Stability of efficient solutions to set optimization problems. (English) Zbl 1475.49028
The authors consider set optimization problems in real topological Hausdorff spaces as well as the Painleve-Kuratowski convergence of Pareto minimal elements. In particular, they study stability properties of solution sets of corresponding perturbed set optimization problems. Finally, they introduce a compact convergence concept in order to study the internal stability of solution sets.
Reviewer: Jan-Joachim Rückmann (Bergen)
MSC:
| 49K40 | Sensitivity, stability, well-posedness |
| 65K10 | Numerical optimization and variational techniques |
| 90C29 | Multi-objective and goal programming |
Keywords:
set optimization problem; Pareto minimal solution; internal and external stability; compact convergence; domination propertyReferences:
| [1] | Anh, LQ; Duy, TQ, Tykhonov well-posedness for lexicographic equilibrium problems, Optimization, 65, 11, 1929-1948 (2016) · Zbl 1353.49034 |
| [2] | Anh, LQ; Duy, TQ; Hien, DV; Kuroiwa, D.; Petrot, N., Convergence of solutions to set optimization problems with the set less order relation, J. Optim. Theory Appl., 185, 416-432 (2020) · Zbl 1439.49055 |
| [3] | Anh, LQ; Duy, TQ; Hien, DV, Stability for parametric vector quasi-equilibrium problems with variable cones, Numer. Func. Anal. Optim., 40, 4, 461-483 (2019) · Zbl 1414.49028 |
| [4] | Anh, LQ; Hung, NV, Gap functions and Hausdorff continuity of solution mappings to parametric strong vector quasiequilibrium problems, J. Ind. Manag. Optim., 14, 1, 65-79 (2018) · Zbl 1412.90145 |
| [5] | Bao, TQ; Mordukhovich, BS; Kusuoka, S.; Maruyama, T., Set-valued optimization in welfare economics, Advances in Mathematical Economics, 113-153 (2010), Tokyo: Springer, Tokyo · Zbl 1203.91074 |
| [6] | Bednarczuk, EM, A note on lower semicontinuity of minimal points, Nonlinear Anal., 50, 3, 285-297 (2002) · Zbl 1041.90049 |
| [7] | Bednarczuk, EM, Continuity of minimal points with applications to parametric multiple objective optimization, Eur. J. Oper. Res., 157, 1, 59-67 (2004) · Zbl 1106.90064 |
| [8] | Beer, G., Topologies on Closed and Closed Convex Sets (1993), Berlin: Springer, Berlin · Zbl 0792.54008 |
| [9] | Chinaie, M.; Fakhar, F.; Fakhar, M.; Hajisharifi, H., Weak minimal elements and weak minimal solutions of a nonconvex set-valued optimization problem, J. Glob. Optim., 75, 1, 131-141 (2019) · Zbl 1428.90149 |
| [10] | Gaydu, M.; Geoffroy, MH; Jean-Alexis, C.; Nedelcheva, D., Stability of minimizers of set optimization problems, Positivity, 21, 1, 127-141 (2017) · Zbl 1369.49021 |
| [11] | Geoffroy, MH, A topological convergence on power sets well-suited for set optimization, J. Glob. Optim., 73, 3, 567-581 (2019) · Zbl 1417.49017 |
| [12] | Gutiérrez, C.; Jiménez, B.; Miglierina, E.; Molho, E., Scalarization in set optimization with solid and nonsolid ordering cones, J. Glob. Optim., 61, 3, 525-552 (2015) · Zbl 1311.49041 |
| [13] | Gutiérrez, C.; Miglierina, E.; Molho, E.; Novo, V., Convergence of solutions of a set optimization problem in the image space, J. Optim. Theory Appl., 170, 2, 358-371 (2016) · Zbl 1346.49021 |
| [14] | Hamel, A.H., Heyde, F., Löhne, A., Rudloff, B., Schrage, C.: Set optimization and applications-the state of the art. In: Springer Proceedings in Mathematics & Statistics, vol. 151, (2015) · Zbl 1335.49003 |
| [15] | Han, Y.; Huang, NJ, Well-posedness and stability of solutions for set optimization problems, Optimization, 66, 1, 17-33 (2017) · Zbl 1365.90233 |
| [16] | Huong, VT; Yao, JC; Yen, ND, Differentiability properties of a parametric consumer problem, J. Nonlinear Convex Anal., 19, 7, 1217-1245 (2018) · Zbl 1454.91123 |
