Abstract
We establish optimization results for set-valued mappings, with the image space given by a topological vector space partially ordered by a cone. Moreover, we obtain stability results relative to parametrized optimization problems. We use a weak semicontinuity concept related to the order structure of the image space and show how compactness assumptions used in previous papers can be lightened.
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References
Aubin, J. P. and Cellina, A.: Differential Inclusions, Springer-Verlag, Berlin, 1984.
Bank, B., Guddat, J., Klatte, D., Kummer, B., and Tammer, K.: Non-Linear Parametric Optimization, Birkhäuser, Basel, 1983.
Berge, C.: Espaces topologiques, Dunod, Paris, 1966.
Corley, H. W.: An existence result for maximizations with respect to cones, J. Optim. Theory Appl. 31 (1980), 277-281.
Corley, H. W.: Existence and lagrangian duality for maximizations of set-valued functions, J. Optim. Theory Appl. 54 (1987), 489-501.
Dolecki, S. and Lechicki, A.: On structure of upper semicontinuity, Pacific J. Math. 88 (1982), 547-554.
Ferro, F.: An optimization result for set-valued mappings and a stability property in vector problems with constraints, J. Optim. Theory Appl. 90 (1996), 63-77.
Hogan, W. W.: Point-to-set maps in mathematical programming, SIAM Rev. 15 (1973), 591-603.
Kelley, J. L.: General Topology, Van Nostrand, Princeton, NJ, 1955.
Luc, D. T.: Theory of Vector Optimization, Springer-Verlag, Berlin, 1989.
Luc, D. T.: An existence theorem in vector optimization, Math. Oper. Res. 14 (1989), 693-699.
Luc, D. T., Lucchetti, R., and Malivert, C.: Convergence of efficient sets, Set-Valued Anal. 2 (1994), 207-218.
Penot, J. P.: Compact nets, filters, and relations, J. Math. Anal. Appl. 93 (1983), 400-417.
Penot, J. P.: Continuity properties of performance functions, in: J. B. Hiriart-Urruty, W. Oettli, and S. Stoer (eds), Optimization: Theory and Algorithms, Dekker, New York, 1983.
Penot, J. P. and Sterna-Karwat, A.: Parametrized multicriteria optimization: continuity and closedness of optimal multifunctions, J. Math. Anal. Appl. 120 (1986), 150-168.
Penot, J. P. and Sterna-Karwat, A.: Parametrized multicriteria optimization: order continuity of the marginal multifunctions, J. Math. Anal. Appl. 144 (1989), 1-15.
Rockafellar, R. T.: Existence theorems for general control problems of Bolza and Lagrange, Adv. Math. 15 (1975), 312-333.
Smithson, R. E.: Subcontinuity for multifunctions, Pacific J. Math. 61 (1975), 283-288.
Tanino, T. and Sawaragi, Y.: Stability of nondominated solutions in multicriteria decisionmaking, J. Optim. Theory Appl. 30 (1980), 229-253.
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Ferro, F. Optimization and Stability Results Through Cone Lower Semicontinuity. Set-Valued Analysis 5, 365–375 (1997). https://doi.org/10.1023/A:1008653120360
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DOI: https://doi.org/10.1023/A:1008653120360