ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

17. Control Theory and Optimization

1991 MS Classification: 49, 52, 90 Belousov, Evgeny G., Moscow State University, Russia On Hausdorff continuity of convex polynomial mappings We consider the following point-to-set mappings

A(λ) = {x ∈ E_{n} | f_{i}(x) ≤ λ_{i}, i ∈ I = {1, …, m}}, λ ∈ Λ,

where Λ = {λ ∈ E_{m} | A(λ) ≠ ∅} and f_{i} : E_{n} → E_{1} are convex polynomials. It is proved here that, generally, such mappings are neither upper nor lower semicontinuous in the sense of Hausdorff at boundary points of set Λ even if these mappings are quasi-bounded, i. e. if the Hausdorff distance between sets A(λ) and A(λ′) is finite for ∀λ, λ′ ∈ Λ. Reference: E. G. Belousov, and V. G. Andronov, Solvability and stability of polynomial-programming problems. Moscow University publishers: Moscow, 1993. (in Russian).