| [17] | Huyen, DTK; Yao, JC; Yen, ND, Sensitivity analysis of a stationary point set map under total perturbations. Part 1: Lipschitzian stability, J. Optim. Theory Appl., 180, 1, 91-116 (2019) · Zbl 1409.49024 |
| [18] | Huyen, DTK; Yao, JC; Yen, ND, Sensitivity analysis of a stationary point set map under total perturbations. Part 2: Robinson stability, J. Optim. Theory Appl., 180, 1, 117-139 (2019) · Zbl 1409.49025 |
| [19] | Jahn, J., Vector Optimization: Theory, Applications, and Extensions (2011), Berlin: Springer, Berlin · Zbl 1401.90206 |
| [20] | Jahn, J.; Ha, TXD, New order relations in set optimization, J. Optim. Theory Appl., 148, 2, 209-236 (2011) · Zbl 1226.90092 |
| [21] | Karuna; Lalitha, C., External and internal stability in set optimization, Optimization, 68, 4, 833-852 (2019) · Zbl 1411.49010 |
| [22] | Khan, AA; Tammer, C.; Zălinescu, C., Set-Valued Optimization (2016), Berlin: Springer, Berlin |
| [23] | Khoshkhabar-amiranloo, S., Stability of minimal solutions to parametric set optimization problems, Appl. Anal., 97, 14, 2510-2522 (2018) · Zbl 1416.49022 |
| [24] | Khushboo; Lalitha, CS, A unified minimal solution in set optimization, J. Glob. Optim., 74, 1, 195-211 (2019) · Zbl 1423.90243 |
| [25] | Kuroiwa, D., The natural criteria in set-valued optimization, RIMS Kokyuroku Kyoto Univ., 1031, 85-90 (1998) · Zbl 0938.90504 |
| [26] | Luc, DT, Theory of Vector Optimization (1989), Berlin: Springer, Berlin |
| [27] | Miglierina, E.; Molho, E., Convergence of minimal sets in convex vector optimization, SIAM J. Optim., 15, 2, 513-526 (2005) · Zbl 1114.90116 |
| [28] | Mordukhovich, BS, Variational Analysis and Applications (2018), Berlin: Springer, Berlin · Zbl 1402.49003 |
| [29] | Nishnianidze, ZG, Fixed points of monotonic multiple-valued operators, Soobshch. Akad. Nauk Gruzin. SSR, 114, 3, 489-491 (1984) · Zbl 0578.47043 |
| [30] | Penot, JP, Differentiability of relations and differential stability of perturbed optimization problems, SIAM J. Control Optim., 22, 4, 529-551 (1984) · Zbl 0552.58006 |
| [31] | Seto, K.; Kuroiwa, D.; Popovici, N., A systematization of convexity and quasiconvexity concepts for set-valued maps, defined by l-type and u-type preorder relations, Optimization, 67, 7, 1077-1094 (2018) · Zbl 1411.52005 |
| [32] | Sisarat, N.; Wangkeeree, R.; Lee, GM, On set containment characterizations for sets described by set-valued maps with applications, J. Optim. Theory Appl., 184, 3, 824-841 (2020) · Zbl 1434.49014 |
| [33] | Sisarat, N.; Wangkeeree, R.; Tanaka, T., Sequential characterizations of approximate solutions in convex vector optimization problems with set-valued maps, J. Glob. Optim., 77, 273-287 (2020) · Zbl 1448.90082 |
| [34] | Tanino, T., Stability and sensitivity analysis in multiobjective nonlinear programming, Ann. Oper. Res., 27, 1, 97-114 (1990) · Zbl 0719.90065 |
| [35] | Thinh, VD; Chuong, TD, Directionally generalized differentiation for multifunctions and applications to set-valued programming problems, Ann. Oper. Res., 269, 1-2, 727-751 (2018) · Zbl 1432.90140 |
| [36] | Tuyen, NV, Convergence of the relative Pareto efficient sets, Taiwan. J. Math., 20, 5, 1149-1173 (2016) · Zbl 1357.90149 |
| [37] | Xu, Y.; Li, S., Continuity of the solution set mappings to a parametric set optimization problem, Optim. Lett., 8, 8, 2315-2327 (2014) · Zbl 1332.90290 |
| [38] | Young, RC, The algebra of many-valued quantities, Math. Ann., 104, 1, 260-290 (1931) · JFM 57.1373.03 |
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