Improvement sets and convergence of optimal points. (English) Zbl 1329.49013
“The aim of this paper is to give sufficient conditions for the existence of optimal points [in vector optimization] with respect to an improvement set, in the framework of Banach spaces.” \(E\) is an improvement set with respect to a convex ordering cone \(K\) in a linear topological space iff the origin does not belong to \(E\), and it is invariant with respect to the sum with \(K\). “The lower and upper convergence of optimal points of a convergent sequence of sets, in finite and infinite dimensional settings, are also considered, improving previous results. Finally, some sufficient conditions for the stability of optimal points are developed, discussing their importance via several examples.”
MSC:
| 49J27 | Existence theories for problems in abstract spaces |
| 49K27 | Optimality conditions for problems in abstract spaces |
| 49J45 | Methods involving semicontinuity and convergence; relaxation |
| 49K40 | Sensitivity, stability, well-posedness |
| 90C31 | Sensitivity, stability, parametric optimization |
References:
| [1] | Chicco, M., Mignanego, F., Pusillo, L., Tijs, S.: Vector optimization problems via improvement sets. J. Optim. Theory Appl. 150(3), 516-529 (2011) · Zbl 1231.90329 · doi:10.1007/s10957-011-9851-1 |
| [2] | Gutiérrez, C., Jiménez, B., Novo, V.: Improvement sets and vector optimization. Eur. J. Oper. Res. 223, 304-311 (2012) · Zbl 1292.90268 · doi:10.1016/j.ejor.2012.05.050 |
| [3] | Zhao, K.Q., Yang, X.M.: A unified stability result with perturbations in vector optimization. Optim. Lett. 7(8), 1913-1919 (2013) · Zbl 1287.90062 · doi:10.1007/s11590-012-0533-1 |
| [4] | Dauer, J.P., Stadler, W.: A survey of vector optimization in infinite-dimensional spaces, part 2. J. Optim. Theory Appl. 51(2), 205-241 (1986) · Zbl 0592.90081 · doi:10.1007/BF00939823 |
| [5] | Deng, S.: Characterizations of the nonemptiness and boundedness of weakly efficient solutions sets of convex vector optimization problems in real reflexive Banach spaces. J. Optim. Theory Appl. 140, 1-7 (2009) · Zbl 1170.90473 · doi:10.1007/s10957-008-9443-x |
| [6] | Attouch, H., Buttazzo, G., Michaille, G.: Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization. MPS/SIAM, Philadelphia (2006) · Zbl 1095.49001 · doi:10.1137/1.9780898718782 |
| [7] | Mordukhovich, B.S.: Multiobjective optimization problems with equilibrium constraints. Math. Program. B 117, 331-354 (2009) · Zbl 1165.90020 · doi:10.1007/s10107-007-0172-y |
| [8] | Aliprantis, C.D., Florenzano, M., Tourky, R.: Production equilibria. J. Math. Econ. 42, 406-421 (2006) · Zbl 1141.91593 · doi:10.1016/j.jmateco.2006.04.006 |
| [9] | Florenzano, M., Gourdel, P., Jofré, A.: Supporting weakly Pareto optimal allocations in infinite dimensional nonconvex economies. Econ. Theory 29(3), 549-564 (2006) · Zbl 1108.91052 · doi:10.1007/s00199-005-0033-y |
| [10] | Tanaka, T., Kuroiwa, D.: The convexity of A and B assures intA \[++ B = int\](A \[++B)\]. Appl. Math. Lett. 6, 83-86 (1993) · Zbl 0810.52003 · doi:10.1016/0893-9659(93)90154-F |
| [11] | Tanaka, T., Kuroiwa, D.: Another observation on conditions assuring intA \[++ B = int\](A \[++B)\]. Appl. Math. Lett. 7, 19-22 (1994) · Zbl 0797.52003 · doi:10.1016/0893-9659(94)90046-9 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
