ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

17. Control Theory and Optimization

1991 MS Classification: 90,49, 26 Mititelu Stefan, Technical University of Civil Engineering of Bucharest, Romania Optimality and duality in nonsmooth vector programming Let A be a nonempty open set of R^{n} and a function φ : A → R. Consider for φ at x ∈ A a subdifferential of the form ∂φ(x) = {ξ ∈

R^{n}|φ^{0}(x; ν) ≥ ξ′ν, ∀v ∈ R^{n}} where φ^{0}(x; ν) = y → x; λ ↓ 0_{lim sup}[f(y + λν) −

f(y)]/λ. Let (P V)x ∈ D_{min}f(x) be a nonsmooth vector program where D = {x ∈ A|g(x) ≤ 0, h(x) = 0, x ∈ X} , f : ArightarrowR^{p}, g : A → R^{m}, h : A → R^{S} and X is a nonempty part of A. Efficiency conditions of Kuhn-Tucker type as x^{0} ∈ D to be a solution Pareto for (P V) are established, using the subdifferentials of the component functions of f, g and h at x^{0}. A dual having as objective the vector Lagrangian of (P V) is associated to (P V) and a duality Wolfe is developed. Reference: S. Mititelu, Nonconvex subsdifferential calculus, Analele Universitãtii “Ovidius” Constanta, ser. Matematica 3 (1995) 1, 118 - 126.