. Y. PengJ. M. PengX. J. Long and J. C. Yao, On the stability of solutions for semi-infinite vector optimization problems, J. Global Optim., 70 (2018), 55-69.  doi: 10.1007/s10898-017-0553-6.

[33]

Z. Y. PengZ. Y. Wang and X. M. Yang, Connectedness of solution sets for weak generalized symmetric Ky Fan inequality problems via addition-invariant sets, J. Optim. Theory Appl., 185 (2020), 188-206.  doi: 10.1007/s10957-020-01633-w.

[34]

Z. Y. PengX. F. Wang and X. M. Yang, Connectedness of approximate efficient solutions for generalized semi-infinite vector optimization problems, Set-Valued Var. Anal., 27 (2019), 103-118.  doi: 10.1007/s11228-017-0423-x.

[35]

G. Salinetti and R. J. B. Wets, On the convergence of sequences of convex sets in finite dimensions, SIAM Rev., 21 (1979), 18-33.  doi: 10.1137/1021002.

[36]

T. Tanaka, Generalized quasiconvexities, cone saddle points, and minimax theorem for vector-valued functions, J. Optim. Theory Appl., 81 (1994), 355-377.  doi: 10.1007/BF02191669.

[37]

H. Tuy, Minimax theorems revisited, Acta Math. Vietnam., 29 (2004), 217-229. 

[38]

N. V. Tuyen, Convergence of the relative Pareto efficient sets, Taiwanese J. Math., 20 (2016), 1149-1173.  doi: 10.11650/tjm.20.2016.6229.

[39]

J. J. WangZ. Y. PengZ. Lin and D. Q. Zhou, On the stability of solutions for the generalized vector quasi-equilibrium problems via free-disposal set, J. Ind. Manag. Optim., 17 (2021), 869-887.  doi: 10.3934/jimo.2020002.

Figure 1.  The free disposal set $ E $ and some infimal sets
Figure 2.  The free disposal set $ E $ and $ E $-infimal sets in Example 6
Figure 3.  The free disposal set $ E $ and $ E $-infimal sets in Example 7
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