Open relation theorem without closedness assumptions

Dolecki, Szymon

doi:10.1090/S0002-9939-1990-1012926-2

Open relation theorem without closedness assumptions
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Proc. Amer. Math. Soc. 109 (1990), 1019-1024 Request permission

Abstract:

It is shown that a convex relation such that the preimage of a bounded set has nonempty interior is lower semicontinuous throughout the algebraic interior of its domain. If, besides, the relation is closed-valued, then it is closed throughout the algebraic interior of its domain.

References

V. M. Alekseev, V. M. Tihomirov, and S. V. Fomin, Optimal′noe upravlenie, “Nauka”, Moscow, 1979 (Russian). MR 566022

Théorie des opérations linéaires

1

Convex relations in optimization and analysis

J. M. Borwein, Automatic continuity and openness of convex relations, Proc. Amer. Math. Soc. 99 (1987), no. 1, 49–55. MR 866428, DOI 10.1090/S0002-9939-1987-0866428-4
Szymon Dolecki, Abstract study of optimality conditions, J. Math. Anal. Appl. 73 (1980), no. 1, 24–48. MR 560934, DOI 10.1016/0022-247X(80)90019-0
Richard B. Holmes, Geometric functional analysis and its applications, Graduate Texts in Mathematics, No. 24, Springer-Verlag, New York-Heidelberg, 1975. MR 0410335, DOI 10.1007/978-1-4684-9369-6
Stephen M. Robinson, Regularity and stability for convex multivalued functions, Math. Oper. Res. 1 (1976), no. 2, 130–143. MR 430181, DOI 10.1287/moor.1.2.130
Stephen M. Robinson, Stability theory for systems of inequalities. II. Differentiable nonlinear systems, SIAM J. Numer. Anal. 13 (1976), no. 4, 497–513. MR 410522, DOI 10.1137/0713043
Corneliu Ursescu, Multifunctions with convex closed graph, Czechoslovak Math. J. 25(100) (1975), no. 3, 438–441. MR 388032, DOI 10.21136/CMJ.1975.101337