Continuity of Solutions for Parametric Set Optimization Problems via Scalarization Methods

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

Received date: 2017-09-15

  Revised date: 2018-06-30

  Online published: 2021-03-11

Supported by

This research was supported by the National Natural Science Foundation of China (Nos. 11301567 and 11571055) and the Fundamental Research Funds for the Central Universities (No. 106112017CDJZRPY0020).

Abstract

The aim of this paper is to investigate the continuity of solution mappings for parametric set optimization problems with upper and lower set less order relations by scalarization methods. First, we recall some linear and nonlinear scalarization properties used to characterize the set order relations. Subsequently, we introduce new monotonicity concepts of the set-valued mapping by linear and nonlinear scalarization methods. Finally, we obtain the semicontinuity and closedness of solution mappings for parametric set optimization problems (both convex and nonconvex cases) under the monotonicity assumption and other suitable conditions. The results achieved do not impose the continuity of the set-valued objective mapping, which are obviously different from the related ones in the literature.

Cite this article

Pei-Pei Liu, Hong-Zhi Wei, Chun-Rong Chen, Sheng-Jie Li . Continuity of Solutions for Parametric Set Optimization Problems via Scalarization Methods[J]. Journal of the Operations Research Society of China, 2021 , 9(1) : 79 -97 . DOI: 10.1007/s40305-018-0230-6

References

[1] Khan, A.A., Tammer, C., Zǎlinescu, C.:Set-Valued Optimization:An Introduction with Applications. Springer, New York (2015)
[2] Miglierina, E., Molho, E.:Scalarization and stability in vector optimization. J. Optim. Theory Appl. 114, 657-670(2002)
[3] Xiang, S.W., Yin, W.S.:Stability results for efficient solutions of vector optimization problems. J. Optim. Theory Appl. 134, 385-398(2007)
[4] Kuroiwa, D., Tanaka, T., Ha, X.T.D.:On cone convexity of set-valued maps. Nonlinear Anal. 30, 1487-1496(1997)
[5] Kuroiwa, D.:On set-valued optimization. Nonlinear Anal. 47, 1395-1400(2001)
[6] Hernández, E., Rodríguez-Marín, L.:Existence theorems for set optimization problems. Nonlinear Anal. 67, 1726-1736(2007)
[7] Kuroiwa, D.:Existence theorems for set optimization with set-valued maps. J. Inf. Optim. Sci. 24, 73-84(2003)
[8] Alonso, M., Rodríguez-Marín, L.:Optimality and conditions for set-valued maps with set optimization. Nonlinear Anal. 70, 3057-3064(2009)
[9] Araya, Y.:Four types of nonlinear scalarization and some application in set optimization. Nonlinear Anal. 75, 3821-3835(2012)
[10] Hernández, E., Rodríguez-Marín, L.:Lagrangian duality in set-valued optimization. J. Optim. Theory Appl. 134, 119-134(2007)
[11] Löhne, A.:Optimization with set relations:conjugate duality. Optimization 54, 265-282(2005)
[12] Gutiérrez, C., Miglierina, E., Molho, E., Novo, V.:Pointwise well-posedness in set optimization with cone proper sets. Nonlinear Anal. 75, o1822-1833(2012)
[13] Zhang, W.Y., Li, S.J., Teo, K.L.:Well-posedness for set optimization problems. Nonlinear Anal. 71, 3769-3778(2009)
[14] Long, X.J., Peng, J.W., Peng, Z.Y.:Scalarization and pointwise well-posedness for set optimization problems. J. Glob. Optim. 62, 763-773(2015)
[15] Han, Y., Huang, N.J.:Well-posedness and stability of solutions for set optimization problems. Optimization 66, 17-33(2017)
[16] Bednarczuk, E.M., Miglierina, E., Molho, E.:A mountain pass-type theorem for vector-valued functions. Set-Valued Anal. 19, 569-587(2011)
[17] Ha, T.X.D.:Some variants of the Ekeland variational principle for a set-valued map. J. Optim. Theory Appl. 124, 187-206(2005)
[18] Xu, Y.D., Li, S.J.:Continuity of the solution set mappings to a parametric set optimization problem. Optim. Lett. 8, 2315-2327(2014)
[19] Xu, Y.D., Zhang, P.P.:On the stability of the solution set mappings to parametric set optimization problems. J. Oper. Res. Soc. China. 4, 255-263(2016)
[20] Xu, Y.D., Li, S.J.:On the solution continuity of parametric set optimization problems. Math. Methods Oper. Res. 84, 223-237(2016)
[21] Jahn, J., Ha, T.X.D.:New order relations in set optimization. J. Optim. Theory Appl. 148, 209-236(2011)
[22] Jahn, J.:Vectorization in set optimization. J. Optim. Theory Appl. 167, 783-795(2015)
[23] Köbis, E., Köbis, M.A.:Treatment of set order relations by means of a nonlinear scalarization functional:a full characterization. Optimization 65, 1805-1827(2016)
[24] Chen, J.W., Ansari, Q.H., Yao, J.-C.:Characterizations of set order relations and constrained set optimization problems via oriented distance function. Optimization 66, 1741-1754(2017)
[25] Chen, C.R., Li, S.J., Teo, K.L.:Solution semicontinuity of parametric generalized vector equilibrium problems. J. Glob. Optim. 45, 309-318(2009)
[26] Peng, Z.Y., Yang, X.M., Peng, J.W.:On the lower semicontinuity of the solution mappings to parametric weak generalized Ky Fan inequality. J. Optim. Theory Appl. 152, 256-264(2012)
[27] Chen, B., Huang, N.J.:Continuity of the solution mapping to parametric generalized vector equilibrium problems. J. Glob. Optim. 56, 1515-1528(2013)
[28] Peng, Z.Y., Zhao, Y., Yang, X.M.:Semicontinuity of approximate solution mappings to parametric set-valued weak vector equilibrium problems. Numer. Funct. Anal. Optim. 36, 481-500(2015)
[29] Chen, C.R., Zuo, X., Lu, F., Li, S.J.:Vector equilibrium problems under improvement sets and linear scalarization with stability applications. Optim. Methods Softw. 31, 1240-1257(2016)
[30] Aubin, J.P., Ekeland, I.:Applied Nonlinear Analysis. Wiley, New York (1984)
[31] Ferro, F.:A minimax theorem for vector-valued functions. J. Optim. Theory Appl. 60, 19-31(1989)
[32] Jahn, J.:Vector Optimization:Theory, Applications, and Extensions, 2nd edn. Springer, Berlin (2011)
[33] Anh,L.Q.,Khanh,P.Q.:Semicontinuityofthesolutionsetofparametricmultivaluedvectorquasiequilibrium problems. J. Math. Anal. Appl. 294, 699-711(2004)
[34] Hiriart-Urruty, J.B.:Tangent cones, generalized gradients and mathematical programming in Banach spaces. Math. Oper. Res. 4, 79-97(1979)
Options
Outlines

/