Duality in vector optimization. II. Vector quasiconcave programming.
Tran Quoc Chien · Czech digital mathematics library · 1984
In this part of the tripaper, on the basis of the abstract theory presented in the first part, a duality theory is developed for the vector quasiconcave programming.In Section 3 some necessary concepts and assertions of (quasi)convexity are introduced.Section 4 deals with the duality theory in vector quasiconcave programming with affine constraints.Finally, in Section 5 a limit approach is proposed to define the dual problems for the vector quasiconcave programming with convex constraints. QUASICONVEXITY OF OPERATORS AND RELATED CONCEPTSIn the following definitions X is a topological linear space, Yis a topological linear space ordered by a convex cone Y+ with int Y+ =t = 0 and Y+ n (-Y+) = {0}.Let D c.X be a convex subset, having at least two points.Given an operator G :is convex for all b e Y (or equivalently: all its strict lower sets {x e D | G(x) < b} are convex).G is convex in D if for all x, y e D and X e (0, 1)G is quasimonotonic if it is both quasiconvex and quasiconcave.G is lower (upper) semicontinuous in D if its lower sets (upper sets) are closed with respect to D. A subset AaX is a polytope if it is an intersection of a finite number of halfspaces.Obviously, if Yis of finite dimension and G is affine, then all lower (upper) sets of G are polytopes.