ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

17. Control Theory and Optimization

1991 MS Classification: 49, 90, 93 Enkhbat, R., Mongolian State University, Mongolia Global Optimality Conditions for some class of Nonconvex Optimization Problems Global optimality conditions for the problems of minimizing and maximizing a quasiconvex function over arbitrary sets have been considered. f(x) ⟶ max(min) , x ∈ D ⊂ R^{n} , (1)((2)) Theorem 1. Let z be a solution of problem (1). Then the following condition holds: ⟨f′(y), x − y⟩ ≤ 0 , ∀x ∈ D , ∀y ∈ R^{n} : f(y) = f(z). If, in addition, f′(y) ≠ 0, ∀y ∈ R^{n} : f(y) = f(z) then this condition becomes sufficient for problem (1). Theorem 2. If z ∈ D is a solution of problem (2), then the following condition holds: ⟨f′(x), x − y⟩ ≥ 0 , ∀x ∈ D , ∀y : f(y) = f(z). If, in addition, inf_{x∈D} ||f′(x+αf′(x))|| ≠ 0 , α ≥ 0 then this condition becomes

sufficient for (2).